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]>
<archimedes>

      <info>

        <author>Guidobaldo del Monte</author>
        <title>Mechanicorum Liber</title>
        <date>1577</date>
        <place>Pisauri</place>
        <editor></editor>        
        <publisher></publisher>
        <translator></translator>
        <lang>LA</lang>
        
      <chunk unit="page*">page</chunk>
      <locator>000000072.xml</locator>
      </info>

<text>        
    <front>       
      <section>
          <pb id="p.0001"/>     
        <p id="id.2.1.1.1.0.0.0" type="head">        

       
          <s id="id.2.1.1.1.2.1.0"> GVIDIV BALDI <lb/>
E MARCHIONIBVS <lb/>
MONTIS <lb/>
MECHANICORVM <lb/>
LIBER. </s>
     <lb/>
       
          
<s> ZZZ head of figure ZZZ </s>
    
        </p>       <figure place="text">       </figure>       
<p id="id.2.1.1.1.4.1.0" type="caption">        
<s id="id.2.1.1.1.4.1.0.capt"> YYY </s>
     <lb/>
       
<s id="id.2.1.1.1.6.1.0"> PISAVRI <lb/>
Apud Hieronymum Concordiam. </s>
     <lb/>
       
<s id="id.2.1.1.1.8.1.0"> M. D. LXXVII. </s>
     <lb/>
       
<s id="id.2.1.1.1.10.1.0"> Cum Licentia Superiorum. </s>
    </p>       
<pb/>
       
<p id="id.2.1.1.3.0.0.0" type="head">        
<s id="id.2.1.1.3.1.1.0"> PRAESENTI OPERE <lb/>
CONTENTA. </s>
    </p>       
<p id="id.2.1.1.4.0.0.0" type="main">        
<s id="id.2.1.1.4.1.1.0"> De Libra. </s>
    </p>       
<p id="id.2.1.1.5.0.0.0" type="main">        
<s id="id.2.1.1.5.1.1.0"> De Vecte. </s>
    </p>       
<p id="id.2.1.1.6.0.0.0" type="main">        
<s id="id.2.1.1.6.1.1.0"> De Trochlea. </s>
    </p>       
<p id="id.2.1.1.7.0.0.0" type="main">        
<s id="id.2.1.1.7.1.1.0"> De Axe in peritrochio. </s>
    </p>       
<p id="id.2.1.1.8.0.0.0" type="main">        
<s id="id.2.1.1.8.1.1.0"> De Cuneo. </s>
    </p>       
<p id="id.2.1.1.9.0.0.0" type="main">        
<s id="id.2.1.1.9.1.1.0"> De Cochlea.  </s>
    </p>       
<p id="id.2.1.1.10.0.0.0" type="head">        
<pb/>
       
<s id="id.2.1.1.11.1.1.0"> AD FRANCISCVM <lb/>
MARIAM II <lb/>
VRBINATVM <lb/>
AMPLISSIMVM DVCEM <lb/>
GVIDIVBALDI <lb/>
E MARCHIONIBVS <lb/>
MONTIS </s>
     <lb/>
       
<s id="id.2.1.1.11.3.1.0"> PRAEFATIO. </s>
    </p>       
<p id="id.2.1.1.12.0.0.0" type="main">        
<s id="id.2.1.1.12.1.1.0"> DVAE res (AMPLISSIME PRIN&shy;<lb/>
CEPS) qu&aelig; ad conciliandas homi<lb/>
nibus facultates, vtilitas <expan abbr="nemp&egrave;">nempe</expan>, &amp; <lb/>
nobilitas, <expan abbr="plurim&ugrave;m">plurimum</expan> valere con&longs;ue<lb/>
uerunt.  </s>
             
<s id="id.2.1.1.12.1.2.0"> ill&aelig; ad exornandam mecha<lb/>
nicam facultatem, &amp; eam pr&aelig; om&shy;<lb/>
nibus alijs appetibilem reddendam con&longs;pira&longs;&longs;e <lb/>
mihi videntur: nam &longs;i nobilitatem (quod pleriq; <lb/>
<expan abbr="mod&ograve;">modo</expan> faciunt) ortuip&longs;o metimur, occurret hinc <lb/>
Geometria, illinc <expan abbr="ver&ograve;">vero</expan> Phi&longs;ica; quorum gemina<lb/>
to complexu nobili&longs;&longs;ima artium prodit mechani&shy;<lb/>
ca.  </s>
             
<s id="id.2.1.1.12.1.3.0"> &longs;i enim nobilitatem magis, <expan abbr="t&ugrave;m">tum</expan> &longs;trat&aelig; materi&aelig;, <lb/>
<expan abbr="t&ugrave;m">tum</expan> argumentorum nece&longs;&longs;itati (quod Ari&longs;tote&shy;<lb/>
les fatetur <expan abbr="aliquand&ograve;">aliquando</expan>) relatam volumus, omnium <lb/>
<expan abbr="proculdubi&ograve;">procul dubio</expan> nobili&longs;&longs;imam per&longs;piciemus.  </s>
             
<s id="id.2.1.1.12.1.4.0"> qu&aelig; 
<pb/>
quidem non &longs;olum geometriam (vt Pappus te&longs;ta<lb/>
tur) ab&longs;oluit, &amp; perficit; <expan abbr="ver&ugrave;m">verum</expan> etiam &amp; phi&longs;ica&shy;<lb/>
rum rerum imperium habet: quandoquidem <lb/>
quodcunq; Fabris, Architectis, Baiulis, Agricolis, <lb/>
Nautis, &amp; <expan abbr="qu&agrave;m">quam</expan> plurimis alijs (repugnantibus na&shy;<lb/>
tur&aelig; legibus) opitulatur; id omne mechanicum <lb/>
e&longs;t imperium.  </s>
             
<s id="id.2.1.1.12.1.5.0"> <expan abbr="quipp&egrave;">quippe</expan> quod aduer&longs;us naturam <lb/>
vel eiu&longs;dem emulata leges exercet; &longs;umma id <lb/>
<expan abbr="cert&egrave;">certe</expan> admiratione dignum; veri&longs;&longs;imum tamen, <lb/>
&amp; <expan abbr="&agrave;">a</expan> quocunque liberaliter admi&longs;&longs;um, qui pri&shy;<lb/>
us ab Ari&longs;totele didicerit, omnia mechanica, <lb/>
<expan abbr="t&ugrave;m">tum</expan> problemata, <expan abbr="t&ugrave;m">tum</expan> theoremata ad rotundam <lb/>
machinam reduci, atq; ideo illo niti principio, <lb/>
<expan abbr="n&otilde;">non</expan> minus &longs;en&longs;ui, <expan abbr="qu&agrave;m">quam</expan> rationi noto.  </s>
             
<s id="id.2.1.1.12.1.6.0"> Rotunda ma<lb/>
china e&longs;t mouenti&longs;&longs;ima, &amp; <expan abbr="qu&ograve;">quo</expan> maior, <expan abbr="e&ograve;">eo</expan> mouen&shy;<lb/>
tior.  </s>
             
<s id="id.2.1.1.12.1.7.0"> <expan abbr="Ver&ugrave;m">Verum</expan> huic nobilitati adnexa e&longs;t &longs;umma re <lb/>
rum ad vitam pertinentium vtilitas, qu&aelig; propte&shy;<lb/>
rea omnes alias <expan abbr="&agrave;">a</expan> diuer&longs;is artibus propagatas an&shy;<lb/>
tecellit; <expan abbr="qu&ograve;d">quod</expan> ali&aelig; facultates po&longs;t mundi gene&longs;im <lb/>
longa temporis intercapedine &longs;uos explicarunt <lb/>
v&longs;us; i&longs;ta <expan abbr="ver&ograve;">vero</expan> &amp; in ip&longs;is mundi primordijs ita fuit <lb/>
hominibus nece&longs;&longs;aria, vt ea &longs;ublata Sol de mun&shy;<lb/>
do &longs;ublatus videretur.  </s>
             
<s id="id.2.1.1.12.1.8.0"> nam quacunq; nece&longs;&longs;ita&shy;<lb/>
te Ad&aelig; vita degeretur; &amp; quamuis etiam ca&longs;is <lb/>
contectis &longs;tramine, &amp; angu&longs;tis tugurijs, ac gurgu&shy;<lb/>
&longs;tijs c&oelig;li de fenderet iniurias; &longs;ic &amp; in corporis ve<lb/>
&longs;titu, licet ip&longs;e nihil aliud &longs;pectaret, ni&longs;i vt imbres, 
<pb/>
vt niues, vt ventos; vt Solem, vt frigus arceret; <lb/>
quodcunque tamen id fuit, omne mechanicum <lb/>
fuit.  </s>
             
<s id="id.2.1.1.12.1.9.0"> neq; tamen huic facultati contingit, quod <lb/>
ventis &longs;olet, qui <expan abbr="c&ugrave;m">cum</expan> <expan abbr="vnd&egrave;">vnde</expan> oriuntur, ibi vehe&shy;<lb/>
menti&longs;&longs;imi &longs;int, ad longinqua tamen fracti, <expan abbr="de&shy;bilitatiqu&egrave;">de&shy;<lb/>
bilitatique</expan> perueniunt: &longs;ed quod magnis flumini&shy;<lb/>
bus crebriu&longs; accidit, qu&aelig; <expan abbr="c&ugrave;m">cum</expan> in ip&longs;o ortu parua <lb/>
&longs;int, <expan abbr="perpetu&ograve;">perpetuo</expan> tamen aucta, <expan abbr="e&ograve;">eo</expan> ampliori ferun<lb/>
tur alueo, <expan abbr="qu&ograve;">quo</expan> <expan abbr="&agrave;">a</expan> fontibus &longs;uis longius rece&longs;&longs;e&shy;<lb/>
runt.  </s>
             
<s id="id.2.1.1.12.1.10.0"> Nam &amp; temporis progre&longs;&longs;u mechanica fa <lb/>
cultas &longs;ub iugo &aelig;quum arationis laborem di&shy;<lb/>
&longs;pen&longs;are, atque aratrum agris circumagere c&aelig;&shy;<lb/>
pit.  </s>
             
<s id="id.2.1.1.12.1.11.0"> deinceps bigis, &amp; quadrigis docuit comea<lb/>
tus, merces, onera qu&aelig;libet vehere, <expan abbr="&egrave;">e</expan> finibus <lb/>
no&longs;tri&longs; ad finitimos populos exportare, &amp; ex il<lb/>
lis contra importare ad nos.  </s>
             
<s id="id.2.1.1.12.1.12.0"> pr&aelig;terea <expan abbr="c&ugrave;m">cum</expan> iam <lb/>
res non <expan abbr="tant&ugrave;m">tantum</expan> nece&longs;&longs;itate, <expan abbr="ver&ugrave;m">verum</expan> etiam orna&shy;<lb/>
tu, &amp; commoditate metirentur, mechanic&aelig; <lb/>
fuit &longs;ubtilitatis, <expan abbr="qu&ograve;d">quod</expan> nauigia remo impellere&shy;<lb/>
mus; <expan abbr="qu&ograve;d">quod</expan> gubernaculo exiguo in extrema pup<lb/>
pi collocato ingentes triremium moles inflecte&shy;<lb/>
remus; <expan abbr="qu&ograve;d">quod</expan> vnius <expan abbr="&longs;&aelig;p&egrave;">&longs;&aelig;pe</expan> manu pro multis fabro&shy;<lb/>
rum manibus <expan abbr="mod&ograve;">modo</expan> pondera lapidum, &amp; tra&shy;<lb/>
bium Fabris, &amp; Architectis &longs;ubleuaremus; <expan abbr="mo&shy;d&ograve;">mo&shy;<lb/>
do</expan> tollenonis &longs;pecie aquas <expan abbr="&egrave;">e</expan> puteis olitoribus e&shy;<lb/>
xhauriremus.  </s>
             
<s id="id.2.1.1.12.1.13.0"> hinc etiam <expan abbr="&egrave;">e</expan> liquidorum pr&aelig;lis vi<lb/>
na, olea, vnguenta expre&longs;&longs;a, &amp; quicquid liquo&shy;
<pb/>
ris habent, per&longs;oluere domino compul&longs;a.  </s>
             
<s id="id.2.1.1.12.1.14.0"> hinc <lb/>
magnas <expan abbr="arbor&utilde;">arborum</expan>, &amp; marmorum moles duobus in <lb/>
contrarias partes <expan abbr="di&longs;trah&etilde;tibus">di&longs;trahentibus</expan> vectibus diremp&shy;<lb/>
&longs;imus; hinc militi&aelig; in aggeribus extruendis, in <lb/>
con&longs;erenda manu, in opugnando, propugnan&shy;<lb/>
doq; loca infinit&aelig; <expan abbr="fer&egrave;">fere</expan> redundarunt vtilitates; <lb/>
hinc demum Lignatores, Lapicid&aelig;, Marmorarij <lb/>
Vinitores, Olearij, Vnguentarij, Ferrarij, Auri<lb/>
fices, Metallici, Chirurgi, Ton&longs;ores, Pi&longs;tores, Sar<lb/>
tores, omnes deniq; opifices beneficiarij, tot, tan<lb/>
taq; vit&aelig; human&aelig; &longs;uppeditarunt commoda.  </s>
             
<s id="id.2.1.1.12.1.15.0"> Eant <lb/>
nunc noui logodedali quidam mechanicorum <lb/>
contemptores, perfricent frontem, &longs;i quam ha&shy;<lb/>
bent, &amp; ignobilitatem, <expan abbr="atqu&egrave;">atque</expan> inutilitatem <expan abbr="fal&longs;&ograve;">fal&longs;o</expan> <lb/>
criminari de&longs;inant: <expan abbr="qu&ograve;d">quod</expan> &longs;i &amp; adhuc id <expan abbr="minim&egrave;">minime</expan> <lb/>
velint, eos qu&aelig;&longs;o in in&longs;citia &longs;ua relinquamus: <lb/>
<expan abbr="Ari&longs;totelemqu&egrave;">Ari&longs;totelemque</expan> potius philo&longs;ophorum cory&shy;<lb/>
ph&aelig;um imitemur, cuius mechanici amoris ardo <lb/>
rem acuti&longs;&longs;im&aelig; ill&aelig; mechanic&aelig; qu&aelig;&longs;tiones po&longs;te <lb/>
ris tradit&aelig; &longs;atis declarant: qua quidem laude <lb/>
Platonem <expan abbr="magnific&egrave;">magnifice</expan> &longs;uperauit; qui (vt te&longs;tatur <lb/>
Plutarcus) Architam, &amp; Eudoxum mechanic&aelig; <lb/>
vtilitatem impen&longs;ius colentes ab in&longs;tituto deter<lb/>
ruit; <expan abbr="qu&ograve;d">quod</expan> nobili&longs;&longs;imam philo&longs;ophorum po&longs;&longs;e&longs;&shy;<lb/>
&longs;ionem in vulgus indicarent, ac publicarent; &amp; <lb/>
velut arcana philo&longs;ophi&aelig; my&longs;teria proderent.  </s>
             
<s id="id.2.1.1.12.1.16.0"> <lb/>
res <expan abbr="&longs;an&egrave;">&longs;ane</expan> meo quidem iudicio pro&longs;us vituperan&shy;
<pb/>
da, ni&longs;i <expan abbr="fort&egrave;">forte</expan> velimus tam nobilis di&longs;ciplin&aelig; con<lb/>
templationem quidem ocio&longs;am laudare; fructum <lb/>
<expan abbr="ver&ograve;">vero</expan>, &amp; v&longs;um, arti&longs;q; finem improbare.  </s>
             
<s id="id.2.1.1.12.1.17.0"> &longs;ed pr&aelig; <lb/>
omnibus mathematicis vnus Archimedes ore <lb/>
laudandus e&longs;t pleniore, quem voluit Deus in me&shy;<lb/>
chanicis velut ideam &longs;ingularem e&longs;&longs;e, quam om&shy;<lb/>
nes earum &longs;tudio&longs;i ad imitandum &longs;ibi propone&shy;<lb/>
rent.  </s>
             
<s id="id.2.1.1.12.1.18.0"> is enim C&oelig;le&longs;tem globum exiguo admo&shy;<lb/>
dum, fragili <expan abbr="qu&egrave;">que</expan> vitreo orbe conclu&longs;um ita efin&shy;<lb/>
xit, &longs;imulatis a&longs;tris viuum natur&aelig; opus, ac iura <lb/>
poli motibus certis <expan abbr="ade&ograve;">adeo</expan> pr&aelig;&longs;eferentibus; vt <lb/>
&aelig;mula natur&aelig; manus tale de &longs;e encomium &longs;it <lb/>
promerita: &longs;ic manus naturam, vt natura ma&shy;<lb/>
num ip&longs;a immitata putetur.  </s>
             
<s id="id.2.1.1.12.1.19.0"> is poli&longs;pa&longs;tu manu <lb/>
leua, &amp; &longs;ola, quinquies millenum modiorum <lb/>
pondus attraxit.  </s>
             
<s id="id.2.1.1.12.1.20.0"> nauem in &longs;iccum litus eductam, <lb/>
ac grauius oneratam &longs;olus machinis &longs;uis ad &longs;e <lb/>
<expan abbr="perind&egrave;">perinde</expan> pertraxit, ac &longs;i in mari remis, <expan abbr="veli&longs;u&egrave;">veli&longs;ue</expan> <lb/>
impul&longs;a moueretur, <expan abbr="qu&atilde;">quam</expan> &amp; po&longs;tea in litore (quod <lb/>
omnes Sicili&aelig; vires non potuerunt) in mare de&shy;<lb/>
duxit.  </s>
             
<s id="id.2.1.1.12.1.21.0"> ab i&longs;to etiam ea extiterunt bellica tor&shy;<lb/>
menta, quibus Syracu&longs;&aelig; aduer&longs;us Marcellum <lb/>
ita defen&longs;&aelig; &longs;unt, vt pa&longs;&longs;im eorum machinator <lb/>
Briareus, &amp; centimanus <expan abbr="&agrave;">a</expan> Romanis appellare&shy;<lb/>
tur.  </s>
             
<s id="id.2.1.1.12.1.22.0"> demum hac arte confi&longs;us <expan abbr="e&ograve;">eo</expan> proce&longs;&longs;it au&shy;<lb/>
daci&aelig;, vt eam vocem natur&aelig; legibus <expan abbr="ade&ograve;">adeo</expan> re&shy;<lb/>
pugnantem protulerit.  </s>
             
<s id="id.2.1.1.12.1.23.0"> Da mihi, vbi &longs;i&longs;tam, ter 
<pb/>
ramq; mouebo.  </s>
             
<s id="id.2.1.1.12.1.24.0"> quod tamen non <expan abbr="mod&ograve;">modo</expan> nos <lb/>
vecte <expan abbr="tant&ugrave;m">tantum</expan> fieri potui&longs;&longs;e in pr&aelig;&longs;enti libro doce<lb/>
mus; <expan abbr="ver&ugrave;m">verum</expan> etiam, &amp; omnis antiquitas (quod <lb/>
multis <expan abbr="forta&longs;&longs;&egrave;">forta&longs;&longs;e</expan> mirabile videbitur) id penitus <lb/>
credidi&longs;&longs;e mihi videtur; qu&aelig; Neptuno tri&shy;<lb/>
dentem tanquam vectem attribuit; cuius ope <lb/>
terr&aelig; concu&longs;&longs;or vbiq; nuncupatur <expan abbr="&agrave;">a</expan> poetis.  </s>
             
<s id="id.2.1.1.12.1.25.0"> ad <lb/>
quod etiam a&longs;piciens celeberrimus no&longs;ter poeta <lb/>
Neptunum inducit i&longs;ta machina &longs;yrtes, <expan abbr="qu&ograve;">quo</expan> ma&shy;<lb/>
gis apparerent Troianis, &longs;ubleuantem. </s>
    </p>       
<p id="id.2.1.1.13.0.0.0" type="main">        
<s id="id.2.1.1.13.1.1.0"> &ldquo;Leuat ip&longs;e tridenti <lb/>
&amp; va&longs;tas aperit &longs;yrtes.&rdquo; </s>
    </p>       
<p id="id.2.1.1.14.0.0.0" type="main">        
<s id="id.2.1.1.14.1.1.0"> Mechanici pr&aelig;terea fuerunt Heron, Cte&longs;ibius, <lb/>
&amp; Pappus, qui licet ad mechanic&aelig; apicem, perin&shy;<lb/>
de atq; Archimedes, euecti <expan abbr="forta&longs;&longs;&egrave;">forta&longs;&longs;e</expan> <expan abbr="minim&egrave;">minime</expan> &longs;int; <lb/>
mechanicam tamen facultatem <expan abbr="egregi&egrave;">egregie</expan> percal&shy;<lb/>
luerunt; tale&longs;q; fuerunt, &amp; pr&aelig;&longs;ertim Pappus, vt <lb/>
eum me ducem &longs;equentem nemo (vt opinor) cul<lb/>
pauerit.  </s>
             
<s id="id.2.1.1.14.1.2.0"> quod &amp; propterea libentius feci, <expan abbr="qu&ograve;d">quod</expan> <lb/>
<expan abbr="n&egrave;">ne</expan> latum quidem vnguem ab Archimedeis prin&shy;<lb/>
cipijs Pappus recedat.  </s>
             
<s id="id.2.1.1.14.1.3.0"> ego enim in hac pr&aelig;&longs;ertim <lb/>
facultate Archimedis ve&longs;tigijs h&aelig;rere &longs;emper vo <lb/>
lui: &amp; licet eius lucubrationes ad <expan abbr="mechanic&atilde;">mechanicam</expan> per&shy;
<pb/>
tinentes multis ab hinc annis pa&longs;&longs;im &longs;oleant do&shy;<lb/>
ctis de&longs;iderari: eruditi&longs;&longs;imus tamen libellus de &aelig;&shy;<lb/>
queponderantibus pr&aelig; manibus <expan abbr="homin&utilde;">hominum</expan> adhuc <lb/>
ver&longs;atur, in <expan abbr="qu&ograve;">quo</expan> tanquam in copio&longs;i&longs;&longs;ima p&oelig;nu <lb/>
omnia <expan abbr="fer&egrave;">fere</expan> mechanica dogmata repo&longs;ita mihi vi&shy;<lb/>
dentur; quem <expan abbr="&longs;an&egrave;">&longs;ane</expan> libellum, &longs;i &aelig;tatis no&longs;tr&aelig; mathe<lb/>
matici &longs;ibi magis familiarem adhibui&longs;&longs;ent; reperi&longs;<lb/>
&longs;ent <expan abbr="&longs;an&egrave;">&longs;ane</expan> <expan abbr="&longs;ent&etilde;tias">&longs;ententias</expan> multas, quas <expan abbr="mod&oacute;">modo</expan> ip&longs;i firmas, <lb/>
&amp; ratas e&longs;&longs;e docent; <expan abbr="&longs;ubtili&longs;&longs;im&egrave;">&longs;ubtili&longs;&longs;ime</expan>, <expan abbr="atqu&egrave;">atque</expan> <expan abbr="veri&longs;&shy;&longs;im&egrave;">veri&longs;&shy;<lb/>
&longs;ime</expan> conuul&longs;as, &amp; labefactatas. &longs;ed hoc vi&shy;<lb/>
derint ip&longs;i. </s>
             
<s id="id.2.1.1.14.1.4.0"> [&longs;ed hoc vi&shy;<lb/>
derint ip&longs;i.]  </s>
             
<s id="id.2.1.1.14.1.5.0"> ego enim ad Pappum redeo, qui <lb/>
ad v&longs;um mathematicarum vberiorem, <expan abbr="emulu&shy;mentorumqu&egrave;">emulu&shy;<lb/>
mentorumque</expan> acce&longs;&longs;iones amplificandas peni&shy;<lb/>
tus conuer&longs;us, de quinque principibus machi&shy;<lb/>
nis, Vecte <expan abbr="nemp&egrave;">nempe</expan>, Trochlea, Axe in peri&shy;<lb/>
trochio, Cuneo, &amp; Cochlea, multa <expan abbr="egre&shy;gi&egrave;">egre&shy;<lb/>
gie</expan> philo&longs;ophatus e&longs;t; demon&longs;trauit <expan abbr="qu&egrave;">que</expan> quicquid <lb/>
in machinis, aut cogitari <expan abbr="perit&egrave;">perite</expan>, aut <expan abbr="acut&egrave;">acute</expan> <lb/>
definiri, aut <expan abbr="cert&ograve;">certo</expan> &longs;tatui pote&longs;t, id omne <expan abbr="quin&shy;qu&egrave;">quin&shy;<lb/>
que</expan> illis infinita vi pr&aelig;ditis machinis referen&shy;<lb/>
dum e&longs;&longs;e.  </s>
             
<s id="id.2.1.1.14.1.6.0"> <expan abbr="atqu&egrave;">atque</expan> vtinam iniuria temporis ni&shy;<lb/>
hil <expan abbr="&egrave;">e</expan> tanti viri &longs;criptis abra&longs;i&longs;&longs;et: nec enim tam <lb/>
den&longs;a in&longs;citi&aelig; caligo vniuer&longs;um <expan abbr="prop&egrave;">prope</expan> terra&shy;<lb/>
rum orbem obtexi&longs;&longs;et, neque tanta mechani<lb/>
c&aelig;facultatis e&longs;&longs;et ignoratio con&longs;ecuta, vt ma&shy;<lb/>
thematicarum proceres exi&longs;timarentur illi, qui <lb/>
<expan abbr="mod&ograve;">modo</expan> inepti&longs;&longs;ima quadam di&longs;tinctione, <expan abbr="diffi&shy;|cultate">diffi&shy;cultate</expan>
<pb/>
s nonnullas, nec illas tamen &longs;atis ar&shy;<lb/>
duas, &amp; ob&longs;curas <expan abbr="&egrave;">e</expan> medio tollunt.  </s>
             
<s id="id.2.1.1.14.1.7.0"> reperiun&shy;<lb/>
tur enim aliqui, no&longs;traq; &aelig;tate emunct&aelig; naris <lb/>
mathematici, qui mechanicam, <expan abbr="t&ugrave;m">tum</expan> <expan abbr="mathe&shy;matic&egrave;">mathe&shy;<lb/>
matice</expan> &longs;eor&longs;um, <expan abbr="t&ugrave;m">tum</expan> <expan abbr="phi&longs;ic&egrave;">phi&longs;ice</expan> con&longs;iderari po&longs;&shy;<lb/>
&longs;e affirmant; ac &longs;i aliquando, vel &longs;ine demon<lb/>
&longs;trationibus geometricis, vel &longs;ine vero motu <lb/>
res mechanic&aelig; con&longs;iderari po&longs;&longs;int: qua <expan abbr="&longs;an&egrave;">&longs;ane</expan> di&shy;<lb/>
&longs;tinctione (vt leuius cum illis agam) nihil aliud mi&shy;<lb/>
hi commini&longs;ci videntur, <expan abbr="qu&agrave;m">quam</expan> vt dum &longs;e, <expan abbr="t&ugrave;m">tum</expan> <lb/>
phi&longs;icos, <expan abbr="t&ugrave;m">tum</expan> mathematicos proferant, vtra&shy;<lb/>
que (quod aiunt) &longs;ella excludantur.  </s>
             
<s id="id.2.1.1.14.1.8.0"> <expan abbr="nequ&egrave;">neque</expan> <lb/>
enim amplius mechanica, &longs;i <expan abbr="&agrave;">a</expan> machinis ab&longs;tra<lb/>
hatur, &amp; &longs;eiungatur, mechanica pote&longs;t appel<lb/>
lari.  </s>
             
<s id="id.2.1.1.14.1.9.0"> Emicuit tamen inter i&longs;tas tenebras (quam&shy;<lb/>
uis alij <expan abbr="quoqu&egrave;">quoque</expan> nonnulli fuerint pr&aelig;clari&longs;&longs;imi) <lb/>
Solis in&longs;tar Federicus Commandinus, qui multis <lb/>
docti&longs;&longs;imis elucubrationibus ami&longs;&longs;um mathema<lb/>
ticarum patrimonium non <expan abbr="mod&ograve;">modo</expan> re&longs;taurauit, <lb/>
<expan abbr="ver&ugrave;m">verum</expan> etiam <expan abbr="aucti&ugrave;s">auctius</expan>, &amp; <expan abbr="locupleti&ugrave;s">locupletius</expan> effecit.  </s>
             
<s id="id.2.1.1.14.1.10.0"> <lb/>
erat enim &longs;ummus i&longs;te vir omnibus <expan abbr="ade&ograve;">adeo</expan> facul&shy;<lb/>
tatibus mathematicis ornatus, vt in eo Archi&shy;<lb/>
tas, Eudoxus, Heron, Euclides, Theon, Ari&shy;<lb/>
&longs;tarcus, Diophantus, Theodo&longs;ius, Ptolem&aelig;us <lb/>
Apollonius, Serenus, Pappus, quin &amp; ip&shy;<lb/>
&longs;emet Archimedes (&longs;iquidem ip&longs;ius in Archi&shy;<lb/>
medem &longs;cripta Archimedis olent lucernam) re 
<pb/>
uixi&longs;&longs;e viderentur.  </s>
             
<s id="id.2.1.1.14.1.11.0"> &amp; ecce <expan abbr="repent&egrave;">repente</expan> <expan abbr="&egrave;">e</expan> tenebris (vt <lb/>
confidimus) ac vinculis corporis in lucem, li&shy;<lb/>
bertatem <expan abbr="qu&egrave;">que</expan> productus mathematicas alieni&longs;&shy;<lb/>
&longs;imo tempore optimo, &amp; pr&aelig;&longs;tanti&longs;&longs;imo patre <lb/>
orbatas, nos <expan abbr="ver&ograve;">vero</expan> ita con&longs;ternatos reliquit, vt e&shy;<lb/>
ius de&longs;iderium vix longo &longs;ermone mitigare <lb/>
po&longs;&longs;e videamur.  </s>
             
<s id="id.2.1.1.14.1.12.0"> Ille tamen <expan abbr="perpetu&ograve;">perpetuo</expan> in alia&shy;<lb/>
rum mathematicarum explicationem ver&longs;ans, <lb/>
mechanicam facultatem, aut penitus pr&aelig;ter&shy;<lb/>
mi&longs;it, aut <expan abbr="modic&egrave;">modice</expan> attigit.  </s>
             
<s id="id.2.1.1.14.1.13.0"> Quapropter in hoc <lb/>
&longs;tudium <expan abbr="ardenti&ugrave;s">ardentius</expan> ego incumbere c&aelig;pi, nec me <lb/>
vnquam per omne mathematum genus vagan<lb/>
tem ea &longs;olicitudo de&longs;eruit; ecquid ex vno <lb/>
<expan abbr="quoqu&egrave;">quoque</expan> decerpi, ac delibari po&longs;&longs;it; quo ad me<lb/>
chanicam expoliendam, &amp; exornandam acco&shy;<lb/>
modatior e&longs;&longs;e po&longs;&longs;em.  </s>
             
<s id="id.2.1.1.14.1.14.0"> Nunc <expan abbr="ver&ograve;">vero</expan> <expan abbr="c&ugrave;m">cum</expan> mihi <lb/>
videar, noni ea quidem omnia, qu&aelig; ad mecha<lb/>
nicam pertinent, perfeci&longs;&longs;e; &longs;ed <expan abbr="e&ograve;">eo</expan> v&longs;q; tamen <lb/>
progre&longs;&longs;us, vtijs, qui ex Pappo, ex Vitruuio, <lb/>
&amp; ex alijs didicerint, quid &longs;it Vectis, quid Tro&shy;<lb/>
chlea, quid Axis in peritrochio, quid Cuneus, <lb/>
quid Cochlea; quomodoq; vt pondera moueri <lb/>
po&longs;&longs;int, aptari debeant; adhuc tamen acciden&shy;<lb/>
tia permulta, qu&aelig; inter potentiam, &amp; pondus <lb/>
vectis virtute illis in&longs;unt in&longs;trumentis, perdi&longs;ce&shy;<lb/>
re cupiunt, opis aliquid adferre po&longs;&longs;im; putaui <lb/>
tempus iam po&longs;tulare, vt prodirem; &amp; nauat&aelig; 
<pb/>
in hoc genere oper&aelig; &longs;pecimen aliquod darem.  </s>
             
<s id="id.2.1.1.14.1.15.0"> <lb/>
<expan abbr="Ver&ugrave;m">Verum</expan> <expan abbr="qu&ograve;">quo</expan> facilius totius operis &longs;ub&longs;tructio <lb/>
ad fa&longs;tigium &longs;uum per duceretur, nonnulla <expan abbr="quo&shy;qu&egrave;">quo&shy;<lb/>
que</expan> de libra fuerunt pertractanda, &amp; pr&aelig;&longs;er&shy;<lb/>
tim dum vnico pondere alterum &longs;olum ip&longs;ius <lb/>
brachium penitus deprimitur: que in re mi&shy;<lb/>
rum e&longs;t quantas fecerint ruinas Iordanus (qui <lb/>
inter recentiores maxim&aelig; fuit auctoritatis) &amp; <lb/>
alij; qui hanc rem &longs;ibi di&longs;cutiendam propo&longs;ue<lb/>
runt.  </s>
             
<s id="id.2.1.1.14.1.16.0"> opus <expan abbr="&longs;an&egrave;">&longs;ane</expan> arduum, &amp; for&longs;an viribus no&shy;<lb/>
&longs;tris impar aggre&longs;si &longs;umus; in eo tamen digni, vt <lb/>
no&longs;tros conatus, &amp; indu&longs;triam ad pr&aelig;clara ten<lb/>
dentem bonorum omnium perpetuus applau&shy;<lb/>
&longs;us, approbatioq; comitetur; <expan abbr="qu&ograve;d">quod</expan> ad &longs;tudium <lb/>
<expan abbr="t&agrave;m">tam</expan> illu&longs;tre, tam magnificum, tam laudabile <lb/>
contulimus quicquid habuimus virium.  </s>
             
<s id="id.2.1.1.14.1.17.0"> quod <lb/>
<expan abbr="&longs;an&egrave;">&longs;ane</expan> qualecunq; &longs;it, tibi celeberrime PRINCEPS <lb/>
nuncupandum cen&longs;uimus; cuius <expan abbr="&longs;an&egrave;">&longs;ane</expan> con&longs;ilij, <lb/>
atq; in&longs;tituti no&longs;tri rationes multas reddere in <lb/>
promptu e&longs;t: &amp; <expan abbr="prim&ugrave;m">primum</expan> h&aelig;reditaria tibi in fa&shy;<lb/>
miliam no&longs;tram promerita, quibus nos ita de&shy;<lb/>
uictos habes; vt <expan abbr="facil&egrave;">facile</expan> intelligamus ad fortunas <lb/>
non <expan abbr="mod&ograve;">modo</expan> no&longs;tras, <expan abbr="ver&ugrave;m">verum</expan> &amp; ad &longs;anguinem, &amp; <lb/>
vitam quoq; pro tua dignitate propendendam <lb/>
parati&longs;&longs;imos e&longs;&longs;e debere.  </s>
             
<s id="id.2.1.1.14.1.18.0"> Pr&aelig;terea illud non <lb/>
parui quoq; ponderis accedit, <expan abbr="qu&ograve;d">quod</expan> <expan abbr="&agrave;">a</expan> pueri&shy;<lb/>
tia literarum omnium, &longs;ed <expan abbr="pr&aelig;cipu&egrave;">pr&aelig;cipue</expan> mathe&shy;
<pb/>
maticarum de&longs;iderio ita fueris incen&longs;us, vt ni&shy;<lb/>
&longs;i illis adeptis vitam tibi acerbam, atq; in&longs;ua&shy;<lb/>
uem &longs;tatueres.  </s>
             
<s id="id.2.1.1.14.1.19.0"> proinde in earum &longs;tudio infi&shy;<lb/>
xus primam &aelig;tatis partem in illis percipiendis <lb/>
exegi&longs;ti, <expan abbr="eamqu&egrave;">eamque</expan> &longs;&aelig;pius <expan abbr="ver&egrave;">vere</expan> principe dignam <lb/>
vocem protuli&longs;ti, te propterea mathematicis <lb/>
pr&aelig;&longs;ertim delectari, <expan abbr="qu&ograve;d">quod</expan> i&longs;t&aelig; <expan abbr="maxim&egrave;">maxime</expan> ex do&shy;<lb/>
me&longs;tico illo, &amp; vmbratili vit&aelig; genere in Solem <lb/>
(quod dicitur) &amp; puluerem prodire po&longs;sint: cu<lb/>
ius <expan abbr="&longs;an&egrave;">&longs;ane</expan> rei tuum flagranti&longs;simum ab ineunte &aelig;ta <lb/>
te periti&aelig; militaris de&longs;iderium, exploratum in&shy;<lb/>
dicium poterat e&longs;&longs;e, ni&longs;i nimis emendicat&aelig; men&shy;<lb/>
tis e&longs;&longs;et ea proponere, qu&aelig; <expan abbr="&agrave;">a</expan> te &longs;perari po&longs;&longs;ent; <lb/>
quando tu penitus adole&longs;cens, egregia multa fa<lb/>
cinora proficere matura&longs;ti.  </s>
             
<s id="id.2.1.1.14.1.20.0"> Tu enim <expan abbr="c&ugrave;m">cum</expan> iam <lb/>
<expan abbr="&agrave;">a</expan> &longs;ancti&longs;&longs;imo Pontifice Pio V &longs;aluberrim&aelig; Prin&shy;<lb/>
cipum Chri&longs;tianorum coniunctionis fundamen&shy;<lb/>
ta iacta e&longs;&longs;ent, alacer admodum ad debellan&shy;<lb/>
dos Chri&longs;ti ho&longs;tes profectus, &longs;olidi&longs;&longs;imam, ac ve&shy;<lb/>
ri&longs;&longs;imam gloriam tibi compara&longs;ti.  </s>
             
<s id="id.2.1.1.14.1.21.0"> Tu quoties de <lb/>
&longs;umma rerum deliberatum e&longs;t, eas &longs;ententias <lb/>
dixi&longs;ti, qu&aelig; &longs;ummam prudentiam <expan abbr="c&ugrave;m">cum</expan> &longs;umma <lb/>
animi excel&longs;itate coniunctam indicarent.  </s>
             
<s id="id.2.1.1.14.1.22.0"> ommit&shy;<lb/>
taminterim pleraq; alia illis temporibus <expan abbr="egre&shy;gi&egrave;">egre&shy;<lb/>
gie</expan>, viriliter <expan abbr="qu&egrave;">que</expan> <expan abbr="&agrave;">a</expan> te ge&longs;ta, ne tibi ip&longs;i ea, qu&aelig; <lb/>
omnibus &longs;unt manife&longs;ta, <expan abbr="pal&agrave;m">palam</expan> facere videar: 
<pb/>
qu&aelig; <expan abbr="c&ugrave;m">cum</expan> omnia magna, &amp; pr&aelig;clara &longs;int; <expan abbr="mul&shy;t&ograve;">mul&shy;<lb/>
to</expan> tamen <expan abbr="&agrave;">a</expan> te maiora, &amp; pr&aelig;clara expectant <lb/>
adhuc homines.  </s>
             
<s id="id.2.1.1.14.1.23.0"> Vale interim pr&aelig;&longs;tanti&longs;&longs;imum <lb/>
orbis decus, &amp; &longs;i quando aliquid otij nactus <lb/>
fueris has meas vigiliolas a&longs;picere ne dedi&shy;<lb/>
gneris.  </s>
    </p>       
        <p id="id.2.1.1.15.0.0.0" type="head">        
          <pb n="1"/>
       
          <s id="id.2.1.1.16.1.1.0"> GVIDIVBALDI <lb/>
E MARCHIONIBVS <lb/>
MONTIS. </s>
     <lb/>
       
          <s id="id.2.1.1.16.3.1.0"> MECHANICORVM <lb/>
LIBER. </s>
     <lb/>
       
          
<s> ZZZ head of figure ZZZ </s>
    </p>       <figure place="text">       </figure>       
        <p id="id.2.1.1.16.5.1.0" type="caption">        
          <s id="id.2.1.1.16.5.1.0.capt"> YYY </s>

        </p>
      </section>
      </front>       
        <body>        
          <chap>        
            <p id="id.id.2.1.1.16.5.1.0.a">
              <s id="id.2.1.1.16.7.1.0"> DEFINITIONES. </s>

        </p>
            <p id="id.2.1.1.17.0.0.0" type="main">        
              <s id="id.2.1.1.17.1.1.0"> Centrvm grauitatis vniu&longs;cu&shy;<lb/>
iu&longs;q; corporis e&longs;t punctum quod&shy;<lb/>
dam intra po&longs;itum, <expan abbr="&agrave;">a</expan> quo &longs;i gra&shy;<lb/>
ue appen&longs;um mente concipiatur, <lb/>
dum fertur, quie&longs;cit; &amp; &longs;eruat eam, <lb/>
quam in principio habebat po&longs;i&shy;<lb/>
tionem: neq; in ip&longs;a latione circumuertitur. </s>
    </p>       
            <p id="id.2.1.1.18.0.0.0" type="main">        
              <s id="id.2.1.1.18.1.1.0"> Hanc centri grauitatis definitionem Pappus Alexandrinus in <lb/>
octauo Mathematicarum collectionum libro tradidit.  </s>
             
              <s id="id.2.1.1.18.1.2.0"> Federicus <lb/>
<expan abbr="ver&ograve;">vero</expan> Commandinus in libro de centro grauitatis &longs;olidorum idem <lb/>
centrum de&longs;cribendo ita explicauit. </s>
    </p>       
            <p id="id.2.1.1.19.0.0.0" type="main">        
              <s id="id.2.1.1.19.1.1.0"> Centrum grauitatis vniu&longs;cuiu&longs;q; &longs;olid&aelig; figu&shy;<lb/>
r&aelig; e&longs;t punctum illud intra po&longs;itum, circa quod <lb/>
vndiq; partes &aelig;qualium momentorum con&longs;i&shy;<lb/>
&longs;tunt.  </s>
             
              <s id="id.2.1.1.19.1.2.0"> &longs;i enim per tale centrum ducatur planum <lb/>
figuram quomodocunq; &longs;ecans &longs;emper in par&shy;<lb/>
tes &aelig;queponderantes ip&longs;am diuidet.  </s>
    </p>       
            <pb/>
       
<p id="id.2.1.1.21.0.0.0" type="head">        
<s id="id.2.1.1.21.1.1.0"> COMMVNES NOTIONES. </s>
     <lb/>
       
<s id="id.2.1.1.21.3.1.0"> I </s>
    </p>       
<p id="id.2.1.1.22.0.0.0" type="main">        
<s id="id.2.1.1.22.1.1.0"> Si ab &aelig;queponderantibus &aelig;queponderantia au&shy;<lb/>
ferantur, reliqua &aelig;queponderabunt.  </s>
    </p>       
<p id="id.2.1.1.23.0.0.0" type="head">        
<s id="id.2.1.1.23.1.1.0"> II </s>
    </p>       
<p id="id.2.1.1.24.0.0.0" type="main">        
<s id="id.2.1.1.24.1.1.0"> Si &aelig;queponderantibus &aelig;queponderantia adii&shy;<lb/>
ciantur, tota &longs;imul &aelig;queponderabunt.  </s>
    </p>       
<p id="id.2.1.1.25.0.0.0" type="head">        
<s id="id.2.1.1.25.1.1.0"> III </s>
    </p>       
<p id="id.2.1.1.26.0.0.0" type="main">        
<s id="id.2.1.1.26.1.1.0"> Qu&aelig; eidem &aelig;queponderant, inter &longs;e <expan abbr="&aelig;qu&egrave;">&aelig;que</expan> &longs;unt <lb/>
grauia.  </s>
    </p>       
<p id="id.2.1.1.27.0.0.0" type="head">        
<s id="id.2.1.1.27.1.1.0"> SVPPOSITIONES. </s>
     <lb/>
       
<s id="id.2.1.1.27.3.1.0"> I </s>
    </p>       
<p id="id.2.1.1.28.0.0.0" type="main">        
<s id="id.2.1.1.28.1.1.0"> Vnius corporis vnum <expan abbr="tant&ugrave;m">tantum</expan> e&longs;t centrum gra&shy;<lb/>
uitatis.  </s>
    </p>       
<p id="id.2.1.1.29.0.0.0" type="head">        
<s id="id.2.1.1.29.1.1.0"> II </s>
    </p>       
<p id="id.2.1.1.30.0.0.0" type="main">        
<s id="id.2.1.1.30.1.1.0"> Vnius corporis centrum grauitatis &longs;emper in <lb/>
eodem e&longs;t &longs;itu re&longs;pectu &longs;ui corporis.  </s>
    </p>       
<p id="id.2.1.1.31.0.0.0" type="head">        
<s id="id.2.1.1.31.1.1.0"> III </s>
    </p>       
<p id="id.2.1.1.32.0.0.0" type="main">        
<s id="id.2.1.1.32.1.1.0"> <expan abbr="Secund&ugrave;m">Secundum</expan> grauitatis centrum pondera deor&shy;<lb/>
&longs;um feruntur.  </s>
    </p>       
      </chap>
      <pb n="2"/>
      <chap>

<p id="id.2.1.1.33.0.0.0" type="head">        
<s id="id.2.1.1.34.1.1.0"> DE LIBRA. </s>
    </p>       
<p id="id.2.1.1.35.0.0.0" type="main">        
<s id="id.2.1.1.35.1.1.0"> Anteqvam de libra &longs;ermo ha<lb/>
beatur, vtres clarior eluce&longs;cat, &longs;it <lb/>
libra AB recta linea; CD <expan abbr="ver&ograve;">vero</expan> <lb/>
trutina, qu&aelig; &longs;ecundum commu&shy;<lb/>
nem con&longs;uetudinem horizonti <lb/>
&longs;emper e&longs;t perpendicularis.  </s>
             
<s id="id.2.1.1.35.1.2.0"> pun&shy;<lb/>
ctum autem C immobile, circa quod vertitur li&shy;<lb/>
bra, centrum libr&aelig; <lb/>
vocetur.  </s>
             
<s id="id.2.1.1.35.1.3.0"> itidemque <lb/>
(quamuis tamen im&shy;<lb/>
proprie) &longs;iue &longs;upra, <lb/>
&longs;iue infra libram fue<lb/>
rit con&longs;titutum.  </s>
             
<s id="id.2.1.1.35.1.4.0"> CA <lb/>
<expan abbr="ver&ograve;">vero</expan>, &amp; CB, tum di<lb/>
&longs;tanti&aelig;, tum libr&aelig; <lb/>
brachia nuncupen&shy;<lb/>
tur.  </s>
             
<s id="id.2.1.1.35.1.5.0"> &amp; &longs;i <expan abbr="&agrave;">a</expan> centro li&shy;<lb/>
br&aelig; &longs;upra, vel infra <lb/>
<arrow.to.target n="fig1"></arrow.to.target><lb/>
libram con&longs;tituto ip&longs;i AB perpendicularis duca&shy;<lb/>
tur, h&aelig;c perpendiculum vocetur, qu&aelig; libram AB <lb/>
&longs;ub&longs;tinebit; &amp; quocunque modo moueatur libra, <lb/>
ip&longs;i &longs;emper perpendicularis exi&longs;tet.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig1" place="text">       </figure>       
<p id="id.2.1.1.35.2.1.0" type="caption">        
<s id="id.2.1.1.35.2.1.0.capt"> YYY </s>
    </p>       
<pb/>
       
<p id="id.2.1.1.37.0.0.0" type="head">        
<s id="id.2.1.1.37.1.1.0"> LEMMA. </s>
    </p>       
<p id="id.2.1.1.38.0.0.0" type="main">        
<s id="id.2.1.1.38.1.1.0"> Sit linea AB horizonti perpendicularis, &amp; dia <lb/>
metro AB circulus de&longs;cribatur AEBD, cuius <lb/>
centrum C.  </s>
     
<s id="id.2.1.1.38.1.1.0.a"> Dico punctum B infimum e&longs;&longs;e lo&shy;<lb/>
cum circumferenti&aelig; circuli AEBD; punctum <lb/>
<expan abbr="ver&ograve;">vero</expan> A &longs;ublimiorem; &amp; qu&aelig;libet puncta, vt DE <lb/>
&aelig;qualiter <expan abbr="&agrave;">a</expan> puncto A di&longs;tantia &aelig;qualiter e&longs;&longs;e <lb/>
deor&longs;um; qu&aelig; <expan abbr="ver&ograve;">vero</expan> propius &longs;unt ip&longs;i A eis, qu&aelig; <lb/>
magis di&longs;tant, &longs;ublimiora e&longs;&longs;e. </s>
    </p>       
<p id="id.2.1.1.39.0.0.0" type="main">        
<s id="id.2.1.1.39.1.1.0"> Producatur AB v&longs;q; ad mundi cen&shy;<lb/>
trum, quod &longs;it F; deinde in circuli circum&shy;<lb/>
<arrow.to.target n="note1"></arrow.to.target> ferentia quoduis accipiatur punctum G; <lb/>
connectanturq; FG FD FE.   </s>
             
<s id="id.2.1.1.39.1.2.0"> Quoniam <lb/>
n. BF minima e&longs;t omnium, qu&aelig; <expan abbr="&agrave;">a</expan> puncto <lb/>
F ad circumferentiam AEBD ducun&shy;<lb/>
tur; erit BF ip&longs;a FG minor.  </s>
             
<s id="id.2.1.1.39.1.3.0"> quare punctum <lb/>
B propius erit puncto F, <expan abbr="qu&agrave;m">quam</expan> G.  </s>
     
<s id="id.2.1.1.39.1.3.0.a"> hacq; <lb/>
ratione o&longs;tendetur punctum B quouis alio <lb/>
puncto circumferenti&aelig; circuli AEDB <lb/>
mundi centro propius e&longs;&longs;e.  </s>
             
<s id="id.2.1.1.39.1.4.0"> erit igitur pun&shy;<lb/>
ctum B circumferenti&aelig; circuli AEBD <lb/>
infimus locus.  </s>
             
<s id="id.2.1.1.39.1.5.0"> Deinde quoniam AF per <lb/>
centrum ducta maior e&longs;t ip&longs;a GF; erit <lb/>
punctum A non <expan abbr="&longs;ol&utilde;">&longs;olum</expan> ip&longs;o G, verum etiam <lb/>
quouis alio puncto circumferenti&aelig; circuli <lb/>
AEBD &longs;ublimius.  </s>
             
<s id="id.2.1.1.39.1.6.0"> Pr&aelig;terea quoniam DF <lb/>
FE &longs;unt &aelig;quales; puncta DE &aelig;qualiter <lb/>
<arrow.to.target n="fig2"></arrow.to.target><lb/>
mundi centro di&longs;tabunt.  </s>
             
<s id="id.2.1.1.39.1.7.0"> &amp; cum DF maior &longs;it FG; erit pun&shy;<lb/>
ctum D ip&longs;i A propius puncto G &longs;ublimius. qu&aelig; omnia demon&shy;<lb/>
&longs;trare oportebat. </s>
             
<s id="id.2.1.1.39.1.8.0"> [qu&aelig; omnia demon&shy;<lb/>
&longs;trare oportebat.]  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig2" place="text">       </figure>       
<p id="id.2.1.1.39.2.1.0" type="caption">        
<s id="id.2.1.1.39.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.2.1.0.0.0" type="margin">        
<s id="id.2.1.2.1.1.1.0"> <margin.target id="note1"></margin.target>8. <emph type="italics"/>Tertil.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.3.1.0.0.0" type="head">        
<pb n="3"/>
       
<s id="id.2.1.3.1.2.1.0"> PROPOSITIO I. </s>
    </p>       
<p id="id.2.1.3.2.0.0.0" type="main">        
<s id="id.2.1.3.2.1.1.0"> Si Pondus in eius centro grauitatis a recta &longs;u&shy;<lb/>
&longs;tineatur linea, nunquam manebit, ni&longs;i eadem li&shy;<lb/>
nea horizonti fuerit perpendicularis. </s>
    </p>       
<p id="id.2.1.3.3.0.0.0" type="main">        
<s id="id.2.1.3.3.1.1.0"> Sit pondus A, cuius centrum gra<lb/>
uitatis B, quod <expan abbr="&agrave;">a</expan> linea CE &longs;u&longs;ti&shy;<lb/>
neatur.  </s>
             
<s id="id.2.1.3.3.1.2.0"> Dico pondus nunquam <lb/>
perman&longs;urum, ni&longs;i CB horizonti <lb/>
perpendicularis exi&longs;tat.  </s>
             
<s id="id.2.1.3.3.1.3.0"> &longs;it pun&shy;<lb/>
ctum C immobile, quod vt pon<lb/>
dus &longs;u&longs;tineatur, nece&longs;&longs;e e&longs;t.  </s>
             
<s id="id.2.1.3.3.1.4.0"> &amp; cum <lb/>
punctum C &longs;it immobile, &longs;i pon&shy;<lb/>
dus A mouebitur, punctum B cir<lb/>
culi circumferentiam de&longs;cribet, <lb/>
cuius &longs;emidiameter erit CB. qua<lb/>
re centro C, &longs;patio <expan abbr="ver&ograve;">vero</expan> BC, cir&shy;<lb/>
culus de&longs;cribatur BFDE.  </s>
     
<s id="id.2.1.3.3.1.4.0.a"> &longs;itq; <lb/>
<arrow.to.target n="fig3"></arrow.to.target><lb/>
primum BC horizonti <expan abbr="perpendicular&iacute;s">perpendicularis</expan>, qu&aelig; v&longs;q; ad D produca&shy;<lb/>
tur; atq; punctum C &longs;it infra punctum B.  </s>
     
<s id="id.2.1.3.3.1.4.0.b"> Quoniam enim pondus <arrow.to.target n="note2"></arrow.to.target><lb/>
A &longs;ecundum grauitatis centrum B deor&longs;um mouetur; punctum <lb/>
B deor&longs;um in centrum mundi, <expan abbr="qu&ograve;">quo</expan> naturaliter tendit, per re&shy;<lb/>
ctam lineam BD mouebitur: totum ergo pondus A eius cen&shy;<lb/>
tro grauitatis B &longs;uper rectam lineam BC graue&longs;cet.  </s>
             
<s id="id.2.1.3.3.1.5.0"> cum au&shy;<lb/>
tem pondus <expan abbr="&agrave;">a</expan> linea CB &longs;u&longs;tineatur, linea CB totum &longs;u&longs;ti&shy;<lb/>
nebit pondus A; &longs;uper quam deor&longs;um moueri non pote&longs;t, cum <lb/>
ab ip&longs;a prohibeatur: per definitionem igitur centri grauitatis pun<lb/>
ctum B, pondu&longs;q; A in hoc &longs;itu manebunt.  </s>
             
<s id="id.2.1.3.3.1.6.0"> &amp; quamquam B quo&shy;<lb/>
cunq; alio puncto circuli &longs;it &longs;ublimius, ab hoc tamen &longs;itu deor&longs;um <lb/>
per circuli circumferentiam nequaquam mouebitur non enim ver&shy;<lb/>
&longs;us F magis, <expan abbr="qu&agrave;m">quam</expan> ver&longs;us E inclinabitur, cum ex vtraq; parte &aelig;qua&shy;<lb/>
lis &longs;it de&longs;cen&longs;us; neq; pondus A in vnam magis, <expan abbr="qu&agrave;m">quam</expan> in alteram <lb/>
partem propen&longs;ionem habeat: quod non accidit in quouis alio <lb/>
puncto circumferenti&aelig; circuli (pr&aelig;ter D) &longs;it ponderis eiu&longs;dem 
<pb/>
centrum grauitatis, vt in F; cum ex <lb/>
puncto F ver&longs;us D &longs;it de&longs;cen&longs;us, at <lb/>
<expan abbr="ver&ograve;">vero</expan> ver&longs;us B a&longs;cen&longs;us.  </s>
             
<s id="id.2.1.3.3.1.7.0"> quare pun&shy;<lb/>
ctum F deor&longs;um mouebitur. </s>
             
<s id="id.2.1.3.3.1.8.0"> &amp; quo<lb/>
niam per rectam lineam in centrum <lb/>
mundi moueri non pote&longs;t, cum <expan abbr="&agrave;">a</expan> <lb/>
puncto C immobili propter lineam <lb/>
CF prohibeatur; deor&longs;um tamen <lb/>
&longs;icuti eius natura po&longs;tulat, &longs;emper <lb/>
mouebitur.  </s>
             
<s id="id.2.1.3.3.1.9.0"> &amp; cum infimus locus &longs;it <lb/>
D, per <expan abbr="circumferenti&atilde;">circumferentiam</expan> FD mouebi<lb/>
tur, donec in D perueniat, in quo <lb/>
&longs;itu manebit, <expan abbr="p&otilde;du&longs;q">pondu&longs;q</expan>; immobile exi <lb/>
<arrow.to.target n="fig4"></arrow.to.target><lb/>
&longs;tet.  </s>
             
<s id="id.2.1.3.3.1.10.0"> tum quia deor&longs;um amplius moueri non pote&longs;t, cum ex pun&shy;<lb/>
cto C &longs;it appen&longs;um; tum etiam, quia in eius centro grauitatis &longs;u&longs;ti<lb/>
netur.  </s>
             
<s id="id.2.1.3.3.1.11.0"> Quando autem F erit in D, erit quoq; linea FC in DC, <lb/>
&longs;imulq; horizonti perpendicularis.  </s>
             
<s id="id.2.1.3.3.1.12.0"> pondus ergo nunquam mane<lb/>
bit, donec linea CF horizonti perpendicularis non exi&longs;tat. quod <lb/>
o&longs;tendere oportebat.  </s>
             
<s id="id.2.1.3.3.1.13.0"> [quod <lb/>
o&longs;tendere oportebat.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig3" place="text">       </figure>       
<p id="id.2.1.3.3.2.1.0" type="caption">        
<s id="id.2.1.3.3.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig4" place="text">       </figure>       
<p id="id.2.1.3.3.2.3.0" type="caption">        
<s id="id.2.1.3.3.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.4.1.0.0.0" type="margin">        
<s id="id.2.1.4.1.1.1.0"> <margin.target id="note2"></margin.target><emph type="italics"/>Supp.<emph.end type="italics"/> 3. <emph type="italics"/>huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.5.1.0.0.0" type="main">        
<s id="id.2.1.5.1.1.1.0"> Ex hoc elici pote&longs;t, pondus quocunq; modo <lb/>
in dato puncto &longs;u&longs;tineatur, nunquam manere; ni <lb/>
&longs;i quando a centro grauitatis ponderis ad id pun<lb/>
ctum ducta linea horizonti &longs;it perpendicularis. </s>
    </p>       
<p id="id.2.1.5.2.0.0.0" type="main">        
<s id="id.2.1.5.2.1.1.0"> Vt ii&longs;dem po&longs;itis, &longs;u&longs;tineatur <lb/>
pondus <expan abbr="&agrave;">a</expan> lineis CG CH.  </s>
     
<s id="id.2.1.5.2.1.1.0.a"> Dico <lb/>
&longs;i ducta BC horizonti &longs;it perpen&shy;<lb/>
dicularis, pondus A manere.  </s>
             
<s id="id.2.1.5.2.1.2.0"> &longs;i <expan abbr="ver&ograve;">vero</expan> <lb/>
ducta CF non &longs;it horizonti per&shy;<lb/>
pendicularis, punctum F deor&longs;um <lb/>
v&longs;q; ad D moueri; in quo &longs;itu pon&shy;<lb/>
dus manebit, ductaq; CD horizon<lb/>
ti perpendicularis exi&longs;tet.  </s>
             
<s id="id.2.1.5.2.1.3.0"> qu&aelig; om&shy;<lb/>
nia eadem ratione o&longs;tendentur. <arrow.to.target n="fig5"></arrow.to.target> </s>
     
<pb n="4"/>
       
<s id="id.2.1.5.2.3.1.0"> PROPOSITIO II. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig5" place="text">       </figure>       
<p id="id.2.1.5.2.4.1.0" type="caption">        
<s id="id.2.1.5.2.4.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.5.3.0.0.0" type="main">        
<s id="id.2.1.5.3.1.1.0"> Libra horizonti &aelig;quidi&longs;tans, cuius centrum <lb/>
&longs;it &longs;upra libram, &aelig;qualia in extremitatibus, &aelig;qua <lb/>
literq; <expan abbr="&agrave;">a</expan> perpendiculo di&longs;tantia habens pondera, <lb/>
&longs;i ab eiu&longs;modi moueatur &longs;itu, in eundem rur&longs;us <lb/>
relicta, redibit; <expan abbr="ib&iacute;q">ibiq</expan>; manebit. </s>
    </p>       
<p id="id.2.1.5.4.0.0.0" type="main">        
<s id="id.2.1.5.4.1.1.0"> Sit libra AB recta li&shy;<lb/>
nea horizonti &aelig;quidi&shy;<lb/>
&longs;tans, cuius centrum C <lb/>
&longs;it &longs;upral ibram; &longs;itq; CD <lb/>
<expan abbr="perpendicul&utilde;">perpendiculum</expan>, quod ho&shy;<lb/>
rizonti perpendiculare <lb/>
erit: atq; di&longs;tantia DA &longs;it <lb/>
di&longs;tanti&aelig; DB &aelig;qualis; <lb/>
&longs;intq; in AB pondera &aelig;&shy;<lb/>
qualia, <expan abbr="quor&utilde;">quorum</expan> grauitatis <lb/>
centra &longs;int in AB <expan abbr="p&utilde;ctis">punctis</expan>.  </s>
             
<s id="id.2.1.5.4.1.2.0"> <lb/>
Moueatur AB libra ab <lb/>
<arrow.to.target n="fig6"></arrow.to.target><lb/>
hoc &longs;itu, <expan abbr="put&aacute;">puta</expan> in EF, deinde relinquatur.  </s>
             
<s id="id.2.1.5.4.1.3.0"> dico libram EF in AB ho<lb/>
rizonti &aelig;quidi&longs;tantem redire, <expan abbr="ib&iacute;q">ibiq</expan>; manere.  </s>
             
<s id="id.2.1.5.4.1.4.0"> Quoniam autem pun<lb/>
ctum C e&longs;t immobile, dum libra mouetur, punctum D circuli cir&shy;<lb/>
cumferentiam de&longs;cribet, cuius &longs;emidiameter erit CD. quare cen&shy;<lb/>
tro C, &longs;patio <expan abbr="ver&ograve;">vero</expan> CD, circulus de&longs;cribatur DGH.  </s>
     
<s id="id.2.1.5.4.1.4.0.a"> Quoniam <lb/>
enim CD ip&longs;i libr&aelig; &longs;emper e&longs;t perpendicularis, dum libra erit in <lb/>
EF, linea CD erit in CG, ita vt CG &longs;it ip&longs;i EF perpendicula&shy;<lb/>
ris.  </s>
             
<s id="id.2.1.5.4.1.5.0"> <expan abbr="C&ugrave;m">Cum</expan> autem AB bifariam <expan abbr="&agrave;">a</expan> puncto D diuidatur, &amp; pondera <lb/>
in AB &longs;int &aelig;qualia; erit magnitudinis ex ip&longs;is AB compo&longs;it&aelig; cen <arrow.to.target n="note3"></arrow.to.target><lb/>
trum grauitatis in medio, hoc e&longs;t in D. &amp; <expan abbr="qu&atilde;do">quando</expan> libra <expan abbr="vn&aacute;">vna</expan> cum pon<lb/>
deribus erit in EF; erit magnitudinis ex vtri&longs;q; EF compo&longs;it&aelig; cen<lb/>
trum grauitatis G.  </s>
     
<s id="id.2.1.5.4.1.5.0.a"> &amp; quoniam CG horizonti non e&longs;t perpendi&shy;<lb/>
cularis;  <arrow.to.target n="note4"></arrow.to.target> magnitudo ex ponderibus EF compo&longs;ita in hoc &longs;itu <expan abbr="mi&shy;nim&egrave;">mi&shy;<lb/>
nime</expan> per&longs;i&longs;tet, &longs;ed deor&longs;um <expan abbr="&longs;ec&utilde;d&ugrave;m">&longs;ecundum</expan> eius centrum grauitatis G per <lb/>
circumferentiam GD mouebitur; donec CG horizonti fiat per&shy;
<pb/>
pendicularis, &longs;cilicet do&shy;<lb/>
nec CG in CD redeat.  </s>
             
<s id="id.2.1.5.4.1.6.0"> <lb/>
Quando autem CG erit <lb/>
in CD, linea EF, <expan abbr="c&ugrave;m">cum</expan> <lb/>
ip&longs;i CG &longs;emper ad rectos <lb/>
&longs;it angulos, erit in AB; in <lb/>
<arrow.to.target n="note5"></arrow.to.target> quo &longs;itu quoq; manebit.  </s>
             
<s id="id.2.1.5.4.1.7.0"> li<lb/>
bra ergo EF in AB hori&shy;<lb/>
zonti <expan abbr="&aelig;quidi&longs;t&atilde;tem">&aelig;quidi&longs;tantem</expan> redi<lb/>
bit, <expan abbr="ib&iacute;q">ibiq</expan>; manebit. quod <lb/>
demon&longs;trare oportebat. </s>
             
<s id="id.2.1.5.4.1.8.0"> [quod <lb/>
demon&longs;trare oportebat.]  </s>
     <lb/>
       

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig6" place="text">       </figure>       
<p id="id.2.1.5.4.2.1.0" type="caption">        
<s id="id.2.1.5.4.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.6.1.0.0.0" type="margin">        
<s id="id.2.1.6.1.1.1.0"> <margin.target id="note3"></margin.target>4. <emph type="italics"/>primi Archimedis de &aelig;queponderantibus.<emph.end type="italics"/> </s>
             
<s id="id.2.1.6.1.1.2.0"> <margin.target id="note4"></margin.target>1. <emph type="italics"/>Huius<emph.end type="italics"/> </s>
             
<s id="id.2.1.6.1.1.3.0"> <margin.target id="note5"></margin.target>1. <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.7.1.0.0.0" type="main">        

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure place="text">       </figure>       
<p id="id.2.1.7.1.1.1.0" type="caption">        
<s id="id.2.1.7.1.1.1.0.capt"> YYY </s>
     <lb/>
       
<s id="id.2.1.7.1.3.1.0"> PROPOSITIO III. </s>
    </p>       
<p id="id.2.1.7.2.0.0.0" type="main">        
<s id="id.2.1.7.2.1.1.0"> Libra horizonti &aelig;quidi&longs;tans &aelig;qualia in extre&shy;<lb/>
mitatibus, &aelig;qualiterq; <expan abbr="&agrave;">a</expan> perpendiculo di&longs;tan&shy;<lb/>
tia habens pondera, centro <expan abbr="infern&egrave;">inferne</expan> collocato, in <lb/>
hoc &longs;itu manebit.  </s>
             
<s id="id.2.1.7.2.1.2.0"> &longs;i <expan abbr="ver&ograve;">vero</expan> inde moueatur, deor&shy;<lb/>
&longs;um relicta, <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> partem decliuiorem mo&shy;<lb/>
uebitur. <arrow.to.target n="fig7"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.7.3.0.0.0" type="main">        
<s id="id.2.1.7.3.1.1.0"> Sit libra AB <expan abbr="rect&aacute;">recta</expan> li&shy;<lb/>
nea horizonti &aelig;quidi&shy;<lb/>
&longs;tans, cuius centrum C <lb/>
&longs;it infra libram; perpen&shy;<lb/>
diculumq; &longs;it CD, quod <lb/>
horizonti perpendiculare <lb/>
erit; &amp; di&longs;tantia AD &longs;it <lb/>
di&longs;tanti&aelig; DB &aelig;qualis; <lb/>
&longs;intq; in AB pondera <lb/>
&aelig;qualia, quorum grauita&shy;<lb/>
tis centra &longs;int in punctis <lb/>
AB.  </s>
     
<s id="id.2.1.7.3.1.1.0.a"> Dico <expan abbr="prim&ugrave;m">primum</expan> libram AB in hoc &longs;itu manere.  </s>
             
<s id="id.2.1.7.3.1.2.0"> Quoniam <lb/>
enim AB bifariam diuiditur <expan abbr="&agrave;">a</expan> puncto D, &amp; pondera in AB &longs;unt <lb/>
&aelig;qualia; erit punctum D centrum grauitatis magnitudinis ex 
<pb n="5"/>
vtri&longs;q; AB ponderibus compo&longs;it&aelig;.  </s>
             
<s id="id.2.1.7.3.1.3.0"> &amp; CD libram &longs;u&longs;tinens ho&shy;<lb/>
rizonti  <arrow.to.target n="note6"></arrow.to.target> e&longs;t perpendicularis, libra ergo AB in hoc &longs;itu manebit. <arrow.to.target n="note7"></arrow.to.target><lb/>
moueatur autem libra AB ab hoc &longs;itu, <expan abbr="put&agrave;">puta</expan> in EF, deinde relinqua<lb/>
tur.  </s>
             
<s id="id.2.1.7.3.1.4.0"> dico libram EF ex parte F moueri.  </s>
             
<s id="id.2.1.7.3.1.5.0"> Quoniam igitur CD <lb/>
ip&longs;i libr&aelig; &longs;emper e&longs;t perpendicularis, dum libra erit in EF, erit <lb/>
CD in CG ip&longs;i EF perpendicularis.  </s>
             
<s id="id.2.1.7.3.1.6.0"> &amp; punctum G magnitudi&shy;<lb/>
nis ex EF compo&longs;it&aelig; centrum grauitatis erit; quod dum moue&shy;<lb/>
tur, circuli circumferentiam de&longs;cribet DGH, cuius &longs;emidiameter <lb/>
CD, &amp; centrum C.  </s>
     
<s id="id.2.1.7.3.1.6.0.a"> Quoniam autem CG horizonti non e&longs;t per&shy;<lb/>
pendicularis, magnitudo ex EF ponderibus compo&longs;ita in hoc &longs;i&shy;<lb/>
tu <expan abbr="minim&egrave;">minime</expan> manebit; &longs;ed <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> eius grauitatis centrum G deor<lb/>
&longs;um per circumferentiam GH mouebitur.  </s>
             
<s id="id.2.1.7.3.1.7.0"> libra ergo EF ex par <lb/>
te F deor&longs;um mouebitur, quod demon&longs;trare oportebat.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig7" place="text">       </figure>       
<p id="id.2.1.7.3.2.1.0" type="caption">        
<s id="id.2.1.7.3.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.8.1.0.0.0" type="margin">        
<s id="id.2.1.8.1.1.1.0"> <margin.target id="note6"></margin.target>4. <emph type="italics"/>Primi Archim. de &aelig;quep.<emph.end type="italics"/> </s>
             
<s id="id.2.1.8.1.1.2.0"> [de &aelig;quep.<emph.end type="italics"/>] </s>
             
<s id="id.2.1.8.1.1.3.0"> <margin.target id="note7"></margin.target>1. <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.9.1.0.0.0" type="head">        
<s id="id.2.1.9.1.1.1.0"> PROPOSITIO IIII. </s>
    </p>       
<p id="id.2.1.9.2.0.0.0" type="main">        
<s id="id.2.1.9.2.1.1.0"> Libra horizonti &aelig;quidi&longs;tans &aelig;qualia in ex&shy;<lb/>
tremitatibus, &aelig;qualiterq; <expan abbr="&agrave;">a</expan> centro in ip&longs;a libra <lb/>
collocato, di&longs;tantia habens pondera; &longs;iue inde <lb/>
moueatur, &longs;iue minus; vbicunq; relicta, manebit. <arrow.to.target n="fig8"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.9.3.0.0.0" type="main">        
<s id="id.2.1.9.3.1.1.0"> Sit libra recta linea A <lb/>
B horizonti &aelig;quidi&longs;tans, <lb/>
cuius centrum C in ea&shy;<lb/>
dem &longs;it linea AB; di&longs;tan<lb/>
tia <expan abbr="ver&ograve;">vero</expan> CA &longs;it di&longs;tanti&aelig; <lb/>
CB &aelig;qualis: &longs;intq; pon&shy;<lb/>
dera in AB &aelig;qualia, quo&shy;<lb/>
rum centra grauitatis &longs;int <lb/>
in puntis AB.  </s>
     
<s id="id.2.1.9.3.1.1.0.a"> Moueatur <lb/>
libra, vt in DE, <expan abbr="ibiqu&egrave;">ibique</expan> <lb/>
relinquatur.  </s>
             
<s id="id.2.1.9.3.1.2.0"> Dico <expan abbr="prim&ugrave;m">primum</expan> libram DE non moueri, in <expan abbr="eoqu&egrave;">eoque</expan> &longs;itu <lb/>
manere.  </s>
             
<s id="id.2.1.9.3.1.3.0"> Quoniam enim pondera AB &longs;unt &aelig;qualia; erit magni&shy;<lb/>
tudinis ex vtroq; pondere, videlicet A, &amp; B compo&longs;it&aelig; centrum <lb/>
grauitatis C. quare idem punctum C, &amp; centrum libr&aelig;, &amp; <expan abbr="centr&utilde;">centrum</expan> <lb/>
grauitatis totius ponderis erit.  </s>
             
<s id="id.2.1.9.3.1.4.0"> Quoniam autem centrum libr&aelig; 
<pb/>
C, dum libra AB <expan abbr="vn&agrave;">vna</expan> <lb/>
cum ponderibus in DE <lb/>
mouetur, immobile re&shy;<lb/>
manet, centrum quoq; <lb/>
grauitatis, quod e&longs;t idem <lb/>
C, non mouebitur.  </s>
             
<s id="id.2.1.9.3.1.5.0"> nec <lb/>
igitur libra DE mouebi<lb/>
tur, per definitionem <lb/>
centri grauitatis, cum in <lb/>
ip&longs;o &longs;u&longs;pendatur.  </s>
             
<s id="id.2.1.9.3.1.6.0"> Idip&shy;<lb/>
<arrow.to.target n="fig9"></arrow.to.target><lb/>
&longs;um quoq; contingit libra in AB horizonti &aelig;quidi&longs;tante, vel in <lb/>
quocunq; alio &longs;itu exi&longs;tente.  </s>
             
<s id="id.2.1.9.3.1.7.0"> Manebit ergo libra, vbi relinque&shy;<lb/>
tur. quod demon&longs;trare oportebat. </s>
             
<s id="id.2.1.9.3.1.8.0"> [quod demon&longs;trare oportebat.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig8" place="text">       </figure>       
<p id="id.2.1.9.3.2.1.0" type="caption">        
<s id="id.2.1.9.3.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig9" place="text">       </figure>       
<p id="id.2.1.9.3.2.3.0" type="caption">        
<s id="id.2.1.9.3.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.9.4.0.0.0" type="main">        
<s id="id.2.1.9.4.1.1.0"> Cum <expan abbr="ver&ograve;">vero</expan> in iis, qu&aelig; dicta &longs;unt, grauitatis <expan abbr="tant&ugrave;m">tantum</expan> magnitudi<lb/>
num, qu&aelig; in extremitatibus libr&aelig; po&longs;it&aelig; &longs;unt &aelig;quales, ab&longs;q; <expan abbr="l&iacute;&shy;br&aelig;">li&shy;<lb/>
br&aelig;</expan> grauitate con&longs;iderauerimus; quoniam tamen adhuc libr&aelig; bra&shy;<lb/>
chia &longs;unt &aelig;qualia, idcirco idem libr&aelig;, eius grauitate con&longs;iderata, <lb/>
<expan abbr="vn&agrave;">vna</expan> cum ponderibus, vel &longs;ine ponderibus eueniet.  </s>
             
<s id="id.2.1.9.4.1.2.0"> idem enim cen<lb/>
trum grauitatis fine ponderibus libr&aelig; <expan abbr="tant&ugrave;m">tantum</expan> grauitatis centrum <lb/>
erit.  </s>
             
<s id="id.2.1.9.4.1.3.0"> Similiter &longs;i pondera in libr&aelig; extremitatibus appendantur, vt <lb/>
fieri &longs;olet, idem cueniet; dummodo ex &longs;u&longs;pen&longs;ionum punctis ad <lb/>
centra grauitatum ponderum duct&aelig; line&aelig; (quocunq; modo mo&shy;<lb/>
ueatur libra) &longs;i protrahantur, in centrum mundi concurrant.  </s>
             
<s id="id.2.1.9.4.1.4.0"> vbi <lb/>
enim pondera hoc modo &longs;unt appen&longs;a, ibi graue&longs;cunt, ac&longs;i in ii&longs;&shy;<lb/>
dem punctis centra grauitatum haberent.  </s>
             
<s id="id.2.1.9.4.1.5.0"> pr&aelig;terea, qu&aelig; &longs;equun&shy;<lb/>
tur, eodem pror&longs;us modo con&longs;iderare poterimus. </s>
    </p>       
<p id="id.2.1.9.5.0.0.0" type="main">        
<s id="id.2.1.9.5.1.1.0"> <arrow.to.target n="note8"></arrow.to.target>Quoniam autem huic determinationi vltim&aelig; multa <expan abbr="&agrave;">a</expan> nonnullis <lb/>
aliter &longs;entientibus dicta officere videntur; idcirco in hac parte ali&shy;<lb/>
<arrow.to.target n="note9"></arrow.to.target> quantulum immorari oportebit; &amp; pro viribus, non &longs;olum pro&shy;<lb/>
priam &longs;ententiam, &longs;ed Archimedem ip&longs;um, qui in hac eadem e&longs;&longs;e <lb/>
<arrow.to.target n="note10"></arrow.to.target> &longs;ententia videtur, defendere conabor. 
<pb n="6"/>
<arrow.to.target n="fig10"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.9.6.0.0.0" type="main">        
<s id="id.2.1.9.6.1.1.0"> Ii&longs;dem po&longs;itis, duca&shy;<lb/>
tur FCG ip&longs;i AB, &amp; <lb/>
horizonti perpendicula&shy;<lb/>
ris; &amp; centro C, <expan abbr="&longs;patio&shy;qu&egrave;">&longs;patio&shy;<lb/>
que</expan> CA, circulus de&longs;cri<lb/>
batur ADFBEG. erunt <lb/>
puncta ADBE in circu<lb/>
li circumferentia; cum li&shy;<lb/>
br&aelig; brachia &longs;int &aelig;qualia.  </s>
             
<s id="id.2.1.9.6.1.2.0"> <lb/>
&amp; quoniam in vnam con<lb/>
ueniunt &longs;ententiam, a&longs;&longs;e&shy;<lb/>
rentes &longs;cilicet libram DE <lb/>
neq; in FG moueri, ne&shy;<lb/>
que in DE manere, &longs;ed in AB horizonti &aelig;quidi&longs;tantem <expan abbr="redir&eacute;">redire</expan>.  </s>
             
<s id="id.2.1.9.6.1.3.0"> <lb/>
hanc eorum &longs;ententiam nullo modo con&longs;i&longs;tere po&longs;&longs;e o&longs;tendam.  </s>
             
<s id="id.2.1.9.6.1.4.0"> <lb/>
Non enim, &longs;ed &longs;i quod aiunt, euenerit, vel ideo erit, quia pondus <lb/>
D pondere E grauius fuerit, vel &longs;i pondera &longs;unt &aelig;qualia, di&longs;tanti&aelig;, <lb/>
quibus &longs;unt po&longs;ita, non erunt &aelig;quales, hoc e&longs;t CD ip&longs;i CE non erit <lb/>
&aelig;qualis, &longs;ed maior.  </s>
             
<s id="id.2.1.9.6.1.5.0"> <expan abbr="Qu&ograve;d">Quod</expan> autem pondera in DE &longs;int &aelig;qualia, &amp; <lb/>
di&longs;tantia CD &longs;it &aelig;qualis di&longs;tanti&aelig; CE: h&aelig;c ex &longs;uppo&longs;itione pa&shy;<lb/>
tent.  </s>
             
<s id="id.2.1.9.6.1.6.0"> Sed quoniam dicunt pondus in D in eo &longs;itu pondere in E <lb/>
grauius e&longs;&longs;e in altero &longs;itu deor&longs;um: dum pondera &longs;unt in DE, pun&shy;<lb/>
ctum C non erit amplius centrum grauitatis, nam non manent, &longs;i <lb/>
ex C &longs;u&longs;pendantur; &longs;ed erit in linea CD, ex tertia primi Archi&shy;<lb/>
medis de &aelig;queponderantibus.  </s>
             
<s id="id.2.1.9.6.1.7.0"> non autem erit in linea CE, cum pon<lb/>
dus D grauius &longs;it pondere E. &longs;it igitur in H, in quo &longs;i &longs;u&longs;pendan&shy;<lb/>
tur, manebunt.  </s>
             
<s id="id.2.1.9.6.1.8.0"> Quoniam autem centrum grauitatis ponderum <lb/>
in AB connexorum e&longs;t punctum C; ponderum <expan abbr="ver&ograve;">vero</expan> in DE e&longs;t <lb/>
punctum H: dum igitur pondera AB mouentur in DE, centrum <lb/>
grauitatis C ver&longs;us D mouebitur, &amp; ad D propius accedet; quod <lb/>
e&longs;t impo&longs;sibile: cum pondera eandem inter &longs;e &longs;e &longs;eruent di&longs;tantiam.  </s>
             
<s id="id.2.1.9.6.1.9.0"> <lb/>
Vniu&longs;cuiu&longs;q; enim corporis centrum grauitatis in eodem &longs;emper <arrow.to.target n="note11"></arrow.to.target><lb/>
e&longs;t &longs;itu re&longs;pectu &longs;ui corporis.  </s>
             
<s id="id.2.1.9.6.1.10.0"> &amp; quamquam punctum C &longs;it duo&shy;<lb/>
rum corporum AB centrum grauitatis, quia tamen inter &longs;e &longs;e ita <expan abbr="&agrave;">a</expan> <lb/>
libra connexa &longs;unt, vt &longs;emper eodem modo &longs;e &longs;e habeant; Ideo <lb/>
punctum C ita eorum erit centrum grauitatis, ac &longs;i vna tantum 
<pb/>
<arrow.to.target n="note12"></arrow.to.target> e&longs;&longs;et magnitudo.  </s>
             
<s id="id.2.1.9.6.1.11.0"> libra <lb/>
enim vna cum ponderi&shy;<lb/>
bus vnum tantum conti<lb/>
nuum efficit, cuius cen&shy;<lb/>
trum grauitatis erit &longs;em&shy;<lb/>
per in medio.  </s>
             
<s id="id.2.1.9.6.1.12.0"> non igitur <lb/>
pondus in D pondere in <lb/>
E e&longs;t grauius.  </s>
             
<s id="id.2.1.9.6.1.13.0"> Si autem <lb/>
dicerent centrum graui&shy;<lb/>
tatis non in linea CD, <lb/>
&longs;ed in CE e&longs;&longs;e debere; <lb/>
idem eueniet ab&longs;urdum. <arrow.to.target n="fig11"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.9.7.0.0.0" type="main">        
<s id="id.2.1.9.7.1.1.0"> Amplius &longs;i pondus D <lb/>
deor&longs;um mouebitur, pondus E &longs;ur&longs;um mouebit.  </s>
             
<s id="id.2.1.9.7.1.2.0"> pondus igitur gra&shy;<lb/>
uius, <expan abbr="qu&agrave;m">quam</expan> &longs;it E, in eodemmet &longs;itu ponderi D &aelig;queponderabit, &amp; <lb/>
grauia in&aelig;qualia &aelig;quali di&longs;tantia po&longs;ita &aelig;queponderabunt.  </s>
             
<s id="id.2.1.9.7.1.3.0"> Adii&shy;<lb/>
ciatur ergo ponderi E aliquod graue, ita vt ip&longs;i D contraponde&shy;<lb/>
ret, &longs;i ex C &longs;u&longs;pendantur.  </s>
             
<s id="id.2.1.9.7.1.4.0"> &longs;ed cum &longs;upra o&longs;ten&longs;um &longs;it punctum C <lb/>
centrum e&longs;&longs;e grauitatis &aelig;qualium ponderum in DE; &longs;i igitur pon&shy;<lb/>
<arrow.to.target n="note13"></arrow.to.target> dus E grauius fuerit pondere D, erit centrum grauitatis in linea <lb/>
CE. &longs;itq; hoc centrum K. at per definitionem centri grauitatis, &longs;i <lb/>
pondera &longs;u&longs;pendantur ex K, manebunt.  </s>
             
<s id="id.2.1.9.7.1.5.0"> ergo &longs;i &longs;u&longs;pendantur ex <lb/>
C, non manebunt, quod e&longs;t contra hypote&longs;im: &longs;ed pondus E deor<lb/>
&longs;um mouebitur.  </s>
             
<s id="id.2.1.9.7.1.6.0"> <expan abbr="qu&ograve;d">quod</expan> &longs;i ex C quoque &longs;u&longs;pen&longs;a &aelig;queponderarent; <lb/>
<arrow.to.target n="note14"></arrow.to.target> vnius magnitudinis duo e&longs;&longs;ent centra grauitatis; quod e&longs;t impo&longs;si<lb/>
bile.  </s>
             
<s id="id.2.1.9.7.1.7.0"> Non igitur pondus in E grauius eo, quod e&longs;t in D, ip&longs;i D &aelig;que&shy;<lb/>
ponderabit, cum ex puncto C fiat &longs;u&longs;pen&longs;io.  </s>
             
<s id="id.2.1.9.7.1.8.0"> Pondera ergo in DE <lb/>
&aelig;qualia ex eorum grauitatis centro C &longs;u&longs;pen&longs;a, &aelig;queponderabunt, <lb/>
<expan abbr="manebuntqu&egrave;">manebuntque</expan>. quod demon&longs;trare fuerat propo&longs;itum. </s>
             
<s id="id.2.1.9.7.1.9.0"> [quod demon&longs;trare fuerat propo&longs;itum.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig10" place="text">       </figure>       
<p id="id.2.1.9.7.2.1.0" type="caption">        
<s id="id.2.1.9.7.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig11" place="text">       </figure>       
<p id="id.2.1.9.7.2.3.0" type="caption">        
<s id="id.2.1.9.7.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.10.1.0.0.0" type="margin">        
<s id="id.2.1.10.1.1.1.0"> <margin.target id="note8"></margin.target><emph type="italics"/>Iordanus de Ponderibus.<emph.end type="italics"/> </s>
             
<s id="id.2.1.10.1.1.2.0"> <margin.target id="note9"></margin.target><emph type="italics"/>Hyerommus Carda nus de &longs;ubtilitate.<emph.end type="italics"/> </s>
             
<s id="id.2.1.10.1.1.3.0"> <margin.target id="note10"></margin.target><emph type="italics"/>Nicolaus Tartalea de qu&aelig;&longs;itis, ac inuentionibus.<emph.end type="italics"/> </s>
             
<s id="id.2.1.10.1.1.4.0"> <margin.target id="note11"></margin.target>2. <emph type="italics"/>Sup. huius.<emph.end type="italics"/>  </s>
             
<s id="id.2.1.10.1.1.5.0"> [huius.<emph.end type="italics"/>] </s>
             
<s id="id.2.1.10.1.1.6.0"> <margin.target id="note12"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 4. <emph type="italics"/>primi Archim de Aequep.<emph.end type="italics"/> </s>
             
<s id="id.2.1.10.1.1.7.0"> <margin.target id="note13"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 3. <emph type="italics"/>primi Archim de Aequep.<emph.end type="italics"/> </s>
             
<s id="id.2.1.10.1.1.8.0"> <margin.target id="note14"></margin.target>1. <emph type="italics"/>Suppo&longs;. huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.10.1.1.9.0"> [huius.<emph.end type="italics"/>] </s>
    </p>       
<p id="id.2.1.11.1.0.0.0" type="main">        
<s id="id.2.1.11.1.1.1.0"> <arrow.to.target n="note15"></arrow.to.target> Huic autem po&longs;tremo inconuenienti occurrunt dicentes, im&shy;<lb/>
po&longs;sibile e&longs;&longs;e addere ip&longs;i E pondus adeo minimum, quin adhuc &longs;i <lb/>
ex C &longs;u&longs;pendantur, pondus E &longs;emper deor&longs;um ver&longs;us G moueatur.  </s>
             
<s id="id.2.1.11.1.1.2.0"> <lb/>
quod nos fieri po&longs;&longs;e &longs;uppo&longs;uimus, at que fieri po&longs;&longs;e credebamus.  </s>
             
<s id="id.2.1.11.1.1.3.0"> ex&shy;<lb/>
ce&longs;&longs;um enim ponderis D &longs;upra pondus E, cum quantitatis ratio&shy;<lb/>
nem habeat, non &longs;olum minimum e&longs;&longs;e, verum in infinitum diuidi <lb/>
po&longs;&longs;e immaginabamur, quod quidem ip&longs;i, non &longs;olum minimum, 
<pb n="7"/>
&longs;ed ne minimum quidem e&longs;&longs;e, cum reperiri non po&longs;sit, hoc mo&shy;<lb/>
do demon&longs;trare nituntur. <arrow.to.target n="fig12"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.11.2.0.0.0" type="main">        
<s id="id.2.1.11.2.1.1.0"> Exponantur eadem.  </s>
             
<s id="id.2.1.11.2.1.2.0"> <lb/>
<expan abbr="&agrave;">a</expan> <expan abbr="puncti&longs;qu&egrave;">puncti&longs;que</expan> DE hori&shy;<lb/>
zonti <expan abbr="perp&etilde;diculares">perpendiculares</expan> du <lb/>
<expan abbr="c&atilde;tur">cantur</expan> DHEK, atq; alius <lb/>
&longs;it circulus LDM, cu&shy;<lb/>
ius <expan abbr="centr&utilde;">centrum</expan> N, qui FDG <lb/>
in puncto D contingat, <lb/>
ip&longs;iq; FDG &longs;it &aelig;qualis: <lb/>
erit NC recta linea.  </s>
             
<s id="id.2.1.11.2.1.3.0"> &amp; <arrow.to.target n="note16"></arrow.to.target><lb/>
quoniam angulus KEC <lb/>
angulo HDN e&longs;t &aelig;qua <arrow.to.target n="note17"></arrow.to.target><lb/>
lis, angulusq; CEG an&shy;<lb/>
gulo NDM e&longs;t etiam <lb/>
&aelig;qualis; cum <expan abbr="&agrave;">a</expan> &longs;emidiametris, &aelig;qualibusq; circumferentiis conti&shy;<lb/>
neatur; erit reliquus <expan abbr="mixtu&longs;qu&egrave;">mixtu&longs;que</expan> angulus KEG reliquo <expan abbr="mixtoqu&egrave;">mixtoque</expan> <lb/>
HDM &aelig;qualis.  </s>
             
<s id="id.2.1.11.2.1.4.0"> &amp; quia &longs;upponunt, <expan abbr="qu&ograve;">quo</expan> minor e&longs;t angulus linea <lb/>
horizonti perpendiculari, &amp; circumferentia contentus, <expan abbr="e&ograve;">eo</expan> pondus <lb/>
in eo &longs;itu grauius e&longs;&longs;e.  </s>
             
<s id="id.2.1.11.2.1.5.0"> vt <expan abbr="qu&ograve;">quo</expan> minor e&longs;t angulus HD, &amp; circumfe<lb/>
rentia DG contentus angulo KEG, hoc e&longs;t angulo HDM; ita &longs;e<lb/>
cundum hanc proportionem pondus in D grauius e&longs;&longs;e pondere in <lb/>
E.  </s>
     
<s id="id.2.1.11.2.1.5.0.a"> Proportio autem anguli MDH ad angulum HDG minor e&longs;t <lb/>
qualibet proportione, qu&aelig; &longs;it inter maiorem, &amp; minorem quanti<lb/>
tatem: ergo proportio ponderum DE omnium proportionum mi<lb/>
nima erit.  </s>
             
<s id="id.2.1.11.2.1.6.0"> immo neq; erit <expan abbr="fer&egrave;">fere</expan> proportio, cum &longs;it omnium pro <lb/>
portionum minima.  </s>
             
<s id="id.2.1.11.2.1.7.0"> <expan abbr="qu&ograve;d">quod</expan> autem proportio MDH ad HDG &longs;it <lb/>
omnium minima, ex hac nece&longs;sitate o&longs;tendunt; quia MDH exce<lb/>
dit HDG angulo curuilineo MDG, qui quidem angulus omnium <lb/>
angulorum rectilineorum minimus exi&longs;tit: ergo cum non po&longs;sit da <lb/>
ri angulus minor MDG, erit proportio MDH ad HDG <expan abbr="omni&utilde;">omnium</expan> <lb/>
proportionum minima.  </s>
             
<s id="id.2.1.11.2.1.8.0"> qu&aelig; ratio inutilis valde videtur e&longs;&longs;e; quia <lb/>
quamquam angulus MDG &longs;it omnibus rectilineis angulis minor, <lb/>
non idcirco &longs;equitur, <expan abbr="ab&longs;olut&egrave;">ab&longs;olute</expan>, &longs;impliciterq; omnium e&longs;&longs;e <expan abbr="angulor&utilde;">angulorum</expan> <lb/>
minimum: nam ducatur <expan abbr="&agrave;">a</expan> puncto D linea DO ip&longs;i NC perpendicu<lb/>
laris, h&aelig;c vtra&longs;q; tanget circumferentias LDM FDG in puncto <arrow.to.target n="note18"></arrow.to.target>
<pb/>
D. quia <expan abbr="ver&ograve;">vero</expan> circumfe<lb/>
renti&aelig; &longs;unt &aelig;quales, erit <lb/>
angulus MDO mixtus <lb/>
angulo ODG mixto <lb/>
&aelig;qualis; alter ergo an<lb/>
gulus, vt ODG minor <lb/>
erit MDG, hoc e&longs;t mi <lb/>
nor minimo.  </s>
             
<s id="id.2.1.11.2.1.9.0"> angulus <lb/>
deinde OGH minor <lb/>
erit angulo MDH; qua <lb/>
re ODH ad angulum <lb/>
<arrow.to.target n="note19"></arrow.to.target> HDG minorem habe<lb/>
bit <expan abbr="proportion&etilde;">proportionem</expan>, <expan abbr="qu&agrave;m">quam</expan> <lb/>
<arrow.to.target n="fig13"></arrow.to.target><lb/>
MDH ad eundem HDG. dabitur ergo <expan abbr="quoqu&egrave;">quoque</expan> proportio mi&shy;<lb/>
nor minima, quam in infinitum adhuc minorem ita o&longs;tende&shy;<lb/>
mus.  </s>
             
<s id="id.2.1.11.2.1.10.0"> De&longs;cribatur circulus DR, cuius centrum E, &amp; &longs;emidiame&shy;<lb/>
<arrow.to.target n="note20"></arrow.to.target> ter ED. continget circumferentia DR circumferentiam DG in <lb/>
<arrow.to.target n="note21"></arrow.to.target> puncto D, <expan abbr="lineamqu&egrave;">lineamque</expan> DO in puncto D; quare minor erit angu&shy;<lb/>
lus RDG angulo ODG. &longs;imiliter &amp; angulus RDH angulo <lb/>
ODH.  </s>
     
<s id="id.2.1.11.2.1.10.0.a"> minorem igitur proportionem habebit RDH ad HDG, <lb/>
<expan abbr="qu&agrave;m">quam</expan> ODH ad HDG.  </s>
     
<s id="id.2.1.11.2.1.10.0.b"> Accipiatur deinde inter EC vtcun&shy;<lb/>
que punctum P, ex quo in di&longs;tantia PD alia de&longs;cribatur circum&shy;<lb/>
ferentia DQ, qu&aelig; circumferentiam DR, <expan abbr="circumferentiamqu&egrave;">circumferentiamque</expan> <lb/>
DG in puncto D continget; &amp; angulus QDH minor erit <lb/>
angulo RDH: ergo QDH ad HDG minorem habebit propor<lb/>
tionem, <expan abbr="qu&agrave;m">quam</expan> RDH ad HDG. <expan abbr="eodemqu&egrave;">eodemque</expan> pror&longs;us modo, &longs;i <lb/>
inter PC aliud accipiatur punctum, &amp; inter hoc &amp;C aliud, &amp; &longs;ic <lb/>
deinceps, infinit&aelig; de&longs;cribentur circumferenti&aelig; inter DO, &amp; cir<lb/>
cumferentiam DG; ex quibus proportionem in infinitum &longs;emper <lb/>
minorem inueniemus.  </s>
             
<s id="id.2.1.11.2.1.11.0"> atque ideo proportionem ponderis in D <lb/>
ad pondus in E non adeo minorem e&longs;&longs;e &longs;equitur, quin ad infini <lb/>
tum ip&longs;a &longs;emper minorem reperiri po&longs;sit.  </s>
             
<s id="id.2.1.11.2.1.12.0"> &amp; quia angulus MDG <lb/>
in infinitum diuidi pote&longs;t; exce&longs;&longs;us quoque grauitatis D &longs;upra E <lb/>
diuidi ad infinitum poterit.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig12" place="text">       </figure>       
<p id="id.2.1.11.2.2.1.0" type="caption">        
<s id="id.2.1.11.2.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig13" place="text">       </figure>       
<p id="id.2.1.11.2.2.3.0" type="caption">        
<s id="id.2.1.11.2.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.12.1.0.0.0" type="margin">        
<s id="id.2.1.12.1.1.1.0"> <margin.target id="note15"></margin.target><emph type="italics"/>Tartalea &longs;exta propo&longs;itione octaui libri.<emph.end type="italics"/> </s>
             
<s id="id.2.1.12.1.1.2.0"> <margin.target id="note16"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 12. <emph type="italics"/>tertii.<emph.end type="italics"/> </s>
             
<s id="id.2.1.12.1.1.3.0"> <margin.target id="note17"></margin.target>29. <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.12.1.1.4.0"> <margin.target id="note18"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 18. <emph type="italics"/>Ter tii.<emph.end type="italics"/> </s>
             
<s id="id.2.1.12.1.1.5.0"> <margin.target id="note19"></margin.target>8. <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.12.1.1.6.0"> <margin.target id="note20"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 11. <emph type="italics"/>tertit.<emph.end type="italics"/> </s>
             
<s id="id.2.1.12.1.1.7.0"> <margin.target id="note21"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 18. <emph type="italics"/>tertii.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.13.1.0.0.0" type="main">        
<pb n="8"/>
       
<s id="id.2.1.13.1.2.1.0"> Sed neque pr&aelig;tereundum <lb/>
e&longs;t, ip&longs;os in demon&longs;tratio&shy;<lb/>
ne angulum KEG maiorem <lb/>
e&longs;&longs;e angulo HDG, tanquam <lb/>
notum accepi&longs;&longs;e.  </s>
             
<s id="id.2.1.13.1.2.2.0"> quod e&longs;t <lb/>
quidem verum, &longs;i DHEK <lb/>
inter &longs;e &longs;e &longs;int &aelig;quidi&longs;tan&shy;<lb/>
tes.  </s>
             
<s id="id.2.1.13.1.2.3.0"> Quoniam autem (vt <lb/>
ip&longs;i quoque &longs;upponunt) li&shy;<lb/>
ne&aelig; DHEK in centrum <lb/>
mundi conueniunt; line&aelig; <lb/>
DHEK &aelig;quidi&longs;tantes nun<lb/>
quam erunt, &amp; angulus KEG <lb/>
angulo HDG non &longs;olum <lb/>
maior erit, &longs;ed minor.  </s>
             
<s id="id.2.1.13.1.2.4.0"> vt <lb/>
exempli gratia, producatur <lb/>
FG v&longs;que ad centrum mun<lb/>
di, quod &longs;it S; <expan abbr="connectan&shy;turqu&eacute;">connectan&shy;<lb/>
turque</expan> DSES. o&longs;tenden&shy;<lb/>
dum e&longs;t angulum SEG mi<lb/>
norem e&longs;&longs;e angulo SDG.  </s>
     
<s id="id.2.1.13.1.2.4.0.a"> du<lb/>
<arrow.to.target n="fig14"></arrow.to.target><lb/>
catur <expan abbr="&agrave;">a</expan> puncto E linea ET circulum DGEF contingens, ab eo <lb/>
<expan abbr="demqu&eacute;">demque</expan> puncto ip&longs;i DS &aelig;quidi&longs;tans ducatur EV.  </s>
     
<s id="id.2.1.13.1.2.4.0.b"> Quoniam igi<lb/>
tur EVDS inter &longs;e &longs;e &longs;unt &aelig;quidi&longs;tantes: &longs;imiliter ETDO &aelig;qui <lb/>
di&longs;tantes: erit angulus VET angulo SDO &aelig;qualis.  </s>
             
<s id="id.2.1.13.1.2.5.0"> &amp; angulus <lb/>
TEG angulo ODM e&longs;t &aelig;qualis; cum <expan abbr="&agrave;">a</expan> lineis contingentibus, <lb/>
<expan abbr="circumferentii&longs;qu&eacute;">circumferentii&longs;que</expan> &aelig;qualibus contineatur: totus ergo angulus <lb/>
VEG angulo SDM &aelig;qualis erit.  </s>
             
<s id="id.2.1.13.1.2.6.0"> Auferatur ab angulo SDM <lb/>
angulus curuilineus MDG; ab angulo autem VEG angulus au&shy;<lb/>
feratur VES; &amp; angulus VES rectilineus maior e&longs;t curuilineo <lb/>
MDG; erit reliquus angulus SEG minor angulo SDG.  </s>
     
<s id="id.2.1.13.1.2.6.0.a"> <lb/>
Quare ex ip&longs;orum &longs;uppo&longs;itionibus non &longs;olum pondus in D gra&shy;<lb/>
uius erit pondere in E; <expan abbr="ver&ugrave;m">verum</expan> <expan abbr="&egrave;">e</expan> conuer&longs;o, pondus in E ip&longs;o D <lb/>
grauius exi&longs;tet.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig14" place="text">       </figure>       
<p id="id.2.1.13.1.3.1.0" type="caption">        
<s id="id.2.1.13.1.3.1.0.capt"> YYY </s>
    </p>       
<pb/>
       
<p id="id.2.1.13.3.0.0.0" type="main">        
<s id="id.2.1.13.3.1.1.0"> Rationes tamen af<lb/>
ferunt, quibus demon<lb/>
&longs;trare nituntur, libram <lb/>
DE in AB horizon&shy;<lb/>
ti &aelig;quidi&longs;tantem ex <lb/>
nece&longs;sitate redire.  </s>
             
<s id="id.2.1.13.3.1.2.0"> <expan abbr="Pri&shy;m&ugrave;m">Pri&shy;<lb/>
mum</expan> quidem o&longs;ten&shy;<lb/>
dunt, idem pondus <lb/>
grauius e&longs;&longs;e in A, <lb/>
<expan abbr="qu&agrave;min">quamin</expan> alio &longs;itu, quem <lb/>
&aelig;qualitatis &longs;itum no&shy;<lb/>
minant, cum linea <lb/>
AB &longs;it horizonti &aelig;&shy;<lb/>
<arrow.to.target n="fig15"></arrow.to.target><lb/>
quidi&longs;tans.  </s>
             
<s id="id.2.1.13.3.1.3.0"> deinde <expan abbr="qu&ograve;">quo</expan> propius e&longs;t ip&longs;i A, quouis alio remotiori <lb/>
grauius e&longs;&longs;e.  </s>
             
<s id="id.2.1.13.3.1.4.0"> Vt pondus in A grauius e&longs;&longs;e, <expan abbr="qu&agrave;m">quam</expan> in D; &amp; in D, <lb/>
<expan abbr="qu&agrave;m">quam</expan> in L. &longs;imiliter in A grauius, quam in N; &amp; in N grauius, <lb/>
<expan abbr="qu&agrave;m">quam</expan> in M.  </s>
     
<s id="id.2.1.13.3.1.4.0.a"> Vnum <expan abbr="tant&ugrave;m">tantum</expan> con&longs;iderando pondus in altero libr&aelig; <lb/>
<arrow.to.target n="note22"></arrow.to.target> brachio &longs;ur&longs;um deor&longs;umq; moto.  </s>
             
<s id="id.2.1.13.3.1.5.0"> Quia (inquiunt) po&longs;itat rutina <lb/>
in CF, pondus in A longius e&longs;t <expan abbr="&agrave;">a</expan> trutina, <expan abbr="qu&agrave;m">quam</expan> in D: &amp; in D <lb/>
longius, <expan abbr="qu&agrave;m">quam</expan> in L. ductis enim DO LP ip&longs;i CF perpendicula&shy;<lb/>
<arrow.to.target n="note23"></arrow.to.target> ribus, li&lt;*&gt;ea AC maior e&longs;t, <expan abbr="qu&agrave;m">quam</expan> DO, &amp; DO ip&longs;a LP. quod <lb/>
<arrow.to.target n="note24"></arrow.to.target> idem euenit in punctis NM.  </s>
     
<s id="id.2.1.13.3.1.5.0.a"> deinde ex quo loco (aiunt) pon<lb/>
dus velocius mouetur, ibi grauius e&longs;t; velocius autem ex A, <expan abbr="qu&agrave;m">quam</expan> <lb/>
ab alio &longs;itu mouetur; ergo in A grauius e&longs;t.  </s>
             
<s id="id.2.1.13.3.1.6.0"> &longs;imili modo, <expan abbr="qu&ograve;">quo</expan> <lb/>
propius e&longs;t ip&longs;i A, velocius quoque mouetur; ergo in D gra&shy;<lb/>
<arrow.to.target n="note25"></arrow.to.target> uius erit, <expan abbr="qu&agrave;m">quam</expan> in L.  </s>
     
<s id="id.2.1.13.3.1.6.0.a"> Altera deinde cau&longs;a, quam ex rectiori, &amp; obli <lb/>
<arrow.to.target n="note26"></arrow.to.target> quiori motu deducunt, e&longs;t; <expan abbr="qu&ograve;">quo</expan> pondus in arcubus &aelig;qualibus re&shy;<lb/>
ctius de&longs;cendit, grauius e&longs;&longs;e videtur; cum pondus liberum, atq; <lb/>
<arrow.to.target n="note27"></arrow.to.target> &longs;olutum <expan abbr="&longs;uapt&egrave;">&longs;uapte</expan> natura <expan abbr="rect&egrave;">recte</expan> moueatur; &longs;ed in A rectius de&longs;cen<lb/>
dit; ergo in A grauius erit.  </s>
             
<s id="id.2.1.13.3.1.7.0"> hocq; o&longs;tendunt accipiendo arcum <lb/>
AN arcui LD &aelig;qualem; <expan abbr="&agrave;">a</expan> puncti&longs;q; NL line&aelig; FG (quam <lb/>
etiam directionis vocant) &aelig;quidi&longs;tantes ducantur NRLQ, qu&aelig; <lb/>
lineas AB DO &longs;ecent in QR; &amp; <expan abbr="&agrave;">a</expan> puncto N ip&longs;i FG perpen<lb/>
dicularis ducatur NT. <expan abbr="rect&egrave;q">recteq</expan>; demon&longs;trant LQ ip&longs;i PO &aelig;qua<lb/>
lem e&longs;&longs;e, &amp; NR ip&longs;i CT; lineamq; NR ip&longs;a LQ maiorem e&longs;&longs;e.  </s>
             
<s id="id.2.1.13.3.1.8.0"> <lb/>
Quoniam autem de&longs;cen&longs;u; ponderis ex A v&longs;q; ad N per circum&shy;
<pb n="9"/>
ferentiam AN maiorem portionem line&aelig; FG pertran&longs;it (quod <lb/>
ip&longs;i vocant capere de directo) <expan abbr="qu&agrave;m">quam</expan> de&longs;cen&longs;us ex L in D per cir<lb/>
cumferentiam LD; <expan abbr="c&ugrave;m">cum</expan> de&longs;cen&longs;us AN lineam CT pertran&longs;eat, <lb/>
de&longs;cen&longs;us <expan abbr="ver&ograve;">vero</expan> LD lineam PO; &amp; CT maior e&longs;t PO; rectior erit <lb/>
de&longs;cen&longs;us AN, <expan abbr="qu&aacute;m">quam</expan> de&longs;cen&longs;us LD.  </s>
     
<s id="id.2.1.13.3.1.8.0.a"> grauius ergo erit pondus <lb/>
in A, <expan abbr="qu&agrave;m">quam</expan> in L, &amp; in quouis alio &longs;itu.  </s>
             
<s id="id.2.1.13.3.1.9.0"> eodemq; pror&longs;us <lb/>
modo o&longs;tendunt, <expan abbr="qu&ograve;">quo</expan> propius e&longs;t ip&longs;i A, grauius e&longs;&longs;e.  </s>
             
<s id="id.2.1.13.3.1.10.0"> <lb/>
Vt &longs;int circumferenti&aelig; LD DA inter &longs;e &longs;e &aelig;quales, &amp; <expan abbr="&agrave;">a</expan> puncto <lb/>
D ip&longs;i AB perpendicularis ducatur DR; erit DR ip&longs;i CO &aelig;qua <arrow.to.target n="note28"></arrow.to.target><lb/>
lis.  </s>
             
<s id="id.2.1.13.3.1.11.0"> lineam deinde DR ip&longs;a LQ maiorem e&longs;&longs;e demon&longs;trant.  </s>
             
<s id="id.2.1.13.3.1.12.0"> di&shy;<lb/>
cuntq; de&longs;cen&longs;um DA magis capere de directo de&longs;cen&longs;u LD, ma<lb/>
ior enim e&longs;t linea CO, <expan abbr="qu&agrave;m">quam</expan> OP; quare pondus grauius erit <lb/>
in D, <expan abbr="qu&agrave;m">quam</expan> in L. quod ip&longs;um euenit in punctis NM.  </s>
     
<s id="id.2.1.13.3.1.12.0.a"> Suppo&shy;<lb/>
&longs;itionem itaq;, qua libram DE in AB redire demon&longs;trant, vt <arrow.to.target n="note29"></arrow.to.target><lb/>
notam, manife&longs;tamq; proferunt.  </s>
             
<s id="id.2.1.13.3.1.13.0"> <expan abbr="Nemp&egrave;">Nempe</expan> <expan abbr="Secund&ugrave;m">Secundum</expan> &longs;itum pon<lb/>
dus grauius e&longs;&longs;e, quanto in eodem &longs;itu minus obliquus e&longs;t de&longs;cen<lb/>
&longs;us.  </s>
             
<s id="id.2.1.13.3.1.14.0"> huiu&longs;q; reditus cau&longs;am eam e&longs;&longs;e dicunt; Quoniam &longs;cilicet <arrow.to.target n="note30"></arrow.to.target><lb/>
de&longs;cen&longs;us ponderis in D rectior e&longs;t de&longs;cen&longs;u ponderis in E, <expan abbr="c&ugrave;m">cum</expan> <lb/>
minus capiat de directo pondus in E de&longs;cendendo, <expan abbr="qu&agrave;m">quam</expan> pon<arrow.to.target n="note31"></arrow.to.target><lb/>
dus in D &longs;im liter de&longs;cendendo.  </s>
             
<s id="id.2.1.13.3.1.15.0"> Vt &longs;i arcus EV &longs;it ip&longs;i DA <lb/>
&aelig;qualis, ducanturq; VH ET ip&longs;i FG perpendiculares; maior <lb/>
erit DR, <expan abbr="qu&agrave;m">quam</expan> TH. quare per &longs;uppo&longs;itionem pondus in D ra<lb/>
tione &longs;itus grauius erit pondere in E.  </s>
     
<s id="id.2.1.13.3.1.15.0.a"> pondus ergo in D, <expan abbr="c&ugrave;m">cum</expan> &longs;it <lb/>
grauius, deor&longs;um mouebitur; pondus <expan abbr="ver&ograve;">vero</expan> in E &longs;ur&longs;um, donec li <lb/>
bra DE in AB redeat. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig15" place="text">       </figure>       
<p id="id.2.1.13.3.2.1.0" type="caption">        
<s id="id.2.1.13.3.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.14.1.0.0.0" type="margin">        
<s id="id.2.1.14.1.1.1.0"> <margin.target id="note22"></margin.target><emph type="italics"/>Cardanus primo de &longs;ubtilitate.<emph.end type="italics"/> </s>
             
<s id="id.2.1.14.1.1.2.0"> <margin.target id="note23"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 15. <emph type="italics"/>tertii.<emph.end type="italics"/> </s>
             
<s id="id.2.1.14.1.1.3.0"> <margin.target id="note24"></margin.target><emph type="italics"/>Cardanus.<emph.end type="italics"/> </s>
             
<s id="id.2.1.14.1.1.4.0"> <margin.target id="note25"></margin.target><emph type="italics"/>Cardanus.<emph.end type="italics"/> </s>
             
<s id="id.2.1.14.1.1.5.0"> <margin.target id="note26"></margin.target><emph type="italics"/>Iordanus propo&longs;itio ne<emph.end type="italics"/> 4. </s>
             
<s id="id.2.1.14.1.1.6.0"> <margin.target id="note27"></margin.target><emph type="italics"/>Tartalea propo&longs;itione<emph.end type="italics"/> 5. </s>
             
<s id="id.2.1.14.1.1.7.0"> <margin.target id="note28"></margin.target>34 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.14.1.1.8.0"> <margin.target id="note29"></margin.target><emph type="italics"/>Iordanus &longs;uppo&longs;itione<emph.end type="italics"/> 4. </s>
             
<s id="id.2.1.14.1.1.9.0"> <margin.target id="note30"></margin.target><emph type="italics"/>Iordanus propo&longs;itio ne<emph.end type="italics"/> 3. </s>
             
<s id="id.2.1.14.1.1.10.0"> <margin.target id="note31"></margin.target><emph type="italics"/>Tartalea propo&longs;itio ne<emph.end type="italics"/> 5. </s>
    </p>       
<p id="id.2.1.15.1.0.0.0" type="main">        
<s id="id.2.1.15.1.1.1.0"> Altera huius quoq; reditus ratio e&longs;t, <expan abbr="c&ugrave;m">cum</expan> trutina &longs;upra libram <arrow.to.target n="note32"></arrow.to.target><lb/>
e&longs;t in CF; linea CG e&longs;t meta.  </s>
             
<s id="id.2.1.15.1.1.2.0"> &amp; quoniam angulus GCD ma<lb/>
ior e&longs;t angulo GCE, &amp; maior <expan abbr="&agrave;">a</expan> meta angulus grauius reddit <lb/>
pondus; trutina igitur &longs;uperius exi&longs;tente, grauius erit pondus in <lb/>
D, <expan abbr="qu&agrave;m">quam</expan> in E. idcirco D in A, &amp; E in B redibit. </s>
    </p>       
<p id="id.2.1.16.1.0.0.0" type="margin">        
<s id="id.2.1.16.1.1.1.0"> <margin.target id="note32"></margin.target><emph type="italics"/>Cardanus.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.17.1.0.0.0" type="main">        
<s id="id.2.1.17.1.1.1.0"> His itaq; rationibus conantur o&longs;tendere libram DE in AB re<lb/>
dire; qu&aelig; meo quidem iuditio facile &longs;olui po&longs;&longs;unt.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.17.3.0.0.0" type="main">        
<s id="id.2.1.17.3.1.1.0"> <expan abbr="Prim&ugrave;m">Primum</expan> itaq; quan<lb/>
tum attinet ad ratio&shy;<lb/>
nes pondus in A gra<lb/>
uius e&longs;&longs;e, <expan abbr="qu&agrave;m">quam</expan> in a&shy;<lb/>
lio &longs;itu o&longs;tendentes, <lb/>
quas ex longiori, &amp; <lb/>
propinquiori <expan abbr="di&longs;t&atilde;tia">di&longs;tantia</expan> <expan abbr="&agrave;">a</expan> <lb/>
linea FG, &amp; ex velo&shy;<lb/>
ciori, &amp; rectiori mo <lb/>
tu <expan abbr="&agrave;">a</expan> puncto A dedu&shy;<lb/>
cunt; <expan abbr="prim&ugrave;m">primum</expan> quidem <lb/>
non demon&longs;trant, cur <lb/>
pondus ex A velocius <lb/>
<arrow.to.target n="fig16"></arrow.to.target><lb/>
moueatur, <expan abbr="qu&agrave;m">quam</expan> ex alio &longs;itu.  </s>
             
<s id="id.2.1.17.3.1.2.0"> nec quia CA e&longs;t DO maior, <lb/>
&amp; DO ip&longs;a LP, propterea &longs;equitur tanquam ex vera cau&longs;a, pon<lb/>
dus in A grauius e&longs;&longs;e, <expan abbr="qu&agrave;m">quam</expan> in D; &amp; in D, <expan abbr="qu&agrave;m">quam</expan> in L.  </s>
     
<s id="id.2.1.17.3.1.2.0.a"> neq; <lb/>
enim intellectus quie&longs;cit, ni&longs;i alia huius o&longs;tendatur cau&longs;a; <expan abbr="c&ugrave;m">cum</expan> po<lb/>
tius &longs;ignum, <expan abbr="qu&agrave;m">quam</expan> vera cau&longs;a e&longs;&longs;e videatur.  </s>
             
<s id="id.2.1.17.3.1.3.0"> id ip&longs;um quoq; al&shy;<lb/>
teri rationi contintingit, quam ex rectiori &amp; obliquiori motu de&shy;<lb/>
ducunt.  </s>
             
<s id="id.2.1.17.3.1.4.0"> Pr&aelig;terea qu&aelig;cunq; ex velociori, &amp; rectiori motu per&shy;<lb/>
&longs;uadent pondus in A grauius e&longs;&longs;e, <expan abbr="qu&agrave;m">quam</expan> in D; non ideo de&shy;<lb/>
mon&longs;trant pondus in A, quatenus e&longs;t in A, grauius e&longs;&longs;e pon<lb/>
dere in D, quatenus e&longs;t in D; &longs;ed quatenus <expan abbr="&agrave;">a</expan> punctis DA rece<lb/>
dit.  </s>
             
<s id="id.2.1.17.3.1.5.0"> Idcirco <expan abbr="antequ&agrave;m">antequam</expan> vlterius progrediar, o&longs;tendam <expan abbr="prim&ugrave;m">primum</expan> <lb/>
pondus, <expan abbr="qu&ograve;">quo</expan> propius e&longs;t ip&longs;is FG, minus grauitare; tum qua&shy;<lb/>
tenus in eo &longs;itu, in quo reperitur, manet: tum quatenus ab eo <lb/>
recedit.  </s>
             
<s id="id.2.1.17.3.1.6.0"> &longs;imulq; fal&longs;um e&longs;&longs;e, pondus in A grauius e&longs;&longs;e, <expan abbr="qu&agrave;m">quam</expan> in <lb/>
alio &longs;itu.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig16" place="text">       </figure>       
<p id="id.2.1.17.3.2.1.0" type="caption">        
<s id="id.2.1.17.3.2.1.0.capt"> YYY </s>
    </p>       
<pb n="10"/>
       
<p id="id.2.1.17.5.0.0.0" type="main">        
<s id="id.2.1.17.5.1.1.0"> Producatur FG v&longs;q; ad mundi cen<lb/>
trum, quod &longs;it S. &amp; <expan abbr="&agrave;">a</expan> puncto S circu<lb/>
lum AFBG contingens ducatur.  </s>
             
<s id="id.2.1.17.5.1.2.0"> neq; <lb/>
enim linea <expan abbr="&agrave;">a</expan> puncto S circulum con&shy;<lb/>
tingere pote&longs;t in A; nam ducta AS <lb/>
triangulum ACS duos haberet angu<lb/>
los rectos, <expan abbr="nemp&egrave;">nempe</expan> SAC ACS, quod <arrow.to.target n="note33"></arrow.to.target><lb/>
e&longs;t impo&longs;sibile.  </s>
             
<s id="id.2.1.17.5.1.3.0"> neq; &longs;upra punctum A <lb/>
in circumferentia AF continget; cir<lb/>
culum enim &longs;ecatet.  </s>
             
<s id="id.2.1.17.5.1.4.0"> tanget igitur in&shy;<lb/>
fra, &longs;itq; SO. connectantur deinde SD <lb/>
SL, qu&aelig; circumferentiam AOG in <lb/>
punctis KH &longs;ecent. &amp; Ck CH con <lb/>
iungantur. </s>
             
<s id="id.2.1.17.5.1.5.0"> [&amp; Ck CH con<lb/>
iungantur.]  </s>
             
<s id="id.2.1.17.5.1.6.0"> Et quoniam pondus, quanto <lb/>
propius e&longs;t ip&longs;i F, magis quoque inni&shy;<lb/>
titur centro; vt pondus in D magis ver&shy;<lb/>
&longs;ionis puncto C innititur tanquam <lb/>
centro; hoc e&longs;t in D magis &longs;upra li&shy;<lb/>
neam CD grauitat, <expan abbr="qu&agrave;m">quam</expan> &longs;i e&longs;&longs;et in A <lb/>
&longs;upra lineam CA; &amp; adhuc magis in <lb/>
L &longs;upra lineam CL; Nam <expan abbr="c&ugrave;m">cum</expan> tres <lb/>
anguli cuiu&longs;cunq; trianguli duobus re&shy;<lb/>
<arrow.to.target n="fig17"></arrow.to.target><lb/>
ctis &longs;int &aelig;quales, &amp; trianguli DCk &aelig;quicruris angulus DCk <lb/>
minor &longs;it angulo LCH &aelig;quicruris trianguli LCH: erunt reli&shy;<lb/>
qui ad ba&longs;im &longs;cilicet CDk CkD &longs;imul &longs;umpti reliquis CLH <lb/>
CHL maiores.  </s>
             
<s id="id.2.1.17.5.1.7.0"> &amp; horum dimidii; hoc e&longs;t angulus CDS angu<lb/>
lo CLS maior erit.  </s>
             
<s id="id.2.1.17.5.1.8.0"> <expan abbr="c&ugrave;m">cum</expan> itaq; CLS &longs;it minor, linea CL ma<lb/>
gis adh&aelig;rebit motui naturali ponderis in L pror&longs;us &longs;oluti.  </s>
             
<s id="id.2.1.17.5.1.9.0"> hoc <lb/>
e&longs;t line&aelig; LS, <expan abbr="qu&agrave;m">quam</expan> CD motui DS.  </s>
     
<s id="id.2.1.17.5.1.9.0.a"> pondus enim in L libe&shy;<lb/>
berum, atq; &longs;olutum in centrum mundi per LS moueretur, pon&shy;<lb/>
dusq; in D per DS.  </s>
     
<s id="id.2.1.17.5.1.9.0.b"> quoniam <expan abbr="ver&ograve;">vero</expan> pondus in L totum &longs;uper LS <lb/>
grauitat, in D <expan abbr="ver&ograve;">vero</expan> &longs;uper DS: pondus in L magis &longs;upra lineam <lb/>
CL grauitabit, <expan abbr="qu&agrave;m">quam</expan> exi&longs;tens in D &longs;upra lineam DC. ergo <lb/>
linea CL pondus magis &longs;u&longs;tentabit, <expan abbr="qu&agrave;m">quam</expan> linea CD.  </s>
     
<s id="id.2.1.17.5.1.9.0.c"> <expan abbr="Eodem&shy;qu&eacute;">Eodem&shy;<lb/>
que</expan> modo, <expan abbr="qu&ograve;">quo</expan> pondus propius fuerit ip&longs;i F, magis ob hanc cau&shy;<lb/>
&longs;am <expan abbr="&agrave;">a</expan> linea CL &longs;u&longs;tineri o&longs;tendetur-&longs;emper enim angulus CLS 
<pb/>
minor e&longs;&longs;et.  </s>
             
<s id="id.2.1.17.5.1.10.0"> quod etiam patet; quia &longs;i <lb/>
line&aelig; CL, &amp; LS in vnam coinciderent <lb/>
lineam, quod euenit in FCS; tunc linea <lb/>
CF totum &longs;u&longs;tineret pondus in F, im&shy;<lb/>
mobilemq; redderet: neq; vllam pror&shy;<lb/>
&longs;us grauitatem in circumferentia circu&shy;<lb/>
li haberet.  </s>
             
<s id="id.2.1.17.5.1.11.0"> Idem ergo pondus propter <lb/>
&longs;ituum diuer&longs;itatem grauius, leuiu&longs;q; erit.  </s>
             
<s id="id.2.1.17.5.1.12.0"> <lb/>
non autem quia ratione &longs;itus interdum <lb/>
maiorem re vera acquirat grauitatem, <lb/>
interdum <expan abbr="ver&ograve;">vero</expan> amittat, <expan abbr="c&ugrave;m">cum</expan> eiu&longs;dem &longs;it <lb/>
&longs;emper grauitatis, vbicunque reperiatur; <lb/>
&longs;ed quia magis, <expan abbr="minu&longs;u&egrave;">minu&longs;ue</expan> in circumferen&shy;<lb/>
tia grauitat, vt in D magis &longs;upra circum<lb/>
ferentiam DA grauitat, <expan abbr="qu&agrave;m">quam</expan> in L &longs;upra <lb/>
circumferentiam LD.  </s>
     
<s id="id.2.1.17.5.1.12.0.a"> hoc e&longs;t, &longs;i pon<lb/>
dus <expan abbr="&agrave;">a</expan> circumferentiis, recti&longs;q; lineis &longs;u<lb/>
&longs;tineatur; circumferentia AD magis &longs;u<lb/>
&longs;tinebit pondus in D, <expan abbr="qu&agrave;m">quam</expan> circumfe<lb/>
rentia DL pondere exi&longs;tente in <emph type="italics"/>L.<emph.end type="italics"/> mi <lb/>
nus enim coadiuuat CD, <expan abbr="qu&agrave;m">quam</expan> CL.  </s>
     
<s id="id.2.1.17.5.1.12.0.b"> <lb/>
Pr&aelig;terea quando pondus e&longs;t in L, &longs;i e&longs;&shy;<lb/>
<arrow.to.target n="fig18"></arrow.to.target><lb/>
&longs;et omnino liberum, penitu&longs;q; &longs;olutum, deor&longs;um per LS moueretur; <lb/>
ni&longs;i <expan abbr="&agrave;">a</expan> linea CL prohiberetur, qu&aelig; pondus in L vltra lineam LS per <lb/>
<expan abbr="circumferenti&atilde;">circumferentiam</expan> LD moueri cogit; ip&longs;umq; quodammodo impellit, <lb/>
impellendoq; pondus partim &longs;u&longs;tentabit.  </s>
             
<s id="id.2.1.17.5.1.13.0"> ni&longs;i enim &longs;u&longs;tineret, ip&longs;iq; <lb/>
reniteretur, deor&longs;um per lineam LS moueretur, non autem per <lb/>
circumferentiam LD. &longs;imiliter CD ponderi in D renititur, <expan abbr="c&ugrave;m">cum</expan> <lb/>
illud per circumferentiam DA moueri cogat.  </s>
             
<s id="id.2.1.17.5.1.14.0"> eodemq; modo <lb/>
exi&longs;tente pondere in A, linea CA pondus vltra lineam AS per <lb/>
circumferentiam AO moueri compellet.  </s>
             
<s id="id.2.1.17.5.1.15.0"> e&longs;t enim angulus CAS <lb/>
acutus; <expan abbr="c&ugrave;m">cum</expan> angulus ACS &longs;it rectus.  </s>
             
<s id="id.2.1.17.5.1.16.0"> line&aelig; igitur CA CD ali <lb/>
qua ex parte, non tamen ex &aelig;quo ponderi renituntur.  </s>
             
<s id="id.2.1.17.5.1.17.0"> &amp; quotie&longs; <lb/>
cunque angulus in circumferentia circuli <expan abbr="&agrave;">a</expan> lineis <expan abbr="&agrave;">a</expan> centro <lb/>
mundi S, &amp; centro C prodeuntibus, fuerit acutus; idem eue&shy;<lb/>
nire &longs;imiliter o&longs;tendemus.  </s>
             
<s id="id.2.1.17.5.1.18.0"> Quoniam autem mixtus angulus CLD 
<pb n="11"/>
&aelig;qualis e&longs;t angulo CDA, <expan abbr="c&ugrave;m">cum</expan> <expan abbr="&agrave;">a</expan> &longs;emidiametris, eademq; circumfe<lb/>
rentia contineantur; &amp; angulus C<emph type="italics"/>L<emph.end type="italics"/>S angulo CDS e&longs;t minor; <lb/>
erit reliquus <emph type="italics"/>s<emph.end type="italics"/>LD reliquo SDA maior.  </s>
             
<s id="id.2.1.17.5.1.19.0"> quare circumferentia <lb/>
DA, hoc e&longs;t de&longs;cen&longs;us ponderis in D propior erit motui natu&shy;<lb/>
rali ponderis in D &longs;oluti, line&aelig; &longs;cilicet DS, <expan abbr="qu&agrave;m">quam</expan> circumferen<lb/>
tia LD line&aelig; LS.  </s>
     
<s id="id.2.1.17.5.1.19.0.a"> minus igitur linea CD ponderi in D reniti&shy;<lb/>
tur, <expan abbr="qu&agrave;m">quam</expan> linea CL ponderi in L.  </s>
     
<s id="id.2.1.17.5.1.19.0.b"> linea ideo CD minus &longs;u&longs;tinet, <lb/>
<expan abbr="qu&agrave;m">quam</expan> CL; pondu&longs;q; magis liberum erit in D, <expan abbr="qu&agrave;m">quam</expan> in L: <lb/>
<expan abbr="c&ugrave;m">cum</expan> pondus naturaliter magis per DA moueatur, <expan abbr="qu&agrave;m">quam</expan> per LD. <lb/>
quare grauius erit in D, <expan abbr="qu&agrave;m">quam</expan> in L. &longs;imiliter o&longs;tendemus CA <lb/>
minus &longs;u&longs;tinere, <expan abbr="qu&agrave;m">quam</expan> CD: pondu&longs;q; magis in A, <expan abbr="qu&agrave;m">quam</expan> in Dli <lb/>
berum, grauiu&longs;q, e&longs;&longs;e.  </s>
             
<s id="id.2.1.17.5.1.20.0"> Ex parte deinde inferiori ob ea&longs;dem cau&longs;as, <lb/>
<expan abbr="qu&ograve;">quo</expan> pondus propius fuerit ip&longs;i G, magis detinebitur, vt in H ma<lb/>
gis <expan abbr="&agrave;">a</expan> linea CH, <expan abbr="qu&agrave;m">quam</expan> in K <expan abbr="&agrave;">a</expan> linea CK. nam <expan abbr="c&ugrave;m">cum</expan> angulus CHS <lb/>
maior &longs;it angulo CkS, ad rectitudinem magis appropinquabunt <arrow.to.target n="note34"></arrow.to.target><lb/>
&longs;e &longs;e line&aelig; CHHS, <expan abbr="qu&agrave;m">quam</expan> Ck kS; atq; ob id pondus magis deti&shy;<lb/>
nebitur <expan abbr="&agrave;">a</expan> CH, <expan abbr="qu&agrave;m">quam</expan> <expan abbr="&agrave;">a</expan> Ck &longs;i enim CH HS in vnam conuenirent <lb/>
lineam vt euenit pondere exi&longs;tente in G; tunc linea CG totum &longs;u<lb/>
&longs;tineret' pondus in G, ita vt immobilis per&longs;i&longs;teret.  </s>
             
<s id="id.2.1.17.5.1.21.0"> <expan abbr="qu&ograve;">quo</expan> igitur <lb/>
minor erit angulus linea CH, &amp; de&longs;cen&longs;u ponderis &longs;oluti, &longs;cilicet <lb/>
HS contentus, <expan abbr="e&ograve;">eo</expan> minus quoq; eiu&longs;modi linea pondus detinebit.  </s>
             
<s id="id.2.1.17.5.1.22.0"> <lb/>
&amp; vbiminus detinebitur, ibi magis liberum, grauiu&longs;q; exi&longs;tet.  </s>
             
<s id="id.2.1.17.5.1.23.0"> <lb/>
Pr&aelig;terea &longs;i pondus in k liberum e&longs;&longs;et, atq; &longs;olutum, per lineam <lb/>
k S moueretur; <expan abbr="&agrave;">a</expan> linea <expan abbr="ver&ograve;">vero</expan> Ck prohibetur, qu&aelig; cogit pondus <lb/>
<expan abbr="citr&agrave;">citra</expan> lineam k S per circumferentiam k H moueri.  </s>
             
<s id="id.2.1.17.5.1.24.0"> ip&longs;um enim <lb/>
quodammodo retrahit, retrahendoq; &longs;u&longs;tinet. ni&longs;i enim &longs;u&longs;tineret.  </s>
             
<s id="id.2.1.17.5.1.25.0"> [ni&longs;i enim &longs;u&longs;tineret.]  </s>
             
<s id="id.2.1.17.5.1.26.0"> <lb/>
pondus deor&longs;um per rectam k S moueretur, non autem per cir<lb/>
cumferentiam k H. &longs;imiliter CH pondus retinet, <expan abbr="c&ugrave;m">cum</expan> per circum<lb/>
<expan abbr="ferenti&atilde;">ferentiam</expan> HG moueri compellat.  </s>
             
<s id="id.2.1.17.5.1.27.0"> <expan abbr="Quoni&atilde;">Quoniam</expan> autem angulus CHS ma&shy;<lb/>
ior e&longs;t angulo CKS, <expan abbr="d&etilde;ptis">demptis</expan> &aelig;qualibus angulis CHG CkH; erit <lb/>
reliquus SHG reliquo SKH maior.  </s>
             
<s id="id.2.1.17.5.1.28.0"> circumferentia igitur k H, hoc <lb/>
e&longs;t de&longs;cen&longs;us ponderis in k, propior erit motui naturali ponderis in <lb/>
k &longs;oluti, hoc e&longs;t line&aelig; k S, <expan abbr="qu&agrave;m">quam</expan> circumferentia HG line&aelig; HS. mi <lb/>
nus idcirco detinet linea Ck, <expan abbr="qu&agrave;m">quam</expan> CH: <expan abbr="c&ugrave;m">cum</expan> pondus naturali&shy;<lb/>
ter magis moueatur per k H, <expan abbr="qu&agrave;m">quam</expan> per HG.  </s>
     
<s id="id.2.1.17.5.1.28.0.a"> &longs;imili ratione o&longs;ten&shy;<lb/>
detur, <expan abbr="qu&ograve;">quo</expan> minor erit angulus SkH, lineam Ck minus &longs;u&longs;tinere.  </s>
             
<s id="id.2.1.17.5.1.29.0"> 
<pb/>
exi&longs;tente igitur pondere in O, quia angu<lb/>
lus SOC non &longs;olum minor e&longs;t angulo <lb/>
CKS, <expan abbr="ver&ugrave;m">verum</expan> etiam omnium angulorum <lb/>
<expan abbr="&agrave;">a</expan> punctis CS prodeuntium, verticemq; <lb/>
in circumferuntia OkG habentium mi&shy;<lb/>
nimus; erit anglus SOK, &amp; angulo SkH, <lb/>
&amp; eiu&longs;modi omnium minimus.  </s>
             
<s id="id.2.1.17.5.1.30.0"> ergo de&shy;<lb/>
&longs;cen&longs;us ponderis in O propior erit motui <lb/>
naturali ip&longs;ius in O &longs;oluti, <expan abbr="qu&agrave;m">quam</expan> in alio <lb/>
&longs;itu circumferenti&aelig; OkG. lineaq; CO <lb/>
minus pondus &longs;u&longs;tinebit, <expan abbr="qu&agrave;m">quam</expan> &longs;i pon&shy;<lb/>
dusin quouis alio fuerit &longs;itu eiu&longs;dem cir<lb/>
cumferenti&aelig; OG.  </s>
     
<s id="id.2.1.17.5.1.30.0.a"> &longs;imiliter quoniam con<lb/>
tingenti&aelig; angulus SOk, &amp; angulo SDA, <lb/>
&amp; SAO, ac quibu&longs;cunq; &longs;imilibus e&longs;t mi <lb/>
nor; erit de&longs;cen&longs;us ponderis in O motui <lb/>
naturali ip&longs;ius ponderis in O &longs;oluti pro&shy;<lb/>
pior, <expan abbr="qu&agrave;m">quam</expan> in alio &longs;itu circumferenti&aelig; <lb/>
ODF.  </s>
     
<s id="id.2.1.17.5.1.30.0.b"> Pr&aelig;te reaquoniam linea GO pon<lb/>
dus in O dum deor&longs;um mouetur, impelle&shy;<lb/>
re nonpote&longs;t, ita vt vltra lineam OS mo<lb/>
ueatur; <expan abbr="c&ugrave;m">cum</expan> linea OS circulum non &longs;ecet, <lb/>
<arrow.to.target n="fig19"></arrow.to.target><lb/>
&longs;ed contingat; angulu&longs;q; SOC &longs;it rectus, &amp; non acutus; pondus <lb/>
in O nihil &longs;upra lineam CO grauitabit. neq; centro innitetur. quem <lb/>
admodum in quouis alio puncto &longs;upra O accideret. </s>
             
<s id="id.2.1.17.5.1.31.0"> [neq; centro innitetur.]  </s>
             
<s id="id.2.1.17.5.1.32.0"> [quem <lb/>
admodum in quouis alio puncto &longs;upra O accideret.]  </s>
             
<s id="id.2.1.17.5.1.33.0"> erit igitur pon<lb/>
dus in O magis ob has cau&longs;as liberum, atq; &longs;olutum in hoc &longs;itu, <lb/>
<expan abbr="qu&agrave;m">quam</expan> in quouis alio circumferenti&aelig; FOG. acidcirco in hoc <lb/>
grauius erit, hoc e&longs;t magis grauitabit, <expan abbr="qu&agrave;m">quam</expan> in alio &longs;itu.  </s>
             
<s id="id.2.1.17.5.1.34.0"> &amp; <expan abbr="qu&ograve;">quo</expan> <lb/>
propius fuerit ip&longs;i O remotiori grauius erit.  </s>
             
<s id="id.2.1.17.5.1.35.0"> lineaq; CO horizonti <lb/>
&aelig;quidi&longs;tans erit.  </s>
             
<s id="id.2.1.17.5.1.36.0"> non tamen puncti C horizonti (vt ip&longs;i exi&longs;ti&shy;<lb/>
mant) &longs;ed ponderis in O con&longs;tituti, <expan abbr="c&ugrave;m">cum</expan> ex centro grauitatis <lb/>
ponderis &longs;ummendus &longs;it horizon. qu&aelig; omnia demon&longs;trare opor&shy;<lb/>
tebat. </s>
             
<s id="id.2.1.17.5.1.37.0"> [qu&aelig; omnia demon&longs;trare opor&shy;<lb/>
tebat.]  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig17" place="text">       </figure>       
<p id="id.2.1.17.5.2.1.0" type="caption">        
<s id="id.2.1.17.5.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig18" place="text">       </figure>       
<p id="id.2.1.17.5.2.3.0" type="caption">        
<s id="id.2.1.17.5.2.3.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig19" place="text">       </figure>       
<p id="id.2.1.17.5.2.5.0" type="caption">        
<s id="id.2.1.17.5.2.5.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.18.1.0.0.0" type="margin">        
<s id="id.2.1.18.1.1.1.0"> <margin.target id="note33"></margin.target>18 <emph type="italics"/>Tertii.<emph.end type="italics"/> </s>
             
<s id="id.2.1.18.1.1.2.0"> <margin.target id="note34"></margin.target>21 <emph type="italics"/>primi.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.19.1.0.0.0" type="main">        
<pb n="12"/>
       
<s id="id.2.1.19.1.2.1.0"> Si autem libr&aelig; brachium ip&longs;o CO <lb/>
fuerit maius, <expan abbr="put&aacute;">puta</expan> quantitate CD; erit <lb/>
quoq; pondus in O grauius.  </s>
             
<s id="id.2.1.19.1.2.2.0"> circulus de&shy;<lb/>
&longs;cribatur OH, cuius centrum &longs;it D, &longs;e <arrow.to.target n="note35"></arrow.to.target><lb/>
midiameterq; DO. tanget circulus OH <lb/>
circulum FOG in puncto O, lineamq; <arrow.to.target n="note36"></arrow.to.target><lb/>
OS, qu&aelig; ponderis in O rectus, natura&shy;<lb/>
li&longs;q; e&longs;t de&longs;cen&longs;us, in eodem puncto con <lb/>
tinget.  </s>
             
<s id="id.2.1.19.1.2.3.0"> &amp; quoniam angulus SOH mi&shy;<lb/>
nor e&longs;t angulo SOG, erit de&longs;cen&longs;us <lb/>
ponderis in O per circumferentiam OH <lb/>
motui naturali OS propior, <expan abbr="qu&agrave;m">quam</expan> per <lb/>
circumferentiam OG.  </s>
     
<s id="id.2.1.19.1.2.3.0.a"> magis ergo li&shy;<lb/>
berum, atq; &longs;olutum, ac per con&longs;equens <lb/>
grauius erit in O, centro libr&aelig; exi&longs;ten<lb/>
te in D, <expan abbr="qu&agrave;m">quam</expan> in C. &longs;imiliter o&longs;ten&shy;<lb/>
detur, <expan abbr="qu&ograve;">quo</expan> maius fuerit brachium DO, <lb/>
pondus in O adhuc grauius e&longs;&longs;e. <arrow.to.target n="fig20"></arrow.to.target> </s>
    </p>       
<pb/>
       
<p id="id.2.1.19.3.0.0.0" type="main">        
<s id="id.2.1.19.3.1.1.0"> <expan abbr="Siver&ograve;">Sivero</expan> idem circulus AFBG, <lb/>
cuius centrum &longs;it R, propius fuerit <lb/>
mundi centro S; <expan abbr="circulumqu&eacute;">circulumque</expan> <expan abbr="&agrave;">a</expan> pun&shy;<lb/>
cto S ducatur contingens ST; punctum <lb/>
T (vbi grauius e&longs;t pondus) magis <lb/>
<expan abbr="&agrave;">a</expan> puncto A di&longs;tabit, <expan abbr="qu&agrave;m">quam</expan> punctum <lb/>
O. ducantur enim <expan abbr="&agrave;">a</expan> punctis OT ip&longs;i <lb/>
CS perpendiculares OMTN; conne<lb/>
ctanturq; RT; &longs;itq; centrum R in li&shy;<lb/>
nea CS; lineaq; ARB ip&longs;i ACB &aelig;qui <lb/>
<arrow.to.target n="note37"></arrow.to.target> di&longs;tans.  </s>
             
<s id="id.2.1.19.3.1.2.0"> Quoniam igitur triangula COS <lb/>
RTS &longs;unt rectangula; erit SC ad CO, <lb/>
vt CO ad CM. &longs;imiliter SR ad RT, <lb/>
vt RT ad RN. <expan abbr="c&ugrave;m">cum</expan> itaq; &longs;it RT ip&shy;<lb/>
<arrow.to.target n="note38"></arrow.to.target> &longs;i CO &aelig;qualis, &amp; SC ip&longs;a SR maior: <lb/>
maiorem habebit proportionem SC <lb/>
ad CO, <expan abbr="qu&agrave;m">quam</expan> SR ad RT. quare ma <lb/>
iorem quoq; proportionem habebit <lb/>
CO ad CM, <expan abbr="qu&agrave;m">quam</expan> RT ad RN.  </s>
     
<s id="id.2.1.19.3.1.2.0.a"> mi <lb/>
<arrow.to.target n="note39"></arrow.to.target> nor ergo erit CM, <expan abbr="qu&agrave;m">quam</expan> RN. &longs;ecetur <lb/>
igitur RN in P, ita vt RP &longs;it ip&longs;i <lb/>
<arrow.to.target n="fig21"></arrow.to.target><lb/>
CM &aelig;qualis; &amp; <expan abbr="&agrave;">a</expan> puncto P ip&longs;is MONT &aelig;quidi&longs;tans ducatur <lb/>
PQ, qu&aelig; circumferentiam AT &longs;ecet in Q: deniq; connectatur <lb/>
<expan abbr="Rq.">Rque</expan> quoniam enim du&aelig; CO CM duabus RQRP &longs;unt &aelig;qua <lb/>
<arrow.to.target n="note40"></arrow.to.target> les, &amp; angulus CMO angulo RPQ e&longs;t &aelig;qualis; erit &amp; angu&shy;<lb/>
lus MCO angulo PRQ &aelig;qualis.  </s>
             
<s id="id.2.1.19.3.1.3.0"> angulus autem MCA rectus <lb/>
<arrow.to.target n="note41"></arrow.to.target> recto PRA e&longs;t &aelig;qualis; ergo reliquus OCA reliquo QRA <lb/>
&aelig;qualis, &amp; circumferentia OA circumferenti&aelig; QA &aelig;qualis quo&shy;<lb/>
que erit.  </s>
             
<s id="id.2.1.19.3.1.4.0"> punctum idcirco T, quia magis <expan abbr="&agrave;">a</expan> puncto A di&longs;tat, <lb/>
<expan abbr="qu&agrave;m">quam</expan> Q; magis quoq; <expan abbr="&agrave;">a</expan> puncto A di&longs;tabit, <expan abbr="qu&agrave;m">quam</expan> punctum O. <lb/>
&longs;imiliter o&longs;tendetur, <expan abbr="qu&ograve;">quo</expan> propius fuerit circulus mundi centro, eun&shy;<lb/>
dem magis di&longs;tare.  </s>
             
<s id="id.2.1.19.3.1.5.0"> atq; ita vt prius demon&longs;trabitur pondus in cir<lb/>
cumferentia TAF centro R inniti, in circumferentia <expan abbr="ver&ograve;">vero</expan> TG <lb/>
<expan abbr="&agrave;">a</expan> linea detineri; atq; in puncto T grauius e&longs;&longs;e.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig20" place="text">       </figure>       
<p id="id.2.1.19.3.2.1.0" type="caption">        
<s id="id.2.1.19.3.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig21" place="text">       </figure>       
<p id="id.2.1.19.3.2.3.0" type="caption">        
<s id="id.2.1.19.3.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.20.1.0.0.0" type="margin">        
<s id="id.2.1.20.1.1.1.0"> <margin.target id="note35"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 11 <emph type="italics"/>Ter tit.<emph.end type="italics"/> </s>
             
<s id="id.2.1.20.1.1.2.0"> <margin.target id="note36"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 18 <emph type="italics"/>Ter tii.<emph.end type="italics"/> </s>
             
<s id="id.2.1.20.1.1.3.0"> <margin.target id="note37"></margin.target><emph type="italics"/>Cor.<emph.end type="italics"/> 8 <emph type="italics"/>&longs;exti<emph.end type="italics"/> </s>
             
<s id="id.2.1.20.1.1.4.0"> <margin.target id="note38"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 8 <emph type="italics"/>quinti<emph.end type="italics"/> </s>
             
<s id="id.2.1.20.1.1.5.0"> <margin.target id="note39"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 10 <emph type="italics"/>quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.20.1.1.6.0"> <margin.target id="note40"></margin.target>7 <emph type="italics"/>Sexti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.20.1.1.7.0"> <margin.target id="note41"></margin.target>26 <emph type="italics"/>Tertii.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.21.1.0.0.0" type="main">        
<pb n="13"/>
       
<s id="id.2.1.21.1.2.1.0"> Si autem punctum G e&longs;&longs;et <lb/>
in centro mundi; tunc <expan abbr="qu&ograve;">quo</expan> <lb/>
pondus propius fuerit ip&longs;i G, <lb/>
grauius erit: &amp; vbicunq; po<lb/>
natur pondus <expan abbr="pr&aelig;terqu&agrave;m">pr&aelig;terquam</expan> in <lb/>
ip&longs;o G, &longs;emper centro C inni<lb/>
tetur, vt in K. nam ducta <lb/>
G k, efficiet h&aelig;c (&longs;ecun&shy;<lb/>
d&ugrave;m quam fit ponderis natu<lb/>
ralis motus) <expan abbr="vn&aacute;">vna</expan> cum libr&aelig; <lb/>
brachio k C angulum acu&shy;<lb/>
tum.  </s>
             
<s id="id.2.1.21.1.2.2.0"> &aelig;quicruris enim trian&shy;<lb/>
guli CkG ad ba&longs;im anguli <lb/>
ad k, &amp; G &longs;unt &longs;emper acuti.  </s>
             
<s id="id.2.1.21.1.2.3.0"> <lb/>
<arrow.to.target n="fig22"></arrow.to.target><lb/>
Conferantur autem inuicem h&aelig;c duo, pondus videlicet in k, &amp; <lb/>
pondus in D: erit pondus in k grauius, <expan abbr="qu&agrave;m">quam</expan> in D. nam iuncta <lb/>
DG, <expan abbr="c&ugrave;m">cum</expan> tres anguli cuiu&longs;cunque trianguli duobus &longs;int rectis <lb/>
&aelig;quales, &amp; trianguli CDG &aelig;quicruris angulus DCG maior &longs;it <lb/>
angulo kCG &aelig;quicruris trianguli CkG: erunt reliqui ad ba&longs;im an<lb/>
guli DGC GDC &longs;imul &longs;umpti reliquis KGCGkC &longs;imul &longs;umptis <lb/>
minores.  </s>
             
<s id="id.2.1.21.1.2.4.0"> horumq; dimidii; angulus &longs;cilicet CDG angulo CKG <lb/>
minor erit.  </s>
             
<s id="id.2.1.21.1.2.5.0"> quare <expan abbr="c&ugrave;m">cum</expan> pondus in k &longs;olutum naturaliter per <lb/>
KG moueatur, pondusq; in D per DG, tanquam per &longs;patia, <lb/>
quibus in centrum mundi feruntur; linea CD, hoc e&longs;t libr&aelig; <lb/>
brachium magis adh&aelig;rebit motui naturali ponderis in D pror&shy;<lb/>
&longs;us &longs;oluti, line&aelig; &longs;cilicet DG; <expan abbr="qu&agrave;m">quam</expan> Ck motui <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> kG <lb/>
effecto.  </s>
             
<s id="id.2.1.21.1.2.6.0"> magis igitur &longs;u&longs;tinebit linea CD, <expan abbr="qu&agrave;m">quam</expan> Ck.  </s>
             
<s id="id.2.1.21.1.2.7.0"> ac pro&shy;<lb/>
pterea pondus in k ex &longs;uperius dictis grauius erit, <expan abbr="qu&agrave;m">quam</expan> in D.  </s>
     
<s id="id.2.1.21.1.2.7.0.a"> <lb/>
Pr&aelig;terea quoniam pondus in K &longs;i e&longs;&longs;et omnino liberum, pror&longs;u&longs;q; <lb/>
&longs;olutum, deor&longs;um per k G moueretur; ni&longs;i <expan abbr="&agrave;">a</expan> linea C k prohibere<lb/>
tur, qu&aelig; pondus vltra lineam KG per circumferentiam KH mo&shy;<lb/>
ueri cogit; linea C k pondus partim &longs;u&longs;tinebit, ip&longs;iq; renitetur; <lb/>
<expan abbr="c&ugrave;m">cum</expan> illud per circumferentiam k H moueri compellat.  </s>
             
<s id="id.2.1.21.1.2.8.0"> &amp; <lb/>
quoniam angulus CDG minor e&longs;t angulo CkG, &amp; angulus CDk <lb/>
angulo CkH e&longs;t &aelig;qualis; erit reliquus GDk reliquo G k H maior.  </s>
             
<s id="id.2.1.21.1.2.9.0"> <lb/>
circumferentia igitur k H motui naturali ponderis in k &longs;oluti, li&shy;
<pb/>
ne&aelig; &longs;cilicet KG propior erit, <lb/>
<expan abbr="qu&agrave;m">quam</expan> circumferentia Dk li&shy;<lb/>
ne&aelig; DG. quare linea CD <lb/>
ponderi in D magis renititur, <lb/>
<expan abbr="qu&agrave;m">quam</expan> linea C k ip&longs;i ponde&shy;<lb/>
ri in K.  </s>
     
<s id="id.2.1.21.1.2.9.0.a"> ergo pondus in k <lb/>
grauius erit, <expan abbr="qu&agrave;m">quam</expan> in D.  </s>
     
<s id="id.2.1.21.1.2.9.0.b"> <lb/>
Similiter o&longs;tendetur pondus, <lb/>
<expan abbr="qu&ograve;">quo</expan> fuerit ip&longs;i F propius, vt <lb/>
in L, minus grauitare: pro&shy;<lb/>
pius <expan abbr="ver&ograve;">vero</expan> ip&longs;i G, vt in H, <lb/>
grauius e&longs;&longs;e. <arrow.to.target n="fig23"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.21.2.0.0.0" type="main">        
<s id="id.2.1.21.2.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> centrum mundi <lb/>
S e&longs;&longs;et inter puncta CG; <lb/>
<expan abbr="prim&ugrave;m">primum</expan> quidem &longs;imili&shy;<lb/>
ter o&longs;tendetur pondus vbi <lb/>
cunq; po&longs;itum centro C <lb/>
initi, vt in H. ductis enim <lb/>
HG HS, angulus ad <lb/>
ba&longs;im GHC &aelig;quicruris tri <lb/>
anguli CHG e&longs;t &longs;emper <lb/>
acutus: quare &amp; SHC ip<lb/>
&longs;o minor erit quoq; &longs;em<lb/>
per acutus.  </s>
             
<s id="id.2.1.21.2.1.2.0"> ducatur au&shy;<lb/>
tem <expan abbr="&agrave;">a</expan> puncto S ip&longs;i CS <lb/>
perpendicularis Sk.  </s>
             
<s id="id.2.1.21.2.1.3.0"> di&shy;<lb/>
<arrow.to.target n="fig24"></arrow.to.target><lb/>
co pondus grauius e&longs;&longs;e in k, <expan abbr="qu&agrave;m">quam</expan> in alio &longs;itu circumferenti&aelig; FKG. <lb/>
&amp; <expan abbr="qu&ograve;">quo</expan> propius fuerit ip&longs;i F, vel G, minus grauitare.  </s>
             
<s id="id.2.1.21.2.1.4.0"> Accipiantur <lb/>
ver&longs;us F puncta DL, connectanturq; LC LS DC DS, produ&shy;<lb/>
canturq; LS DS k SHS v&longs;q; ad circuli circumferentiam in EM <lb/>
NO; connectanturq; CE, CM, CN, CO.  </s>
     
<s id="id.2.1.21.2.1.4.0.a"> Quoniam enim <lb/>
<arrow.to.target n="note42"></arrow.to.target> LE DM &longs;e inuicem &longs;ecant in S; erit rectangulum LSE rectan&shy;<lb/>
<arrow.to.target n="note43"></arrow.to.target> gulo DSM &aelig;quale.  </s>
             
<s id="id.2.1.21.2.1.5.0"> quare vt LS ad DS ita erit SM <lb/>
<arrow.to.target n="note44"></arrow.to.target> ad SE.  </s>
     
<s id="id.2.1.21.2.1.5.0.a"> maior autem e&longs;t LS, <expan abbr="qu&agrave;m">quam</expan> DS; &amp; SM ip&longs;a SE.  </s>
     
<s id="id.2.1.21.2.1.5.0.b"> 
<pb n="14"/>
ergo LS SE &longs;imul &longs;umpt&aelig; ip&longs;is DS SM maiores erunt.  </s>
             
<s id="id.2.1.21.2.1.6.0"> eademq; <arrow.to.target n="note45"></arrow.to.target><lb/>
ratione kN minorem e&longs;&longs;e DM o&longs;tendetur.  </s>
             
<s id="id.2.1.21.2.1.7.0"> rur&longs;us quoniam re<lb/>
ctangulum OSH &aelig;quale e&longs;t rectangulo kSN; ob eandem cau&longs;am <lb/>
HO maior erit kN. eodemq; pror&longs;us modo kN omnibus a&shy;<lb/>
liis per punctum S tran&longs;euntibus minorem e&longs;&longs;e demon&longs;trabitur.  </s>
             
<s id="id.2.1.21.2.1.8.0"> <lb/>
&amp; quoniam &aelig;quicrurium triangulorum CLE DCM latera LC <lb/>
CE lateribus DC CM &longs;unt &aelig;qualia; ba&longs;is <expan abbr="ver&ograve;">vero</expan> LE maior e&longs;t <lb/>
DM: erit angulus LCE angulo DCM maior.  </s>
             
<s id="id.2.1.21.2.1.9.0"> quare ad ba&longs;im <arrow.to.target n="note46"></arrow.to.target><lb/>
anguli C<emph type="italics"/>L<emph.end type="italics"/>E CEL &longs;imul &longs;umpti angulis CDM CMD mi&shy;<lb/>
nores erunt.  </s>
             
<s id="id.2.1.21.2.1.10.0"> &amp; horum dimidii, angulus &longs;cilicet CLS angulo CDS <lb/>
minor erit.  </s>
             
<s id="id.2.1.21.2.1.11.0"> ergo pondus in <emph type="italics"/>L<emph.end type="italics"/> magis &longs;upra lineam LC, <expan abbr="qu&agrave;m">quam</expan> <lb/>
in D &longs;upra DC grauitabit, <expan abbr="magisqu&eacute;">magisque</expan> centro innitetur in L, <expan abbr="qu&agrave;m">quam</expan> <lb/>
in D. &longs;imiliter o&longs;tendetur in D magis <expan abbr="c&etilde;tro">centro</expan> C inniti, <expan abbr="qu&agrave;m">quam</expan> in k.  </s>
             
<s id="id.2.1.21.2.1.12.0"> ergo <lb/>
ponds in k grauius erit, <expan abbr="qu&agrave;m">quam</expan> in D; &amp; in D, <expan abbr="qu&agrave;m">quam</expan> in L. eademq; pror <lb/>
&longs;us ratione quoniam kN minor e&longs;t HO, erit angulus CKS an&shy;<lb/>
gulo CHS maior.  </s>
             
<s id="id.2.1.21.2.1.13.0"> quare pondus in H magis centro C innite&shy;<lb/>
tur, <expan abbr="qu&agrave;m">quam</expan> in k.  </s>
             
<s id="id.2.1.21.2.1.14.0"> &amp; hoc modo o&longs;tendetur, vbicunq; in circum&shy;<lb/>
ferentia FDG fuerit pondus, minus in K centro C inniti, <expan abbr="qu&agrave;m">quam</expan> <lb/>
in alio &longs;itu: &amp; <expan abbr="qu&ograve;">quo</expan> propius fuerit ip&longs;i F, vel G, magis inniti.  </s>
             
<s id="id.2.1.21.2.1.15.0"> dein&shy;<lb/>
de quoniam angulus CkS maior e&longs;t CDS, &amp; CDk &aelig;qualis <lb/>
e&longs;t CkH: erit reliquus SkH reliquo SDk minor.  </s>
             
<s id="id.2.1.21.2.1.16.0"> quare cir&shy;<lb/>
cumferentia k H propior erit motui naturali recto ponderis in K <lb/>
&longs;oluti, line&aelig; &longs;cilicet k S, <expan abbr="qu&agrave;m">quam</expan> circumferentia D k motui DS. &amp; <lb/>
ideo linea CD magis ip&longs;i ponderi in D renititur, <expan abbr="qu&agrave;m">quam</expan> CK <lb/>
ponderi in k con&longs;tituto.  </s>
             
<s id="id.2.1.21.2.1.17.0"> hacq; ratione o&longs;tendetur angulum <lb/>
SHG maiorem e&longs;&longs;e SkH: &amp; per con&longs;equens lineam CH magis <lb/>
ponderi in H reniti, <expan abbr="qu&agrave;m">quam</expan> CK ponderi in K. &longs;imiliter demon&shy;<lb/>
&longs;trabitur lineam C<emph type="italics"/>L<emph.end type="italics"/> magis pondus &longs;u&longs;tinere, <expan abbr="qu&agrave;m">quam</expan> CD: ob <lb/>
ea&longs;demq; cau&longs;as o&longs;tendetur pondus in K minus &longs;upra lineam Ck <lb/>
grauitare, <expan abbr="qu&agrave;m">quam</expan> in quouis alio &longs;itu fuerit circumferenti&aelig; FDG. <lb/>
&amp; <expan abbr="qu&ograve;">quo</expan> propius fuerit ip&longs;i F, vel G, minus grauitare.  </s>
             
<s id="id.2.1.21.2.1.18.0"> grauius ergo <lb/>
erit in k, <expan abbr="qu&agrave;m">quam</expan> in alio &longs;itu: minu&longs;q; graue erit, <expan abbr="qu&ograve;">quo</expan> propius fue&shy;<lb/>
rit ip&longs;i F. vel G. 
<pb/>
 </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig22" place="text">       </figure>       
<p id="id.2.1.21.2.2.1.0" type="caption">        
<s id="id.2.1.21.2.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig23" place="text">       </figure>       
<p id="id.2.1.21.2.2.3.0" type="caption">        
<s id="id.2.1.21.2.2.3.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig24" place="text">       </figure>       
<p id="id.2.1.21.2.2.5.0" type="caption">        
<s id="id.2.1.21.2.2.5.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.22.1.0.0.0" type="margin">        
<s id="id.2.1.22.1.1.1.0"> <margin.target id="note42"></margin.target>35 <emph type="italics"/>Tertii.<emph.end type="italics"/> </s>
             
<s id="id.2.1.22.1.1.2.0"> <margin.target id="note43"></margin.target>16 <emph type="italics"/>Sexti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.22.1.1.3.0"> <margin.target id="note44"></margin.target>7 <emph type="italics"/>Tertii.<emph.end type="italics"/> </s>
             
<s id="id.2.1.22.1.1.4.0"> <margin.target id="note45"></margin.target>25 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.22.1.1.5.0"> <margin.target id="note46"></margin.target>25 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.23.1.0.0.0" type="main">        
<s id="id.2.1.23.1.1.1.0"> Si deniq; centrum C <lb/>
e&longs;&longs;et in centro mundi, <lb/>
pondus vbicunque con&shy;<lb/>
&longs;titutum manere mani&shy;<lb/>
fe&longs;tum e&longs;t.  </s>
             
<s id="id.2.1.23.1.1.2.0"> vt po&longs;ito pon<lb/>
dere in D, linea CD to&shy;<lb/>
tum &longs;u&longs;tinebit pondus; <lb/>
<expan abbr="c&ugrave;m">cum</expan> ip&longs;ius ponderis in D <lb/>
horizonti &longs;it perpendicu <lb/>
<arrow.to.target n="note47"></arrow.to.target> laris.  </s>
             
<s id="id.2.1.23.1.1.3.0"> pondus ergo ma <lb/>
nebit. <arrow.to.target n="fig25"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.23.2.0.0.0" type="main">        
<s id="id.2.1.23.2.1.1.0"> Quoniam autem in his hactenus demon&longs;tratis, nullam de gra<lb/>
uitate brachii libr&aelig; mentionem fecimus, idcirco &longs;i brach&longs;i quoq; <lb/>
grauitatem con&longs;iderare voluerimus, centrum grauitatis magnitu<lb/>
dinis ex pondere, brachioq; compo&longs;it&aelig; inueniri poterit, circulo<lb/>
rumq; circumferenti&aelig; &longs;ecundum di&longs;tantiam <expan abbr="&agrave;">a</expan> centro libr&aelig; ad <lb/>
hoc ip&longs;um grauitatis centrum de&longs;cribentur, ac &longs;i in ip&longs;o (vt re ue<lb/>
ra e&longs;t) pondus con&longs;titutum fuerit; omnia, &longs;icuti ab&longs;q; libr&aelig; bra<lb/>
chii grauitate con&longs;iderata inuenimus; hoc quoq; modo eius con&longs;i<lb/>
derata grauitate reperiemus.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig25" place="text">       </figure>       
<p id="id.2.1.23.2.2.1.0" type="caption">        
<s id="id.2.1.23.2.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.24.1.0.0.0" type="margin">        
<s id="id.2.1.24.1.1.1.0"> <margin.target id="note47"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.25.1.0.0.0" type="main">        
<pb n="15"/>
       
<s id="id.2.1.25.1.2.1.0"> Ex dictis igitur, con&longs;iderando li&shy;<lb/>
bram, vt <expan abbr="long&egrave;">longe</expan> <expan abbr="&agrave;">a</expan> mundi centro a&shy;<lb/>
be&longs;t, quemadmodum ip&longs;i fecere, &longs;i&shy;<lb/>
cuti etiam actu e&longs;t, apparet fal&longs;itas <lb/>
dicentium pondus in A grauius e&longs;&longs;e, <lb/>
<expan abbr="qu&agrave;m">quam</expan> in alio &longs;itu.  </s>
             
<s id="id.2.1.25.1.2.2.0"> &longs;imulq; fal&longs;um e&longs;&longs;e, <lb/>
<expan abbr="qu&ograve;">quo</expan> pondus <expan abbr="&agrave;">a</expan> linea FG magis di&longs;tat <lb/>
grauiuis e&longs;&longs;e.  </s>
             
<s id="id.2.1.25.1.2.3.0"> nam punctum O pro&shy;<lb/>
pius e&longs;t ip&longs;i FG, <expan abbr="qu&agrave;m">quam</expan> punctum A. <lb/>
e&longs;t enim linea <expan abbr="&agrave;">a</expan> puncto O ip&longs;i FG <arrow.to.target n="note48"></arrow.to.target><lb/>
perpendicularis ip&longs;a CA minor.  </s>
             
<s id="id.2.1.25.1.2.4.0"> de&shy;<lb/>
inde ex puncto A pondus velocius mo <lb/>
ueri, <expan abbr="qu&agrave;m">quam</expan> ab alio &longs;itu, e&longs;t quoque <lb/>
fal&longs;um.  </s>
             
<s id="id.2.1.25.1.2.5.0"> ex puncto enim O pondus ve&shy;<lb/>
locius mouebitur, <expan abbr="qu&agrave;m">quam</expan> ex puncto <lb/>
A; <expan abbr="c&ugrave;m">cum</expan> in O &longs;it magis liberum, atq; <lb/>
&longs;olutum, <expan abbr="qu&agrave;m">quam</expan> in alio &longs;itu: de&longs;cen&longs;us <lb/>
<expan abbr="qu&eacute;">que</expan> ex puncto O propior &longs;it motui na&shy;<lb/>
turali recto, <expan abbr="qu&agrave;m">quam</expan> quilibet alius de&shy;<lb/>
&longs;cen&longs;us. <arrow.to.target n="fig26"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.25.2.0.0.0" type="main">        
<s id="id.2.1.25.2.1.1.0"> Pr&aelig;terea <expan abbr="c&ugrave;m">cum</expan> ex re&shy;<lb/>
ctiori, &amp; obliquiori <expan abbr="defc&etilde;">defcem</expan> <lb/>
&longs;u o&longs;tendunt, pondus in <lb/>
A grauiur e&longs;&longs;e, <expan abbr="qu&agrave;m">quam</expan> in <lb/>
D; &amp; in D, <expan abbr="qu&agrave;m">quam</expan> in <lb/>
L; <expan abbr="prim&ugrave;m">primum</expan> quidem fal<lb/>
&longs;um exi&longs;timant, &longs;i pon<lb/>
dus aliquod collocatum <lb/>
fuerit in quocunq; &longs;itu <lb/>
circunferenti&aelig;, vt in D, <lb/>
rectum eius de&longs;cen&longs;um <lb/>
per rectam lineam DR <lb/>
ip&longs;i FG parallelam, tam <lb/>
<expan abbr="qu&agrave;m">quam</expan> <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> <expan abbr="mo&shy;|tum">mo&shy;tum</expan><arrow.to.target n="fig27"></arrow.to.target>
<pb/>
 naturalem fieri de&shy;<lb/>
bere; &longs;icuti prius dictum <lb/>
e&longs;t.  </s>
             
<s id="id.2.1.25.2.1.2.0"> In quocunq; enim <lb/>
&longs;itu pondus aliquod con<lb/>
&longs;tituatur, &longs;i naturalem <lb/>
eius ad propium locum <lb/>
motionem &longs;pectemus, <lb/>
<expan abbr="c&ugrave;m">cum</expan> <expan abbr="rect&aacute;">recta</expan> ad eum <expan abbr="&longs;ua&shy;pt&egrave;">&longs;ua&shy;<lb/>
pte</expan> natura moueatur, &longs;up<lb/>
po&longs;ita totius vniuer&longs;i figu<lb/>
ra, eiu&longs;modi erit; vt <lb/>
&longs;emper <expan abbr="&longs;pati&utilde;">&longs;patium</expan>, per quod <lb/>
naturaliter mouetur, ra&shy;<lb/>
tionem habere videatur <lb/>
<arrow.to.target n="fig28"></arrow.to.target><lb/>
line&aelig; <expan abbr="&agrave;">a</expan> circumferentia ad centrum product&aelig;.  </s>
             
<s id="id.2.1.25.2.1.3.0"> non igitur natura<lb/>
les de&longs;cen&longs;us recti cuiuslibet &longs;oluti ponderis per lineas fieri po&longs;<lb/>
&longs;unt inter &longs;e &longs;e parallelas; <expan abbr="c&ugrave;m">cum</expan> omnes in centrum mundi conue&shy;<lb/>
niant.  </s>
             
<s id="id.2.1.25.2.1.4.0"> &longs;upponunt deinde ponderis ex D in A per rectam lineam <lb/>
ver&longs;us centrum mundi motum eiu&longs;dem e&longs;&longs;e quantitatis, ac &longs;i fui&longs;<lb/>
&longs;et ex O in C: ita vt punctum A &aelig;qualiter <expan abbr="&agrave;">a</expan> centro mundi &longs;it <lb/>
di&longs;tans, vt C. quod e&longs;t etiam fal&longs;um; nam punctum A magis <lb/>
<expan abbr="&agrave;">a</expan> centro mundi di&longs;tat, <expan abbr="qu&agrave;m">quam</expan> C: maior enim e&longs;t linea <expan abbr="&agrave;">a</expan> cen&shy;<lb/>
<arrow.to.target n="note49"></arrow.to.target> tro mundi v&longs;q; ad A, <expan abbr="qu&agrave;m">quam</expan> <expan abbr="&agrave;">a</expan> centro mundi v&longs;q; ad C: <expan abbr="c&ugrave;m">cum</expan> li&shy;<lb/>
nea <expan abbr="&agrave;">a</expan> centro mundi v&longs;q; ad A rectum &longs;ubtendat angulum <expan abbr="&agrave;">a</expan> li&shy;<lb/>
neis AC, &amp; <expan abbr="&agrave;">a</expan> puncto C ad centrum mundi contentum.  </s>
             
<s id="id.2.1.25.2.1.5.0"> ex qui&shy;<lb/>
bus non &longs;olum &longs;uppo&longs;itio illa, qua libram DE in AB redire demon<lb/>
&longs;trant, <expan abbr="ver&ugrave;m">verum</expan> etiam omnes <expan abbr="fer&egrave;">fere</expan> ip&longs;orum demon&longs;trationes ruunt.  </s>
             
<s id="id.2.1.25.2.1.6.0"> <lb/>
ni&longs;i forta&longs;&longs;e dixerint, h&aelig;c omnia propter maximam <expan abbr="&agrave;">a</expan> centro mun<lb/>
di v&longs;q; ad nos di&longs;tantiam adeo in&longs;en&longs;ibilia e&longs;&longs;e, vt propter in&longs;en<lb/>
&longs;ibilitatem tanquam vera &longs;upponi po&longs;sint: <expan abbr="c&ugrave;m">cum</expan> omnes <expan abbr="quid&etilde;">quidem</expan> alii, qui <lb/>
h&aelig;c tractauerunt, tanquam nota &longs;uppo&longs;uerint.  </s>
             
<s id="id.2.1.25.2.1.7.0"> pr&aelig;&longs;ertim quia <lb/>
&longs;en&longs;ibilitas illa non efficit, quin de&longs;cen&longs;us ponderis ex L in D <lb/>
(vt eorum verbis vtar) minus capiat de directo, <expan abbr="qu&agrave;m">quam</expan> de&longs;cen&shy;<lb/>
&longs;us DA. &longs;imiliter arcus DA magis de directo capiet, <expan abbr="qu&agrave;m">quam</expan> cir<lb/>
cumferentia EV. quocirca vera erit &longs;uppo&longs;itio; ali&aelig;q; demon&shy;<lb/>
&longs;trationes in &longs;uo robore permanebunt.  </s>
             
<s id="id.2.1.25.2.1.8.0"> Concedamus etiam pon 
<pb n="16"/>
dus in A grauius e&longs;&longs;e, <expan abbr="qu&agrave;m">quam</expan> in alio &longs;itu; rectumq; ponderis de&shy;<lb/>
&longs;cen&longs;um per rectam lineam ip&longs;i FG parallelam fieri debere; &amp; <lb/>
qu&aelig;libet puncta in lineis horizonti &aelig;quidi&longs;tantibus accepta &aelig;&shy;<lb/>
qualiter <expan abbr="&agrave;">a</expan> centro mundi di&longs;tare: non tamen propterea &longs;equetur, <lb/>
veram e&longs;&longs;e demon&longs;trationem, qua inferunt pondus in A grauius <lb/>
e&longs;&longs;e, <expan abbr="qu&agrave;m">quam</expan> in alio &longs;itu, vt in L. &longs;i enim verum e&longs;&longs;et, <expan abbr="qu&ograve;">quo</expan> pon<lb/>
dus hoc modo rectius de&longs;cendit, ibi grauius e&longs;&longs;e; &longs;equeretur etiam, <lb/>
<expan abbr="qu&ograve;">quo</expan> idem pondus in &aelig;qualibus arcubus &aelig;qualiter <expan abbr="rect&egrave;">recte</expan> de&longs;cende <lb/>
ret, vt in ii&longs;dem locis &aelig;qualem haberet grauitatem, quod fal<lb/>
&longs;um e&longs;&longs;e ita demon&longs;tratur. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig26" place="text">       </figure>       
<p id="id.2.1.25.2.2.1.0" type="caption">        
<s id="id.2.1.25.2.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig27" place="text">       </figure>       
<p id="id.2.1.25.2.2.3.0" type="caption">        
<s id="id.2.1.25.2.2.3.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig28" place="text">       </figure>       
<p id="id.2.1.25.2.2.5.0" type="caption">        
<s id="id.2.1.25.2.2.5.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.26.1.0.0.0" type="margin">        
<s id="id.2.1.26.1.1.1.0"> <margin.target id="note48"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 15 <emph type="italics"/>Tertii.<emph.end type="italics"/> </s>
             
<s id="id.2.1.26.1.1.2.0"> <margin.target id="note49"></margin.target>18 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.27.1.0.0.0" type="main">        
<s id="id.2.1.27.1.1.1.0"> Sint circumferenti&aelig; AL AM inter &longs;e &longs;e &aelig;quales; &amp; conne<lb/>
ctatur LM, qu&aelig; AB &longs;ecet in X: erit LM ip&longs;i FG &aelig;quidi&longs;tans, <lb/>
ip&longs;iq; AB perpendicularis.  </s>
             
<s id="id.2.1.27.1.1.2.0"> &amp; XM ip&longs;i XL &aelig;qualis erit.  </s>
             
<s id="id.2.1.27.1.1.3.0"> &longs;i igi<arrow.to.target n="note50"></arrow.to.target><lb/>
tur pondus ex L moueatur in A per circumferentiam LA, rectus <lb/>
eius motus erit <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> lineam LX. &longs;i <expan abbr="ver&ograve;">vero</expan> moueatur ex A <lb/>
in M per circum&longs;erentiam AM, <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> rectam eius motus <lb/>
erit XM. quare de&longs;cen&longs;us ex L in A &aelig;qualis erit de&longs;cen&longs;ui ex A <lb/>
in M; tum ob circumferentias &aelig;quales, tum propter rectas li <lb/>
neas ip&longs;i AB perpendiculares &aelig;quales.  </s>
             
<s id="id.2.1.27.1.1.4.0"> ergo idem pondus in L <lb/>
<expan abbr="&aelig;qu&egrave;">&aelig;que</expan> graue erit, vt in A, quod e&longs;t fal&longs;um.  </s>
             
<s id="id.2.1.27.1.1.5.0"> cum <expan abbr="long&eacute;">longe</expan> grauius &longs;it <lb/>
in A, <expan abbr="qu&agrave;m">quam</expan> in L. </s>
    </p>       
<p id="id.2.1.28.1.0.0.0" type="margin">        
<s id="id.2.1.28.1.1.1.0"> <margin.target id="note50"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 3 <emph type="italics"/>Tertii.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.29.1.0.0.0" type="main">        
<s id="id.2.1.29.1.1.1.0"> Quamuis autem AMLA &aelig;qualiter <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> ip&longs;os de directo <lb/>
capiant; dicent forta&longs;&longs;e, quia tamen principium de&longs;cen&longs;us ex L <lb/>
&longs;cilicet LD minus de directo capit, <expan abbr="qu&agrave;m">quam</expan> principium de&longs;cen&longs;us <lb/>
ex A, &longs;cilicet AN; pondus in A grauius erit, <expan abbr="qu&agrave;m">quam</expan> in L. nam <lb/>
<expan abbr="c&ugrave;m">cum</expan> circumferentia AN &longs;it ip&longs;i LD (vt &longs;upra po&longs;itum e&longs;t) <lb/>
&aelig;qualis, qu&aelig; <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> ip&longs;os de directo capit CT; LD <expan abbr="ver&ograve;">vero</expan> <lb/>
de directo capit PO. ideo pondus grauius erit in A, <expan abbr="qu&agrave;m">quam</expan> in L. <lb/>
quod &longs;i verum e&longs;&longs;et, &longs;equeretur idem pondus in eodem &longs;itu diuer<lb/>
&longs;o duntaxat modo con&longs;ideratum in habitudine ad eundem &longs;itum, <lb/>
tum grauius, tum leuius e&longs;&longs;e. quod e&longs;t impo&longs;sibile. </s>
             
<s id="id.2.1.29.1.1.2.0"> [quod e&longs;t impo&longs;sibile.]  </s>
             
<s id="id.2.1.29.1.1.3.0"> hoc e&longs;t, &longs;i <lb/>
de&longs;cen&longs;um con&longs;ideremus ponderis in L, quatenus ex L in A de&shy;<lb/>
&longs;cendit, grauius erit, <expan abbr="qu&agrave;m">quam</expan> &longs;i eiu&longs;dem ponderis de&longs;cen&longs;um con&shy;<lb/>
&longs;ideremus ex L in D <expan abbr="tant&ugrave;m">tantum</expan>.  </s>
             
<s id="id.2.1.29.1.1.4.0"> neq; enim negare po&longs;&longs;unt ex ei&longs;&shy;<lb/>
demmet dictis, quin de&longs;cen&longs;us ponderis ex L in A de directo ca <lb/>
piat LX, &longs;iue PC. de&longs;cen&longs;us <expan abbr="ver&ograve;">vero</expan> AM, quin &longs;imiliter de directo 
<pb/>
capiat XM: <expan abbr="c&ugrave;m">cum</expan> ip&longs;i <lb/>
quoq; hoc modo acci&shy;<lb/>
piant, atq; ita accipe&shy;<lb/>
re &longs;it nece&longs;&longs;e.  </s>
             
<s id="id.2.1.29.1.1.5.0"> &longs;i enim li&shy;<lb/>
bram DE in AB redire <lb/>
demon&longs;trare volunt, com<lb/>
parando de&longs;cen&longs;us pon&shy;<lb/>
deris in D cum de&longs;cen&shy;<lb/>
&longs;u ponderis in E, nece&longs;&longs;e <lb/>
e&longs;t, vt o&longs;tendant rectum <lb/>
de&longs;cen&longs;um OC corre&shy;<lb/>
&longs;pondentem circumferen<lb/>
ti&aelig; DA maiorem e&longs;&longs;e re<lb/>
cto de&longs;cen&longs;u TH circum<lb/>
<arrow.to.target n="fig29"></arrow.to.target><lb/>
ferenti&aelig; EV corre&longs;pondente.  </s>
             
<s id="id.2.1.29.1.1.6.0"> &longs;i enim partem <expan abbr="tant&ugrave;m">tantum</expan> totius de&shy;<lb/>
&longs;cen&longs;us ex D in A acciperent, vt D k; o&longs;tenderentq; magis cape&shy;<lb/>
re de directo de&longs;cen&longs;um Dk, <expan abbr="qu&agrave;m">quam</expan> &aelig;qualis portio de&longs;cen&longs;us ex <lb/>
puncto E. &longs;equetur pondus in D <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> ip&longs;os grauius e&longs;&longs;e pon<lb/>
dere in E; &amp; v&longs;q; ad k <expan abbr="tant&ugrave;m">tantum</expan> deor&longs;um moueri: ita vt libra mo<lb/>
ta &longs;it in kI. &longs;imiliter &longs;i libram KI in AB redire demon&longs;trare vo<lb/>
lunt accipiendo portionem de&longs;cen&longs;us ex k in A; hoc e&longs;t k S; <lb/>
o&longs;tenderentq; k S magis de directo capere, <expan abbr="qu&agrave;m">quam</expan> ex aduer&longs;o &aelig;&shy;<lb/>
qualis de&longs;cen&longs;us ex puncto I: &longs;imili modo &longs;equetur pondus in k <lb/>
grauius e&longs;&longs;e, <expan abbr="qu&agrave;m">quam</expan> in I; &amp; v&longs;q; ad S <expan abbr="tant&ugrave;m">tantum</expan> moueri.  </s>
             
<s id="id.2.1.29.1.1.7.0"> &amp; &longs;i rur&longs;us <lb/>
o&longs;tenderent portionem de&longs;cen&longs;us ex S in A, atq; ita deinceps, re<lb/>
ctiorem e&longs;&longs;e &aelig;quali de&longs;cen&longs;u ponderis oppo&longs;iti; &longs;emper &longs;equetur <lb/>
libram SI ad AB propius accedere, nunquam tamen in AB per&shy;<lb/>
uenire demon&longs;trabunt.  </s>
             
<s id="id.2.1.29.1.1.8.0"> &longs;i igitur libram DE in AB redire demon<lb/>
&longs;trare volunt, nece&longs;&longs;e e&longs;t, vt de&longs;cen&longs;um ponderis ex D in A de di <lb/>
recro capere quantitatem line&aelig; ex puncto D ip&longs;i AB ad rectos <lb/>
angulos duct&aelig; accipiant.  </s>
             
<s id="id.2.1.29.1.1.9.0"> atq; ita, &longs;i &aelig;quales de&longs;cen&longs;us DA AN <lb/>
inuicem comparemus, qui &aelig;qualiter de directo capient OC CT, <lb/>
cueniet idem pondus in D <expan abbr="&aelig;qu&egrave;">&aelig;que</expan> graue e&longs;&longs;e, vt in A. &longs;i <expan abbr="ver&ograve;">vero</expan> por<lb/>
tiones tantum ex D A accipiamus; grauius erit in A, <expan abbr="qu&agrave;m">quam</expan> <lb/>
in D. ergo ex diuer&longs;itate <expan abbr="tant&ugrave;m">tantum</expan> modi con&longs;iderandi, idem pon<lb/>
dus, &amp; grauius, &amp; leuius e&longs;&longs;e continget. non autem exip&longs;a na&shy;
<pb n="17"/>
tura rei. </s>
             
<s id="id.2.1.29.1.1.10.0"> [non autem exip&longs;a na&shy;
<pb n="17"/>
tura rei.]  </s>
             
<s id="id.2.1.29.1.1.11.0"> In&longs;uper ip&longs;orum &longs;uppo&longs;itio non a&longs;&longs;erit, pondus &longs;ecun<lb/>
<expan abbr="d&ugrave;m">dum</expan> &longs;itum grauius e&longs;&longs;e, <expan abbr="quant&ograve;">quanto</expan> in eodem &longs;itu minus obliquum <lb/>
e&longs;t principium ip&longs;ius de&longs;cen&longs;us.  </s>
             
<s id="id.2.1.29.1.1.12.0"> Suppo&longs;itio igitur &longs;uperius alla<lb/>
ta, hoc e&longs;t, <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> &longs;itum pondus grauius e&longs;&longs;e, <expan abbr="quant&ograve;">quanto</expan> in eo <lb/>
dem &longs;itu minus obliquus e&longs;t de&longs;cen&longs;us; non &longs;olum ex his, qu&aelig; <lb/>
diximus, vllo modo concedi pote&longs;t; &longs;ed quoniam huius oppo&longs;i<lb/>
tum o&longs;tendere quoq; non e&longs;t difficile: &longs;cilicet idem pondus in <lb/>
&aelig;qualibus circumferentiis, <expan abbr="qu&ograve;">quo</expan> minus obliquus e&longs;t de&longs;cen&longs;us, ibi <lb/>
minus grauitare. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig29" place="text">       </figure>       
<p id="id.2.1.29.1.2.1.0" type="caption">        
<s id="id.2.1.29.1.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.29.2.0.0.0" type="main">        
<s id="id.2.1.29.2.1.1.0"> Sint enim vt prius cir <lb/>
cumferentr&aelig; AL AM <lb/>
inter &longs;e &longs;e &aelig;quales; &longs;itq; <lb/>
punctum L <expan abbr="prop&egrave;">prope</expan> F. &amp; <lb/>
connectatur LM, qu&aelig; <lb/>
ip&longs;i AB perpendicularis <lb/>
erit. &amp; LX ip&longs;i XM <lb/>
&aelig;qualis.  </s>
             
<s id="id.2.1.29.2.1.2.0"> [&amp; LX ip&longs;i XM <lb/>
&aelig;qualis.]  </s>
             
<s id="id.2.1.29.2.1.3.0"> deinde <expan abbr="prop&egrave;">prope</expan> <lb/>
M inter MG quoduis <lb/>
accipiatur punctum P. <lb/>
fiatq; circumferentia PO <lb/>
circumferenti&aelig; AM &aelig;&shy;<lb/>
qualis.  </s>
             
<s id="id.2.1.29.2.1.4.0"> erit punctum O <lb/>
<arrow.to.target n="fig30"></arrow.to.target><expan abbr="prop&egrave;"><lb/>
prope</expan> A. connectanturq; CL, CO, CM, CP, OP. &amp; <expan abbr="&agrave;">a</expan> <lb/>
puncto P ip&longs;i OC perpendicularis ducatur PN.  </s>
     
<s id="id.2.1.29.2.1.4.0.a"> &amp; quoniam cir<lb/>
cumferentia AM circumferenti&aelig; OP e&longs;t &aelig;qualis: erit angu&shy;<lb/>
lus  <arrow.to.target n="note51"></arrow.to.target> ACM &aelig;qualis angulo OCP; &amp; angulus CXM rectus re&shy;<lb/>
cto CNP e&longs;t &aelig;qualis: erit quoq; reliquus XMC trianguli MCX <arrow.to.target n="note52"></arrow.to.target><lb/>
reliquo NPC trianguli PCN &aelig;qualis.  </s>
             
<s id="id.2.1.29.2.1.5.0"> &longs;ed &amp; latus CM lateri <arrow.to.target n="note53"></arrow.to.target><lb/>
CP e&longs;t &aelig;quale: ergo triangulum MCX triangulo PCN &aelig;quale <lb/>
erit. latu&longs;q; MX lateri NP &aelig;quale. quare linea PN ip&longs;i LX &aelig;qua <lb/>
lis erit.  </s>
             
<s id="id.2.1.29.2.1.6.0"> [latu&longs;q; MX lateri NP &aelig;quale.]  </s>
             
<s id="id.2.1.29.2.1.7.0"> [quare linea PN ip&longs;i LX &aelig;qua <lb/>
lis erit.]  </s>
             
<s id="id.2.1.29.2.1.8.0"> ducatur pr&aelig;terea <expan abbr="&agrave;">a</expan> puncto O linea OT ip&longs;i AC &aelig;qui <lb/>
di&longs;tans, qu&aelig; NP &longs;ecet in V. atq; ip&longs;i OT <expan abbr="&agrave;">a</expan> puncto P perpendi<lb/>
cularis ducatur, qu&aelig; quidem inter OV cadere non pote&longs;t; nam <lb/>
<expan abbr="c&ugrave;m">cum</expan> angulus ONV &longs;it rectus; erit OVN acutus.  </s>
             
<s id="id.2.1.29.2.1.9.0"> quare OVP <arrow.to.target n="note54"></arrow.to.target><lb/>
obtu&longs;us erit.  </s>
             
<s id="id.2.1.29.2.1.10.0"> non igitur linea <expan abbr="&agrave;">a</expan> puncto P ip&longs;i OT intra OV 
<pb/>
perpendicularis cadet.  </s>
             
<s id="id.2.1.29.2.1.11.0"> <lb/>
duo enim anguli vnius <lb/>
trianguli, vnus quidem <lb/>
rectus, alter <expan abbr="ver&ograve;">vero</expan> ob&shy;<lb/>
tu&longs;us e&longs;&longs;et. quod e&longs;t im <lb/>
po&longs;sibile. </s>
             
<s id="id.2.1.29.2.1.12.0"> [quod e&longs;t im<lb/>
po&longs;sibile.]  </s>
             
<s id="id.2.1.29.2.1.13.0"> cadet ergo in <lb/>
linea OT in parte VT. <lb/>
&longs;itq; PT. erit PT &longs;ecun<lb/>
<expan abbr="d&ugrave;m">dum</expan> ip&longs;os rectus circum<lb/>
ferenti&aelig; OP de&longs;cen&longs;us.  </s>
             
<s id="id.2.1.29.2.1.14.0"> <lb/>
Quoniam igitur angulus <lb/>
ONV e&longs;t rectus; erit <lb/>
<arrow.to.target n="note55"></arrow.to.target> linea OV ip&longs;a ON ma<lb/>
ior.  </s>
             
<s id="id.2.1.29.2.1.15.0"> quare OT ip&longs;a <lb/>
<arrow.to.target n="fig31"></arrow.to.target><lb/>
quoq; ON maior exi&longs;tet.  </s>
             
<s id="id.2.1.29.2.1.16.0"> <expan abbr="C&ugrave;m">Cum</expan> itaq; <expan abbr="lin&egrave;a">linea</expan> OP angulos &longs;ubten&shy;<lb/>
dat rectos ONP OTP; erit quadratum ex OP quadratis ex <lb/>
<arrow.to.target n="note56"></arrow.to.target> ON NP &longs;imul &longs;umptis &aelig;quale.  </s>
             
<s id="id.2.1.29.2.1.17.0"> &longs;imiliter quadratis ex OT TP <lb/>
&longs;imul &aelig;quale.  </s>
             
<s id="id.2.1.29.2.1.18.0"> quare quadrata &longs;imul ex ON NP quadratis ex <lb/>
OT TP &longs;imul &aelig;qualia erunt.  </s>
             
<s id="id.2.1.29.2.1.19.0"> quadratum autem ex OT maius <lb/>
e&longs;t quadrato ex ON; cum linea OT &longs;it ip&longs;a ON maior.  </s>
             
<s id="id.2.1.29.2.1.20.0"> ergo qua<lb/>
dratum ex NP maius erit quadrato ex TP. ac propterea linea <lb/>
TP minor erit linea PN, &amp; linea LX. minus obliquus igitur e&longs;t <lb/>
de&longs;cen&longs;us arcus LA, <expan abbr="qu&agrave;m">quam</expan> arcus OP.  </s>
     
<s id="id.2.1.29.2.1.20.0.a"> ergo pondus in L, ex ip<lb/>
&longs;orum dictis, grauius erit, <expan abbr="qu&agrave;m">quam</expan> in O. quod ex iis, qu&aelig; &longs;upra di<lb/>
ximus e&longs;t <expan abbr="manife&longs;t&egrave;">manife&longs;te</expan> fal&longs;um, <expan abbr="c&ugrave;m">cum</expan> pondus in O grauius &longs;it, <expan abbr="qu&agrave;m">quam</expan> <lb/>
in L.  </s>
     
<s id="id.2.1.29.2.1.20.0.b"> non igitur ex rectiori, &amp; obliquiori motu ita accepto col&shy;<lb/>
ligi pote&longs;t, <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> &longs;itum pondus grauius e&longs;&longs;e, <expan abbr="quant&ograve;">quanto</expan> in eo <lb/>
dem &longs;itu minus obliquus e&longs;t de&longs;cen&longs;us.  </s>
             
<s id="id.2.1.29.2.1.21.0"> Atq; hinc oritur omnis <lb/>
<expan abbr="ferm&eacute;">ferme</expan> ip&longs;orum error in hacre, atq; deceptio: nam quamuis per <lb/>
accidens interdum ex fal&longs;is &longs;equatur verum, per &longs;e tamen ex fal<lb/>
&longs;is fal&longs;um &longs;equitur, quemadmodum ex veris &longs;emper verum, nil <lb/>
idcirco mirum, &longs;i dum fal&longs;a accipiunt; illi&longs;q; tanquam veri&longs;si&shy;<lb/>
mis innituntur; fal&longs;i&longs;sima <expan abbr="omnin&ograve;">omnino</expan> colligunt, atq; concludunt.  </s>
             
<s id="id.2.1.29.2.1.22.0"> <lb/>
decipiuntur quinetiam, <expan abbr="d&ugrave;m">dum</expan> libr&aelig; contemplationem mathemati<lb/>
<expan abbr="c&egrave;">ce</expan> &longs;impliciter a&longs;&longs;ummunt; <expan abbr="c&ugrave;m">cum</expan> eius con&longs;ideratio &longs;it pror&longs;us me&shy;<lb/>
chanica: nec vllo modo ab&longs;q; vero motu, ac ponderibus (en&shy;
<pb n="18"/>
tibus <expan abbr="omnin&ograve;">omnino</expan> naturalibus) de ip&longs;a &longs;ermo haberi po&longs;sit: &longs;ine qui&shy;<lb/>
bus eorum, qu&aelig; libr&aelig; accidunt, ver&aelig; caul&aelig; reperiri nullo mo <lb/>
do po&longs;sint. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig30" place="text">       </figure>       
<p id="id.2.1.29.2.2.1.0" type="caption">        
<s id="id.2.1.29.2.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig31" place="text">       </figure>       
<p id="id.2.1.29.2.2.3.0" type="caption">        
<s id="id.2.1.29.2.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.30.1.0.0.0" type="margin">        
<s id="id.2.1.30.1.1.1.0"> <margin.target id="note51"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 27 <emph type="italics"/>Ter tii.<emph.end type="italics"/> </s>
             
<s id="id.2.1.30.1.1.2.0"> <margin.target id="note52"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 32 <emph type="italics"/>primi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.30.1.1.3.0"> <margin.target id="note53"></margin.target>26 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.30.1.1.4.0"> <margin.target id="note54"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 13 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.30.1.1.5.0"> <margin.target id="note55"></margin.target>19 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.30.1.1.6.0"> <margin.target id="note56"></margin.target>47 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.31.1.0.0.0" type="main">        
<s id="id.2.1.31.1.1.1.0"> Pr&aelig;terea &longs;i adhuc &longs;up<lb/>
po&longs;itionem conceda&shy;<lb/>
mus; <expan abbr="&agrave;">a</expan> con&longs;ideratione <lb/>
libr&aelig; <expan abbr="long&egrave;">longe</expan> recedunt; <lb/>
dum eo pacto, vt libra <lb/>
DE in AB redire de&shy;<lb/>
beat, di&longs;currunt.  </s>
             
<s id="id.2.1.31.1.1.2.0"> &longs;emper <lb/>
enim alterum pondus <lb/>
&longs;eor&longs;um accipiunt, <expan abbr="put&aacute;">puta</expan> <lb/>
D, vel E; ac &longs;i <expan abbr="mod&ograve;">modo</expan> <expan abbr="vn&utilde;">vnum</expan> <lb/>
<expan abbr="mod&ograve;">modo</expan> alterum in libra <lb/>
con&longs;titutum e&longs;&longs;et, nec <lb/>
vllo modo ambo con&shy;<lb/>
<arrow.to.target n="fig32"></arrow.to.target><lb/>
nexa; cuius tamen oppo&longs;itum <expan abbr="omnin&ograve;">omnino</expan> fieri oportet; neq; alterum <lb/>
&longs;ine altero <expan abbr="rect&egrave;">recte</expan> con&longs;iderari pote&longs;t; <expan abbr="c&ugrave;m">cum</expan> de ip&longs;is in libra con&longs;ti&shy;<lb/>
tutis &longs;ermo habeatur.  </s>
             
<s id="id.2.1.31.1.1.3.0"> <expan abbr="c&ugrave;m">cum</expan> enim dicunt, de&longs;cen&longs;um ponderis in <lb/>
D minus obliquum e&longs;&longs;e de&longs;cen&longs;u ponderis in E; erit pondus in <lb/>
D per &longs;uppo&longs;itionem grauius pondere in E: quare <expan abbr="c&ugrave;m">cum</expan> &longs;it graui&shy;<lb/>
us, nece&longs;&longs;e e&longs;t deor&longs;um moueri, libramq; DE in AB redire: di<lb/>
&longs;cur&longs;us i&longs;te nullius pror&longs;us momenti e&longs;t.  </s>
             
<s id="id.2.1.31.1.1.4.0"> <expan abbr="Prim&ugrave;m">Primum</expan> quidem &longs;em&shy;<lb/>
per argumentantur, ac &longs;i pondera in DE de&longs;cendere debeant, <lb/>
vnius <expan abbr="tant&ugrave;m">tantum</expan> &longs;ine alterius connexione con&longs;iderando de&longs;cen&longs;um.  </s>
             
<s id="id.2.1.31.1.1.5.0"> <lb/>
<expan abbr="po&longs;trem&ograve;">po&longs;tremo</expan> tamen ob ponderum de&longs;cen&longs;uum comparationem colli&shy;<lb/>
gentes inferunt, pondus in D deor&longs;um moueri, &amp; pondus in E <lb/>
&longs;ur&longs;um, vtraq; &longs;imul in libra inuicem connexa accipientes.  </s>
             
<s id="id.2.1.31.1.1.6.0"> <expan abbr="ve&shy;r&ugrave;m">ve&shy;<lb/>
rum</expan> ex ii&longs;demmet, quibus vtuntur, principiis, ac demon&longs;tratio<lb/>
nibus, oppo&longs;itum eius, quod defendere conantur, <expan abbr="facillim&egrave;">facillime</expan> col&shy;<lb/>
ligi pote&longs;t.  </s>
             
<s id="id.2.1.31.1.1.7.0"> Nam &longs;i comparetur de&longs;cen&longs;us ponderis in D cum a&shy;<lb/>
&longs;cen&longs;u ponderis in E, vt ductis EK DH ip&longs;i AB perpendicula&shy;<lb/>
ribus; <expan abbr="c&ugrave;m">cum</expan> angulus DCH &longs;it &aelig;qualis angulo ECk; &amp; angulus <arrow.to.target n="note57"></arrow.to.target><lb/>
DHC rectus &aelig;qualis e&longs;t recto E k C; &amp; latus DC lateri CE &aelig;qua <lb/>
le: erit triangulum CDH triangulo CEk &aelig;quale, &amp; latus DH la-<arrow.to.target n="note58"></arrow.to.target>
<pb/>
teri Ek &aelig;quale.  </s>
             
<s id="id.2.1.31.1.1.8.0"> <expan abbr="c&ugrave;m">cum</expan> <lb/>
autem angulus DCA <lb/>
&longs;it angulo ECB &aelig;qua&shy;<lb/>
lis: erit quoq; circum&shy;<lb/>
ferentia DA cirferen&shy;<lb/>
ti&aelig; BE &aelig;qualis.  </s>
             
<s id="id.2.1.31.1.1.9.0"> dum <lb/>
itaq; pondus in D de&shy;<lb/>
&longs;cendit per circumfe&shy;<lb/>
rentiam DA, pondus <lb/>
in E per circumferen&shy;<lb/>
tiam EB ip&longs;i DA &aelig;&shy;<lb/>
qualem a&longs;cendit.  </s>
             
<s id="id.2.1.31.1.1.10.0"> &amp; de&shy;<lb/>
&longs;cen&longs;us <expan abbr="p&otilde;deris">ponderis</expan> in D de <lb/>
directo (more <expan abbr="ip&longs;or&utilde;">ip&longs;orum</expan>) <lb/>
<arrow.to.target n="fig33"></arrow.to.target><lb/>
capiet DH; a&longs;cen&longs;us <expan abbr="ver&ograve;">vero</expan> ponderis in E de directo capiet Ek ip<lb/>
&longs;i DH &aelig;qualem: erit itaq; de&longs;cen&longs;us ponderis in D a&longs;cen&longs;ui pon<lb/>
deris in E &aelig;qualis, &amp; qualis erit propen&longs;io vnius ad motum deor<lb/>
sum, talis etiam erit re&longs;i&longs;tentia alterius ad motum &longs;ur&longs;um.  </s>
             
<s id="id.2.1.31.1.1.11.0"> re&shy;<lb/>
&longs;i&longs;tentia &longs;cilicet violenti&aelig; ponderis in E in a&longs;cen&longs;u naturali po&shy;<lb/>
tenti&aelig; ponderis in D in de&longs;cen&longs;u <expan abbr="contr&agrave;">contra</expan> nitendo apponitur; <expan abbr="c&ugrave;m">cum</expan> <lb/>
&longs;it ip&longs;i &aelig;qualis.  </s>
             
<s id="id.2.1.31.1.1.12.0"> <expan abbr="qu&ograve;">quo</expan> enim pondus in D naturali potentia deor<lb/>
&longs;um velocius de&longs;cendit, <expan abbr="e&ograve;">eo</expan> tardius pondus in E violenter a&longs;cendit.  </s>
             
<s id="id.2.1.31.1.1.13.0"> <lb/>
quare neutrum ip&longs;orum alteri pr&aelig;ponderabit, <expan abbr="c&ugrave;m">cum</expan> ab &aelig;quali non <lb/>
proueniat actio.  </s>
             
<s id="id.2.1.31.1.1.14.0"> Non igitur pondus in D pondus in E &longs;ur&longs;um <lb/>
mouebit.  </s>
             
<s id="id.2.1.31.1.1.15.0"> &longs;i enim moueret; nece&longs;&longs;e e&longs;&longs;et, pondus in D maiorem <lb/>
habere virtutem de&longs;cendendo, <expan abbr="qu&agrave;m">quam</expan> pondus in E a&longs;cendendo; <lb/>
&longs;ed h&aelig;c &longs;unt &aelig;qualia: ergo pondera manebunt.  </s>
             
<s id="id.2.1.31.1.1.16.0"> &amp; grauitas pon&shy;<lb/>
deris in D grauitati ponderis in E &aelig;qualis erit.  </s>
             
<s id="id.2.1.31.1.1.17.0"> Pr&aelig;terea quoniam <lb/>
&longs;upponunt, <expan abbr="qu&ograve;">quo</expan> pondus <expan abbr="&agrave;">a</expan> linea directionis FG magis di&longs;tat, <expan abbr="e&ograve;">eo</expan> <lb/>
grauius e&longs;&longs;e: Idcirco ductis quoq; <expan abbr="&agrave;">a</expan> punctis DE ip&longs;i FG perpen<lb/>
dicularibus DO EI; &longs;imili modo demon&longs;trabitur, triangulum <lb/>
CDO triangulo CEI &aelig;qualem e&longs;&longs;e: &amp; lineam DO ip&longs;i EI &aelig;qua<lb/>
lem.  </s>
             
<s id="id.2.1.31.1.1.18.0"> tam igitur di&longs;tat <expan abbr="&agrave;">a</expan> linea FG pondus in D, <expan abbr="qu&agrave;m">quam</expan> pondus in <lb/>
E. ex ip&longs;orum igitur rationibus, atq; &longs;uppo&longs;itionibus, pondera <lb/>
in DE <expan abbr="&aelig;qu&egrave;">&aelig;que</expan> grauia erunt.  </s>
             
<s id="id.2.1.31.1.1.19.0"> Amplius quid prohibet, quin libram <lb/>
DE ex nece&longs;sitate in FG moueri &longs;imili ratione o&longs;tendatur?  </s>
             
<s id="id.2.1.31.1.1.20.0"> Pri&shy;
<pb n="19"/>
<expan abbr="m&ugrave;m">mum</expan> quidem ex eorummet demon&longs;trationibus colligi pote&longs;t, a&shy;<lb/>
&longs;cen&longs;um ponderis in E ver&longs;us B rectiorem e&longs;&longs;e a&longs;cen&longs;u ponderis <lb/>
in D ver&longs;us F; hoc e&longs;t minus capere de directo a&longs;cen&longs;um pon&shy;<lb/>
deris in D in arcubus &aelig;qualibus a&longs;cen&longs;u ponderis in E.  </s>
     
<s id="id.2.1.31.1.1.20.0.a"> &longs;uppona<lb/>
tur ergo <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> &longs;itum pondus leuius e&longs;&longs;e, <expan abbr="quant&ograve;">quanto</expan> in eodem &longs;i&shy;<lb/>
tu minus rectus e&longs;t a&longs;cen&longs;us: qu&aelig; quidem &longs;uppo&longs;itio, <expan abbr="ade&ograve;">adeo</expan> ma&shy;<lb/>
nife&longs;ta e&longs;&longs;e videtur, veluti ip&longs;orum altera.  </s>
             
<s id="id.2.1.31.1.1.21.0"> Quoniam igitur a&longs;cen&shy;<lb/>
&longs;us ponderis in E rectior e&longs;t a&longs;cen&longs;u ponderis in D; per &longs;uppo&longs;i&shy;<lb/>
tionem pondus in D leuius erit pondere in E. ergo pondus in <lb/>
D &longs;ur&longs;um <expan abbr="&agrave;">a</expan> pondere in E mouebitur, ita vt libra in FG perue<lb/>
niat. atq; ita demon&longs;trari poterit, libram DE in FG moueri. <lb/>
qu&aelig; quidem demon&longs;tratio inutilis e&longs;t pror&longs;us, ea&longs;demq; patitur <lb/>
difficultates. </s>
             
<s id="id.2.1.31.1.1.22.0"> [atq; ita demon&longs;trari poterit, libram DE in FG moueri.]  </s>
             
<s id="id.2.1.31.1.1.23.0"> [<lb/>
qu&aelig; quidem demon&longs;tratio inutilis e&longs;t pror&longs;us, ea&longs;demq; patitur <lb/>
difficultates.]  </s>
             
<s id="id.2.1.31.1.1.24.0"> licet enim <expan abbr="tanqu&agrave;m">tanquam</expan> verum admittatur pondus in E <lb/>
a&longs;cendendo grauius e&longs;&longs;e pondere in D &longs;imiliter a&longs;cendendo, <lb/>
non tamen ex hoc &longs;equitur, pondus in E de&longs;cendendo grauius <lb/>
e&longs;&longs;e pondere in D a&longs;cendendo.  </s>
             
<s id="id.2.1.31.1.1.25.0"> Neutra igitur harum demon&shy;<lb/>
&longs;trationum libram DE, vel in AB redire, vel in FG moue&shy;<lb/>
ri, o&longs;tendentium, vera e&longs;t. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig32" place="text">       </figure>       
<p id="id.2.1.31.1.2.1.0" type="caption">        
<s id="id.2.1.31.1.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig33" place="text">       </figure>       
<p id="id.2.1.31.1.2.3.0" type="caption">        
<s id="id.2.1.31.1.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.32.1.0.0.0" type="margin">        
<s id="id.2.1.32.1.1.1.0"> <margin.target id="note57"></margin.target>15 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.32.1.1.2.0"> <margin.target id="note58"></margin.target>26 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.33.1.0.0.0" type="main">        
<s id="id.2.1.33.1.1.1.0"> Pr&aelig;terea &longs;i ip&longs;orum &longs;uppo&longs;itionem, eorumq; verborum vim <lb/>
<expan abbr="rect&egrave;">recte</expan> perpendamus; alium <expan abbr="cert&egrave;">certe</expan> habere &longs;en&longs;um con&longs;piciemus.  </s>
             
<s id="id.2.1.33.1.1.2.0"> nam <lb/>
<expan abbr="c&ugrave;m">cum</expan> &longs;emper &longs;patium, per quod naturaliter pondus mouetur, <expan abbr="&agrave;">a</expan> cen<lb/>
tro grauitatis ip&longs;ius ponderis ad centrum mundi, in&longs;tar rect&aelig; li&shy;<lb/>
ne&aelig; <expan abbr="&agrave;">a</expan> centro grauitatis ad centrum mundi product&aelig;, &longs;it &longs;umendum; <lb/>
<expan abbr="tant&ograve;">tanto</expan> huiusmodi ponderis de&longs;cen&longs;us, magis, <expan abbr="minusu&egrave;">minusue</expan> obliquus <lb/>
dicetur; <expan abbr="quant&ograve;">quanto</expan> <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> &longs;patium in&longs;tar pr&aelig;dict&aelig; line&aelig; de&longs;igna <lb/>
tum, magis, aut minus (naturalem tamen locum petens, &longs;emperq; <lb/>
magis ip&longs;i appropinquans) mouebitur; ita vt <expan abbr="tant&ograve;">tanto</expan> obliquior de&shy;<lb/>
&longs;cen&longs;us dicatur, <expan abbr="quant&ograve;">quanto</expan> recedit ab eiu&longs;modi &longs;patio: rectior <expan abbr="ver&ograve;">vero</expan>, <lb/>
<expan abbr="quant&ograve;">quanto</expan> ad idem accedit.  </s>
             
<s id="id.2.1.33.1.1.3.0"> &amp; in hoc &longs;en&longs;u &longs;uppo&longs;itio illa nemini <lb/>
difficultatem parere debet, <expan abbr="ade&ograve;">adeo</expan> enim veritas eius con&longs;picua e&longs;t; <lb/>
rationiq; con&longs;entanea: vt nulla pro&longs;us manife&longs;tatione egere vi&shy;<lb/>
deatur.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.33.3.0.0.0" type="main">        
<s id="id.2.1.33.3.1.1.0"> Si itaq; pondus &longs;olutum in &longs;itu D <lb/>
collocatum ad propium locum mo&shy;<lb/>
ueri debeat; proculdubio po&longs;ito cen&shy;<lb/>
tro mundi S, per lineam DS moue&shy;<lb/>
bitur.  </s>
             
<s id="id.2.1.33.3.1.2.0"> &longs;imiliter pondus in E &longs;olutum <lb/>
per lineam ES mouebitur.  </s>
             
<s id="id.2.1.33.3.1.3.0"> quare &longs;i <lb/>
(vt rei veritas e&longs;t) ponderis de&longs;cen&shy;<lb/>
&longs;us magis, <expan abbr="minu&longs;u&egrave;">minu&longs;ue</expan> obliquus dicetur <lb/>
<expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> rece&longs;&longs;um, &amp; acce&longs;&longs;um ad <lb/>
&longs;patia per lineas DSES de&longs;ignata, <lb/>
iuxta naturales ip&longs;orum ad propria lo <lb/>
ca lationes; con&longs;picuum e&longs;t, minus <lb/>
obliquum e&longs;&longs;e de&longs;cen&longs;um ip&longs;ius E <lb/>
per EG, <expan abbr="qu&agrave;m">quam</expan> ip&longs;ius D per DA: <lb/>
<expan abbr="c&ugrave;m">cum</expan> angulum SEG angulo SDA <lb/>
minorem e&longs;&longs;e &longs;upra o&longs;ten&longs;um &longs;it.  </s>
             
<s id="id.2.1.33.3.1.4.0"> qua <lb/>
re in E pondus magis grauitabit, <lb/>
<expan abbr="qu&agrave;m">quam</expan> in D. quod e&longs;t penitus oppo&shy;<lb/>
&longs;itum eius, quod ip&longs;i o&longs;tendere cona<lb/>
ti &longs;unt.  </s>
             
<s id="id.2.1.33.3.1.5.0"> In&longs;urgent autem forta&longs;&longs;e <lb/>
contranos, &longs;i igitur (dicent) pondus <lb/>
in E grauius e&longs;t pondere in D, libra <lb/>
<arrow.to.target n="fig34"></arrow.to.target><lb/>
DE in hoc &longs;itu <expan abbr="minim&egrave;">minime</expan> per&longs;i&longs;tet, quod <expan abbr="equid&etilde;">equidem</expan> tueri propo&longs;uimus: <lb/>
&longs;ed in FG mouebitur.  </s>
             
<s id="id.2.1.33.3.1.6.0"> quibus re&longs;pondemus, plurimum referre, &longs;iue <lb/>
con&longs;ideremus pondera, quatenus &longs;unt inuicem di&longs;iuncta, &longs;iue quate <lb/>
nus &longs;unt &longs;ibi inuicem connexa.  </s>
             
<s id="id.2.1.33.3.1.7.0"> alia e&longs;t enim ratio ponderis in E &longs;ine <lb/>
connexione ponderis in D, alia <expan abbr="ver&ograve;">vero</expan> eiu&longs;dem alteri ponderi con<lb/>
nexi; ita vt alterum &longs;ine altero moueri non po&longs;sit.  </s>
             
<s id="id.2.1.33.3.1.8.0"> nam ponde<lb/>
ris in E, quatenus e&longs;t &longs;ine alterius ponderis connexione, rectus <lb/>
naturalis de&longs;cen&longs;us e&longs;t per lineam ES; quatenus <expan abbr="ver&ograve;">vero</expan> connexum <lb/>
e&longs;t ponderi in D, eius naturalis de&longs;cen&longs;us non erit amplius per <lb/>
lineam ES, &longs;ed per lineam ip&longs;i CS parallelam.  </s>
             
<s id="id.2.1.33.3.1.9.0"> magnitudo enim <lb/>
ex ponderibus ED, &amp; libra DE compo&longs;ita, cuius grauitatis cen&shy;<lb/>
trum e&longs;t C, &longs;i nullibi &longs;u&longs;tineatur, deor&longs;um eo modo, quo reperi<lb/>
tur, <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> grauitatis centrum per rectam <expan abbr="&agrave;">a</expan> centro grauita<lb/>
tis C ad centrum mundi S ductam naturaliter mouebitur, donec 
<pb n="20"/>
centrum C in centrum S perueniat.  </s>
             
<s id="id.2.1.33.3.1.10.0"> libra igitur DE <expan abbr="vn&aacute;">vna</expan> cum pon<lb/>
deribus eo modo, quo reperitur, deor&longs;um mouebitur, ita vt pun&shy;<lb/>
ctum C per lineam CS moueatur, donec C in S, libraq; DE in <lb/>
Hk perueniat; habeatq; libra in Hk eandem, quam prius habe&shy;<lb/>
bat po&longs;itionem; hoc e&longs;t Hk &longs;it ip&longs;i DE &aelig;quidi&longs;tans. connect antur <lb/>
igitur DH Ek. </s>
             
<s id="id.2.1.33.3.1.11.0"> [connectantur <lb/>
igitur DH Ek.]  </s>
             
<s id="id.2.1.33.3.1.12.0"> manife&longs;tum e&longs;t, dum libra DE in Hk mouetur pun<lb/>
cta DE per lineas DH Ek moueri, quippe exi&longs;tentibus inter &longs;e <arrow.to.target n="note59"></arrow.to.target><lb/>
&longs;e, ip&longs;iq; CS &aelig;qualibus, &amp; &aelig;quidi&longs;tantibus.  </s>
             
<s id="id.2.1.33.3.1.13.0"> Quare pondera in <lb/>
DE, quatenus &longs;unt &longs;ibi inuicem connexa, &longs;i ip&longs;orum naturalem mo <lb/>
tum &longs;pectemus, non <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> lineas DS ES, &longs;ed <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> <lb/>
LDH MEk ip&longs;i CS &aelig;quidi&longs;tantes mouebuntur.  </s>
             
<s id="id.2.1.33.3.1.14.0"> ponderis <expan abbr="ve&shy;r&ograve;">ve&shy;<lb/>
ro</expan> in E liberi, ac &longs;oluti, naturalis propen&longs;io erit per ES: ponderis <lb/>
autem in D &longs;imiliter &longs;oluti erit per DS. ac propterea non e&longs;t incon&shy;<lb/>
ueniens idem pondus <expan abbr="mod&ograve;">modo</expan> in E, <expan abbr="mod&ograve;">modo</expan> in D, grauius e&longs;&longs;e in E, <lb/>
<expan abbr="qu&agrave;m">quam</expan> in D.  </s>
     
<s id="id.2.1.33.3.1.14.0.a"> &longs;i <expan abbr="ver&ograve;">vero</expan> pondera in ED &longs;ibi inuicem connexa, quate&shy;<lb/>
nusq; &longs;unt connexa con&longs;iderauerimus; erit ponderis in E natura&shy;<lb/>
lis propen&longs;io per lineam MEK: grauitas enim alterius ponde&shy;<lb/>
ris in D efficit, <expan abbr="n&egrave;">ne</expan> pondus in E per lineam ES grauitet, &longs;ed per <lb/>
Ek.  </s>
             
<s id="id.2.1.33.3.1.15.0"> quod ip&longs;um quoq; grauitas ponderis in E efficit, <expan abbr="n&egrave;">ne</expan> &longs;cilicet <lb/>
pondus in D per rectam DS degrauet; &longs;ed <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> DH: vtra&shy;<lb/>
que enim &longs;e impediunt, <expan abbr="n&egrave;">ne</expan> ad propria loca permeent.  </s>
             
<s id="id.2.1.33.3.1.16.0"> <expan abbr="C&ugrave;m">Cum</expan> igi<lb/>
tur naturalis de&longs;cen&longs;us rectus ponderum in DE &longs;it <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> <lb/>
LDH MEK: erit &longs;imliter rectus eorum a&longs;cen&longs;us <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> ea&longs; <lb/>
dem lineas HDL KEM. atq; a&longs;cen&longs;us ponderis in E magis, mi<lb/>
<expan abbr="nu&longs;u&egrave;">nu&longs;ue</expan> obliquus dicetur; <expan abbr="quant&ograve;">quanto</expan> <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> &longs;patium magis, <expan abbr="mi&shy;nu&longs;u&egrave;">mi&shy;<lb/>
nu&longs;ue</expan> iuxta lineam Mk mouebitur.  </s>
             
<s id="id.2.1.33.3.1.17.0"> hocq; pror&longs;us modo iuxta li<lb/>
neam LH &longs;ummendus e&longs;t, <expan abbr="t&ugrave;m">tum</expan> de&longs;cen&longs;us, <expan abbr="t&ugrave;m">tum</expan> a&longs;cen&longs;us ponde&shy;<lb/>
ris in D. &longs;i itaq; pondus in E deor&longs;um per EG moueretur; pon<lb/>
dus in D &longs;ur&longs;um per DF moueret.  </s>
             
<s id="id.2.1.33.3.1.18.0"> &amp; quoniam angulus CEK <arrow.to.target n="note60"></arrow.to.target><lb/>
&aelig;qualis e&longs;t angulo CDL, &amp; angulus CEG angulo CDF &aelig;qua&shy;<lb/>
lis; erit reliquus GEK reliquo LDF &aelig;qualis.  </s>
             
<s id="id.2.1.33.3.1.19.0"> <expan abbr="c&ugrave;m">cum</expan> autem &longs;up&shy;<lb/>
po&longs;itio illa, qu&aelig; ait, <expan abbr="&longs;ecund&uacute;m">&longs;ecundum</expan> &longs;itum pondus grauius e&longs;&longs;e, <expan abbr="quan&shy;t&ograve;">quan&shy;<lb/>
to</expan> in eodem &longs;itu minus obliquus e&longs;t de&longs;cen&longs;us; tanquam clara, <lb/>
atq; con&longs;picua admittatur; proculdubio h&aelig;c quoq; accipienda <lb/>
erit; <expan abbr="nemp&egrave;">nempe</expan>, <expan abbr="&longs;ecund&uacute;m">&longs;ecundum</expan> &longs;itum pondus grauius e&longs;&longs;e, <expan abbr="quant&ograve;">quanto</expan> in eo&shy;<lb/>
dem &longs;itu minus obliquus e&longs;t a&longs;cen&longs;us.  </s>
             
<s id="id.2.1.33.3.1.20.0"> <expan abbr="c&ugrave;m">cum</expan> non minus manife&longs;ta, 
<pb/>
rationiq; &longs;it con&longs;entanea.  </s>
             
<s id="id.2.1.33.3.1.21.0"> &aelig;qualis <lb/>
igitur erit de&longs;cen&longs;us ponderis in E <lb/>
a&longs;cen&longs;ui ponderis in D. eandem <lb/>
enim obliquitatem habet de&longs;cen&longs;us <lb/>
ponderis in E, quam habet a&longs;cen&shy;<lb/>
&longs;us ponderis in D; &amp; qualis erit <lb/>
propen&longs;io vnius ad motum deor&longs;um, <lb/>
talis quoq; erit re&longs;i&longs;tentia alterius ad <lb/>
motum &longs;ur&longs;um.  </s>
             
<s id="id.2.1.33.3.1.22.0"> <expan abbr="n&otilde;">non</expan> ergo pondus in E <lb/>
pondus in D &longs;ur&longs;um mouebit.  </s>
             
<s id="id.2.1.33.3.1.23.0"> neq; <lb/>
pondus in D deor&longs;um mouebitur, ita <lb/>
vt &longs;ur&longs;um moueat pondus in E. nam <lb/>
<expan abbr="c&utilde;">cum</expan> angulus CEB &longs;it ip&longs;i CDA &aelig;qua&shy;<lb/>
<arrow.to.target n="note61"></arrow.to.target> lis, &amp; Angulus CEM &longs;it angulo <lb/>
CDH &aelig;qualis; erit reliquus MEB <lb/>
reliquo HDA &aelig;qualis.  </s>
             
<s id="id.2.1.33.3.1.24.0"> de&longs;cen&longs;us <lb/>
igitur ponderis in D a&longs;cen&longs;ui ponde<lb/>
ris in E &aelig;qualis erit.  </s>
             
<s id="id.2.1.33.3.1.25.0"> non ergo pon<lb/>
dus in D pondus in E &longs;ur&longs;um moue<lb/>
bit.  </s>
             
<s id="id.2.1.33.3.1.26.0"> ex quibus &longs;equitur pondera in <lb/>
DE, quatenus &longs;unt &longs;ibi inuicem con<lb/>
nexa, <expan abbr="&aelig;qu&egrave;">&aelig;que</expan> grauia e&longs;&longs;e. <arrow.to.target n="fig35"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.33.4.0.0.0" type="main">        
<s id="id.2.1.33.4.1.1.0"> Alia deinde ratio, li&shy;<lb/>
bram &longs;imiliter DE in AB <lb/>
redire o&longs;tendens, <expan abbr="c&ugrave;m">cum</expan> in&shy;<lb/>
quiunt, exi&longs;tente trutina in <lb/>
CF meta e&longs;t CG.  </s>
     
<s id="id.2.1.33.4.1.1.0.a"> &amp; quo&shy;<lb/>
niam angulus DCG maior <lb/>
e&longs;t angulo ECG; pondus <lb/>
in D grauius erit pondere <lb/>
in E; ergo libra DE in AB <lb/>
redibit: nihil meo iudicio <lb/>
concludit.  </s>
             
<s id="id.2.1.33.4.1.2.0"> figmentumq; <lb/>
hoc de trutina, &amp; meta po&shy;<lb/>
tius omittendum, ac <expan abbr="&longs;ilen&shy;|tio">&longs;ilen&shy;tio</expan><arrow.to.target n="fig36"></arrow.to.target>
<pb n="21"/>
 <expan abbr="pr&aelig;tereund&utilde;">pr&aelig;tereundum</expan> e&longs;&longs;et, <expan abbr="qu&agrave;m">quam</expan> <expan abbr="verb&utilde;">verbum</expan> <expan abbr="vll&utilde;">vllum</expan> in eius confutatione &longs;umen<lb/>
dum; <expan abbr="c&ugrave;m">cum</expan> &longs;it pror&longs;us voluntarium.  </s>
             
<s id="id.2.1.33.4.1.3.0"> nece&longs;sitas enim cur pondus <lb/>
in D ex maiore angulo &longs;it grauius; curq; maior angulus maioris <lb/>
&longs;it cau&longs;a grauitatis; nu&longs;quam apparet.  </s>
             
<s id="id.2.1.33.4.1.4.0"> &longs;i autem comparentur in&shy;<lb/>
uicem anguli, <expan abbr="c&ugrave;m">cum</expan> angulus GCD &longs;it &aelig;qualis angulo FCE; &longs;i angu<lb/>
lus GCD e&longs;t cau&longs;a grauitatis; quare angulus FCE &longs;imiliter gra&shy;<lb/>
uitatis non e&longs;t cau&longs;a?  </s>
             
<s id="id.2.1.33.4.1.5.0"> Huius autem rei eam in medium rationem <lb/>
afferre videntur, quoniam CG e&longs;t meta, &amp; CF trutina.  </s>
             
<s id="id.2.1.33.4.1.6.0"> &longs;i (inquiunt) <lb/>
CG e&longs;&longs;et trutina, &amp; CF meta, tunc angulus FCE grauitatis e&longs;&longs;et <lb/>
cau&longs;a; non autem DCG ip&longs;i &aelig;qualis.  </s>
             
<s id="id.2.1.33.4.1.7.0"> qu&aelig; quidem ratio imma&shy;<lb/>
ginaria pror&longs;us, ac voluntaria e&longs;&longs;e videtur.  </s>
             
<s id="id.2.1.33.4.1.8.0"> quid enim refert, &longs;iue tru<lb/>
tina &longs;it in CF, &longs;iue in CG, <expan abbr="c&ugrave;m">cum</expan> libra DE in eodem &longs;emper pun&shy;<lb/>
cto C &longs;u&longs;tineatur?  </s>
             
<s id="id.2.1.33.4.1.9.0"> Vt autem eorum deceptio clarius appa&shy;<lb/>
reat. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig34" place="text">       </figure>       
<p id="id.2.1.33.4.2.1.0" type="caption">        
<s id="id.2.1.33.4.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig35" place="text">       </figure>       
<p id="id.2.1.33.4.2.3.0" type="caption">        
<s id="id.2.1.33.4.2.3.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig36" place="text">       </figure>       
<p id="id.2.1.33.4.2.5.0" type="caption">        
<s id="id.2.1.33.4.2.5.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.34.1.0.0.0" type="margin">        
<s id="id.2.1.34.1.1.1.0"> <margin.target id="note59"></margin.target>33 <emph type="italics"/>Prmi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.34.1.1.2.0"> <margin.target id="note60"></margin.target>29 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.34.1.1.3.0"> <margin.target id="note61"></margin.target>29 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.35.1.0.0.0" type="main">        
<s id="id.2.1.35.1.1.1.0"> Sit eadem libra AB, cu&shy;<lb/>
ius medium C. &longs;it deinde <lb/>
tota FG trutina.  </s>
             
<s id="id.2.1.35.1.1.2.0"> eaq; im<lb/>
mobilis exi&longs;tat; qu&aelig; libram <lb/>
AB in puncto C &longs;u&longs;tineat.  </s>
             
<s id="id.2.1.35.1.1.3.0"> <lb/>
moueaturq; libra in DE. &amp; <lb/>
quoniam trutina e&longs;t, &amp; &longs;u&shy;<lb/>
pra, &amp; infra libram, quis <lb/>
nam angulus erit cau&longs;a gra&shy;<lb/>
uitatis, <expan abbr="c&ugrave;m">cum</expan> libra DE in <lb/>
<arrow.to.target n="fig37"></arrow.to.target><expan abbr="eod&etilde;"><lb/>
eodem</expan> &longs;emper puncto &longs;u&longs;tineatur?  </s>
             
<s id="id.2.1.35.1.1.4.0"> dicent for&longs;an, &longs;i trutina <expan abbr="&agrave;">a</expan> potentia <lb/>
in F &longs;u&longs;titencatur, tunc CG erit tanquam meta, &amp; angulus <lb/>
DCG grauitatis erit cau&longs;a.  </s>
             
<s id="id.2.1.35.1.1.5.0"> &longs;i <expan abbr="ver&ograve;">vero</expan> &longs;u&longs;tineatur in G, tunc FCE <lb/>
erit cau&longs;a grauitatis, CF <expan abbr="ver&ograve;">vero</expan> tanquam meta erit.  </s>
             
<s id="id.2.1.35.1.1.6.0"> cuius quidem <lb/>
rei nulla videtur e&longs;&longs;e cau&longs;a, ni&longs;i immaginaria.  </s>
             
<s id="id.2.1.35.1.1.7.0"> meta enim (quod <lb/>
aiunt) nullam pror&longs;us vim attractiuam, quandoq; ex maioris an&shy;<lb/>
guli parte, quandoq; ex parte minoris habere videtur.  </s>
             
<s id="id.2.1.35.1.1.8.0"> <expan abbr="Ver&ugrave;m">Verum</expan> <expan abbr="&agrave;">a</expan> dua<lb/>
bus potentiis &longs;u&longs;tineatur trutina, in F &longs;cilicet, &amp; in G, quod pr&aelig; ne<lb/>
ce&longs;sitate fieri pote&longs;t, veluti &longs;i potentia in F &longs;it <expan abbr="ade&ograve;">adeo</expan> debilis, vt ex &longs;e <lb/>
ip&longs;a medietatem <expan abbr="tant&ugrave;m">tantum</expan> ponderis &longs;u&longs;tinere qu&aelig;at: &longs;itq; potentia in <lb/>
Gip&longs;i potenti&aelig; in F &aelig;qualis, vtr&aelig;q; <expan abbr="aut&etilde;">autem</expan> &longs;imul libram <expan abbr="vn&aacute;">vna</expan> cum pon<lb/>
deribus &longs;u&longs;tineant.  </s>
             
<s id="id.2.1.35.1.1.9.0"> tunc quis nam angulus erit cau&longs;a grauitatis?  </s>
             
<s id="id.2.1.35.1.1.10.0"> non 
<pb/>
FCE, quia trutina e&longs;t in <lb/>
CF, &amp; in F &longs;u&longs;tinetur.  </s>
             
<s id="id.2.1.35.1.1.11.0"> neq; <lb/>
DCG, <expan abbr="c&ugrave;m">cum</expan> trutina &longs;it in <lb/>
CG, &amp; in G quoq; &longs;u&longs;ti<lb/>
neatur; non igitur anguli <lb/>
grauitatis cau&longs;a erunt.  </s>
             
<s id="id.2.1.35.1.1.12.0"> ergo <lb/>
neq; libra DE ab hoc &longs;itu <lb/>
ob hanc cau&longs;am mo uebi&shy;<lb/>
<arrow.to.target n="note62"></arrow.to.target> tur.  </s>
             
<s id="id.2.1.35.1.1.13.0"> Hanc autem eorum <lb/>
&longs;ententiam dupliciter con&shy;<lb/>
<arrow.to.target n="fig38"></arrow.to.target><lb/>
firmare videntur.  </s>
             
<s id="id.2.1.35.1.1.14.0"> <expan abbr="prim&ugrave;m">primum</expan> quidem a&longs;&longs;erunt Ari&longs;totelem in qu&aelig;&longs;tio<lb/>
nibus mechanicis has duas <expan abbr="tant&ugrave;m">tantum</expan> qu&aelig;&longs;tiones propo&longs;ui&longs;&longs;e; eiu&longs;q; <lb/>
demon&longs;trationes, tum maiori, &amp; minori angulo, <expan abbr="t&ugrave;m">tum</expan> trutin&aelig; po&longs;i<lb/>
tioni inniti.  </s>
             
<s id="id.2.1.35.1.1.15.0"> Affirmant deinde experientiam hoc idem docere; <lb/>
hoc e&longs;t libram DE trutina exi&longs;tente in CF, in AB horizonti <lb/>
&aelig;quidi&longs;tantem redire.  </s>
             
<s id="id.2.1.35.1.1.16.0"> quando autem trutina e&longs;t in CG, in FG <lb/>
moueri.  </s>
             
<s id="id.2.1.35.1.1.17.0"> <expan abbr="Ver&ugrave;m">Verum</expan> neq; Ari&longs;toteles, neq; experientia huic eorum <lb/>
opinioni fauent, quin potius aduer&longs;antur.  </s>
             
<s id="id.2.1.35.1.1.18.0"> <expan abbr="quant&ugrave;m">quantum</expan> enim atti&shy;<lb/>
net ad experientiam decipiuntur, ip&longs;a quidem experientia ma&shy;<lb/>
nife&longs;tum e&longs;t hoc accidere, quando libr&aelig; quoq; centrum, vel &longs;u&shy;<lb/>
pra, vel infra libram fuerit collocatum: non autem trutina dun<lb/>
taxat &longs;upra, vel infra exi&longs;tente, id contingere.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig37" place="text">       </figure>       
<p id="id.2.1.35.1.2.1.0" type="caption">        
<s id="id.2.1.35.1.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig38" place="text">       </figure>       
<p id="id.2.1.35.1.2.3.0" type="caption">        
<s id="id.2.1.35.1.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.36.1.0.0.0" type="margin">        
<s id="id.2.1.36.1.1.1.0"> <margin.target id="note62"></margin.target><emph type="italics"/>Cardanus.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.37.1.0.0.0" type="main">        
<pb n="22"/>
       
<s id="id.2.1.37.1.2.1.0"> Nam &longs;i libra AB habeat <lb/>
centrum C &longs;upra libram; <lb/>
&longs;itq; trutina CD infra li&shy;<lb/>
bram; moueaturq; libra in <lb/>
EF; tunc EF rur&longs;us in AB <lb/>
horizonti &aelig;quidi&longs;tantem <arrow.to.target n="note63"></arrow.to.target><lb/>
redibit.  </s>
             
<s id="id.2.1.37.1.2.2.0"> &longs;imiliter &longs;i libra <lb/>
centrum C habeat infra li<lb/>
bram, &longs;itq; trutina CD &longs;u<lb/>
pra libram, &amp; moueatur <lb/>
libra in EF; patet libram <arrow.to.target n="note64"></arrow.to.target><lb/>
ex parte F deor&longs;um moue <lb/>
ri, trutina &longs;upra libram e&shy;<lb/>
xi&longs;tente.  </s>
             
<s id="id.2.1.37.1.2.3.0"> &amp; in quocunq; a&shy;<lb/>
lio &longs;itu fuerit trutina, idem <lb/>
&longs;emper eueniet.  </s>
             
<s id="id.2.1.37.1.2.4.0"> non igitur <lb/>
trutina, &longs;ed centrum libr&aelig; <lb/>
harum diuer&longs;itatum cau&shy;<lb/>
&longs;a erit. <arrow.to.target n="fig39"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.37.2.0.0.0" type="main">        
<s id="id.2.1.37.2.1.1.0"> Animaduertendum e&longs;t <lb/>
itaq; in hac parte difficulter materialem libram con&longs;titui po&longs;&longs;e, <lb/>
qu&aelig; in vno <expan abbr="tant&ugrave;m">tantum</expan> puncto &longs;u&longs;tineatur; quemadmodum mente <lb/>
concipimus.  </s>
             
<s id="id.2.1.37.2.1.2.0"> brachiaq; ab eiu&longs;modi centro <expan abbr="ade&ograve;">adeo</expan> &aelig;qualia habeat, <lb/>
non &longs;olum in longitudine, <expan abbr="ver&ugrave;m">verum</expan> etiam in latitudine, &amp; profun<lb/>
ditate, vt omnes partes hinc <expan abbr="ind&eacute;">inde</expan> ad vnguem &aelig;queponderent.  </s>
             
<s id="id.2.1.37.2.1.3.0"> <lb/>
hoc enim materia <expan abbr="difficilim&egrave;">difficilime</expan> patitur.  </s>
             
<s id="id.2.1.37.2.1.4.0"> quocirca &longs;i centrum in ip&longs;a <lb/>
libra e&longs;&longs;e con&longs;iderauerimus, ad &longs;en&longs;um confugiendum non e&longs;t: <lb/>
<expan abbr="c&ugrave;m">cum</expan> artificilia ad &longs;ummum illud perfectionis gradum ab artifice <lb/>
deduci <expan abbr="minim&egrave;">minime</expan> po&longs;sint.  </s>
             
<s id="id.2.1.37.2.1.5.0"> In aliis <expan abbr="ver&ograve;">vero</expan> experientia quidem appa&shy;<lb/>
rentia docere poterit; proptereaquod, quamquam centrum libr&aelig; <lb/>
&longs;it &longs;emper punctum, quando tamen &longs;upra libram fuerit, <expan abbr="par&ugrave;m">parum</expan> re&shy;<lb/>
fert, &longs;i libra in eo puncto adamu&longs;&longs;im <expan abbr="minim&egrave;">minime</expan> &longs;u&longs;tineatur; quia <expan abbr="c&ugrave;m">cum</expan> <lb/>
&longs;it &longs;emper &longs;upra libram, idem &longs;emper eueniet.  </s>
             
<s id="id.2.1.37.2.1.6.0"> &longs;imili quoq; modo <lb/>
quando e&longs;t infra libram: quod tamen non accidit centro in ip&longs;a li&shy;<lb/>
bra exi&longs;tente.  </s>
             
<s id="id.2.1.37.2.1.7.0"> &longs;i enim ad vnguem &longs;emper in illo medio non &longs;u&shy;<lb/>
&longs;tineatur, diuer&longs;itatem efficiet; <expan abbr="c&ugrave;m">cum</expan> facillimum &longs;it, centrum il&shy;
<pb/>
lud, <expan abbr="d&ugrave;m">dum</expan> libra mouetur, proprium mutare &longs;itum. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig39" place="text">       </figure>       
<p id="id.2.1.37.2.2.1.0" type="caption">        
<s id="id.2.1.37.2.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.38.1.0.0.0" type="margin">        
<s id="id.2.1.38.1.1.1.0"> <margin.target id="note63"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.38.1.1.2.0"> <margin.target id="note64"></margin.target>3 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.39.1.0.0.0" type="main">        
<s id="id.2.1.39.1.1.1.0"> <expan abbr="Qu&ograve;d">Quod</expan> autem Ari&longs;toteles duas <expan abbr="tant&ugrave;m">tantum</expan> qu&aelig;&longs;tiones propo&shy;<lb/>
&longs;uerit, cur &longs;cilicet trutina &longs;uperius exi&longs;tente, &longs;i libra non &longs;it <lb/>
horizonti &aelig;quidi&longs;tans in &aelig;quilibrium, hoc e&longs;t horizonti &aelig;qui <lb/>
di&longs;tans redit: &longs;i autem trutina deor&longs;um fuerit con&longs;tituta, non <lb/>
redit; &longs;ed adhuc <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> partem depre&longs;&longs;am mouetur: verum <lb/>
quidem e&longs;t.  </s>
             
<s id="id.2.1.39.1.1.2.0"> non tamen eius demon&longs;trationes maiori, &amp; mino <lb/>
ri angulo, <expan abbr="po&longs;itioniqu&eacute;">po&longs;itionique</expan> trutin&aelig; (vt ip&longs;i dicunt) innituntur.  </s>
             
<s id="id.2.1.39.1.1.3.0"> In <lb/>
hoc enim mentem philo&longs;ophi a&longs;ignantis rationem diuer&longs;itatis <lb/>
motuum libr&aelig; <expan abbr="minim&egrave;">minime</expan> attingunt.  </s>
             
<s id="id.2.1.39.1.1.4.0"> <expan abbr="tant&ugrave;m">tantum</expan> enim abe&longs;t philo&longs;o&shy;<lb/>
phum has diuer&longs;itates in angulos referre, vt potius in cau&longs;a e&longs;&longs;e <lb/>
dicat magnitudinis alterius brachii libr&aelig; exce&longs;&longs;um <expan abbr="&agrave;">a</expan> perpendiculo, <lb/>
<expan abbr="mod&ograve;">modo</expan> ex vna, <expan abbr="mod&ograve;">modo</expan> ex altera parte contingentem. </s>
    </p>       
<p id="id.2.1.39.2.0.0.0" type="main">        
<s id="id.2.1.39.2.1.1.0"> Vt trutina &longs;uperius in <lb/>
CF exi&longs;tente, perpendicu<lb/>
lum erit FCG, quod <expan abbr="&longs;e&shy;cund&ugrave;m">&longs;e&shy;<lb/>
cundum</expan> ip&longs;um in centrum <lb/>
mundi &longs;emper vergit; <lb/>
quod quidem libram mo&shy;<lb/>
tam in DE in partes di&shy;<lb/>
uidit in&aelig;quales; &amp; maior <lb/>
pars e&longs;t ver&longs;us D: id au&shy;<lb/>
tem, quod plus e&longs;t, deor<lb/>
&longs;um fertur; ergo ex par&shy;<lb/>
te D deor&longs;um libra moue<lb/>
bitur, donec in AB re&shy;<lb/>
deat.  </s>
             
<s id="id.2.1.39.2.1.2.0"> &longs;i <expan abbr="ver&ograve;">vero</expan> trutina &longs;it <lb/>
<arrow.to.target n="fig40"></arrow.to.target><lb/>
in CG deor&longs;um, erit GCF perpendiculum, quod libram DE <lb/>
in partes in&aelig;quales &longs;imiliter diuidit: maior autem pars erit ver&longs;us <lb/>
E; quare ex parte E deor&longs;um libra mouebitur.  </s>
             
<s id="id.2.1.39.2.1.3.0"> quod vt <expan abbr="rect&egrave;">recte</expan> in&shy;<lb/>
telligatur, <expan abbr="c&ugrave;m">cum</expan> trutina e&longs;t &longs;upra libram, libr&aelig; quoq; centrum &longs;u&shy;<lb/>
pra libram e&longs;&longs;e intelligendum e&longs;t; &amp; &longs;i deor&longs;um, centrum quoque <lb/>
deor&longs;um: vt infra patebit.  </s>
             
<s id="id.2.1.39.2.1.4.0"> Aliter ip&longs;a Ari&longs;totelis demon&longs;tratio <lb/>
nihil concluderet.  </s>
             
<s id="id.2.1.39.2.1.5.0"> exi&longs;tente enim centro in ip&longs;a libra, vt in C; quo&shy;<lb/>
cunq; modo moueatur libra, nunquam perpendiculum FG libram, 
<pb n="23"/>
ni&longs;i in puncto C, &amp; in partes diuidet &aelig;quales.  </s>
             
<s id="id.2.1.39.2.1.6.0"> quare Ari&longs;totelis <lb/>
&longs;ententia ip&longs;is non &longs;olum non fauet, <expan abbr="ver&ugrave;m">verum</expan> etiam <expan abbr="maxim&egrave;">maxime</expan> aduer&shy;<lb/>
&longs;atur.  </s>
             
<s id="id.2.1.39.2.1.7.0"> <expan abbr="qu&ograve;d">quod</expan> non &longs;olum ex &longs;ecunda, &amp; tertia huius liquet; <expan abbr="ver&ugrave;m">verum</expan> <lb/>
quia exi&longs;tente centro &longs;upra libram pondus eleuatum maiorem <lb/>
propter &longs;itum acquirit grauitatem.  </s>
             
<s id="id.2.1.39.2.1.8.0"> ex <expan abbr="qu&ograve;">quo</expan> contingit redditus li&shy;<lb/>
br&aelig; ad &aelig;qualem horizonti di&longs;tantiam.  </s>
             
<s id="id.2.1.39.2.1.9.0"> <expan abbr="&egrave;">e</expan> contra <expan abbr="ver&ograve;">vero</expan>, quando <lb/>
centrum e&longs;t infra libram.  </s>
             
<s id="id.2.1.39.2.1.10.0"> Qu&aelig; omnia hoc modo o&longs;tendentur; <lb/>
&longs;upponendo ea, qu&aelig; &longs;upra declarata &longs;unt.  </s>
             
<s id="id.2.1.39.2.1.11.0"> &longs;cilicet pondus ex <expan abbr="qu&ograve;">quo</expan> <lb/>
loco rectius de&longs;cendit, grauius fieri.  </s>
             
<s id="id.2.1.39.2.1.12.0"> &amp; ex quo rectius a&longs;cendit, gra<lb/>
uius quoq; reddi. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig40" place="text">       </figure>       
<p id="id.2.1.39.2.2.1.0" type="caption">        
<s id="id.2.1.39.2.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.39.3.0.0.0" type="main">        
<s id="id.2.1.39.3.1.1.0"> Sit libra AB horizonti <lb/>
&aelig;quidi&longs;tans, cuius centrum <lb/>
C &longs;it &longs;upra libram, perpen&shy;<lb/>
diculumq; &longs;it CD. &longs;intq; in <lb/>
AB ponderum &aelig;qualium <lb/>
centra grauitatis po&longs;ita: mo<lb/>
taq; &longs;it libra in EF.  </s>
     
<s id="id.2.1.39.3.1.1.0.a"> Dico <lb/>
pondus in E maiorem ha&shy;<lb/>
bere grauitatem, <expan abbr="qu&agrave;m">quam</expan> pon<lb/>
dus in F. &amp; ob id libram <lb/>
EF in AB redire.  </s>
             
<s id="id.2.1.39.3.1.2.0"> Produ<lb/>
catur <expan abbr="prim&ugrave;m">primum</expan> CD v&longs;q; ad <lb/>
mundi <expan abbr="centr&utilde;">centrum</expan>, quod &longs;it S. de <lb/>
inde AC CB EC CF HS <lb/>
<expan abbr="c&otilde;nectantur">connectantur</expan>, <expan abbr="&agrave;">a</expan> puncti&longs;q; EF <lb/>
ip&longs;i HS &aelig;quidi&longs;tantes du<lb/>
cantur Ek GFL.  </s>
     
<s id="id.2.1.39.3.1.2.0.a"> Quoniam <lb/>
igitur naturalis de&longs;cen&longs;us re<lb/>
ctus totius magnitudinis, <lb/>
libr&aelig; &longs;cilicet EF &longs;ic con&longs;ti&shy;<lb/>
tut&aelig; <expan abbr="vn&aacute;">vna</expan> cum ponderibus, <lb/>
e&longs;t <expan abbr="&longs;cund&ugrave;m">&longs;cundum</expan> grauitatis cen<lb/>
trum H per rectam HS; erit <lb/>
<arrow.to.target n="fig41"></arrow.to.target><lb/>
quoq; ponderum in EF ita po&longs;sitorum de&longs;cen&longs;us <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> re&shy;<lb/>
ctas Ek FL ip&longs;i HS parallelas; &longs;icuti &longs;upra demon&longs;trauimus.  </s>
             
<s id="id.2.1.39.3.1.3.0"> 
<pb/>
De&longs;cen&longs;us igitur, &amp; a&longs;cen&shy;<lb/>
&longs;us ponderum in EF ma&shy;<lb/>
gis, <expan abbr="minu&longs;u&egrave;">minu&longs;ue</expan> obliquus di&shy;<lb/>
cetur <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> acce&longs;&longs;um, <lb/>
&amp; rece&longs;&longs;um iuxta lineas Ek <lb/>
FL de&longs;ignatum.  </s>
             
<s id="id.2.1.39.3.1.4.0"> <expan abbr="Quoni&atilde;">Quoniam</expan> au<lb/>
<expan abbr="t&etilde;">tem</expan> duo latera AD DC duo<lb/>
bus lateribus BD DE &longs;unt <lb/>
&aelig;qualia; anguliq; ad D &longs;unt <lb/>
<arrow.to.target n="note65"></arrow.to.target> recti; erit latus AC lateri <lb/>
CB &aelig;quale.  </s>
             
<s id="id.2.1.39.3.1.5.0"> &amp; <expan abbr="c&ugrave;m">cum</expan> pun&shy;<lb/>
ctum C &longs;it immobile; dum <lb/>
puncta AB mouentur, cir<lb/>
culi circumferentiam de&longs;cri<lb/>
bent, cuius &longs;emidiameter <lb/>
erit AC. quare centro C, <lb/>
circulus de&longs;cribatur AEBF. <lb/>
puncta AB EF in circuli <lb/>
circumferentia erunt.  </s>
             
<s id="id.2.1.39.3.1.6.0"> &longs;ed <lb/>
<expan abbr="c&ugrave;m">cum</expan> EF &longs;it ip&longs;i AB &aelig;qua <lb/>
<arrow.to.target n="note66"></arrow.to.target> lis; erit circumferentia <lb/>
EAF circumferenti&aelig; AFB <lb/>
&aelig;qualis.  </s>
             
<s id="id.2.1.39.3.1.7.0"> quare dempta <lb/>
<arrow.to.target n="fig42"></arrow.to.target><lb/>
communi AF, erit circumferentia EA circumferenti&aelig; FB &aelig;qua <lb/>
lis.  </s>
             
<s id="id.2.1.39.3.1.8.0"> Quoniam autem mixtus angulus CEA e&longs;t &aelig;qualis mixto <lb/>
CFB; &amp; HFB ip&longs;o CFB e&longs;t maior; angulus <expan abbr="ver&ograve;">vero</expan> HEA ip&longs;o <lb/>
CEA minor; erit angulus HFB angulo HEA maior.  </s>
             
<s id="id.2.1.39.3.1.9.0"> <expan abbr="&agrave;">a</expan> quibus <lb/>
<arrow.to.target n="note67"></arrow.to.target> &longs;i auferantur anguli HFG HEk &aelig;quales; erit angulus GFB an <lb/>
gulo kEA maior.  </s>
             
<s id="id.2.1.39.3.1.10.0"> ergo de&longs;cen&longs;us ponderis in E minus obliquus <lb/>
erit a&longs;cen&longs;u ponderis in F. &amp; quamquam pondus in E de&longs;cen<lb/>
dendo, &amp; pondus in F a&longs;cendendo per circumferentias mouean<lb/>
tur &aelig;quales; quia tamen pondus in E ex hoc loco rectius de&longs;cen<lb/>
dit, <expan abbr="qu&agrave;m">quam</expan> pondus in F a&longs;cendit: idcirco naturalis potentia pon<lb/>
deris in E re&longs;i&longs;tentiam violenti&aelig; ponderis F &longs;uperabit.  </s>
             
<s id="id.2.1.39.3.1.11.0"> quare <lb/>
maiorem grauitatem habebit pondus in E, <expan abbr="qu&agrave;m">quam</expan> pondus in F.  </s>
     
<s id="id.2.1.39.3.1.11.0.a"> <lb/>
ergo pondus in E deor&longs;um, pondus <expan abbr="ver&ograve;">vero</expan> in F &longs;ur&longs;um mouebitur: 
<pb n="24"/>
donec libra EF in AB redeat. quod demon&longs;trare oportebat. </s>
             
<s id="id.2.1.39.3.1.12.0"> [quod demon&longs;trare oportebat.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig41" place="text">       </figure>       
<p id="id.2.1.39.3.2.1.0" type="caption">        
<s id="id.2.1.39.3.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig42" place="text">       </figure>       
<p id="id.2.1.39.3.2.3.0" type="caption">        
<s id="id.2.1.39.3.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.40.1.0.0.0" type="margin">        
<s id="id.2.1.40.1.1.1.0"> <margin.target id="note65"></margin.target>4 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.40.1.1.2.0"> <margin.target id="note66"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 28 <emph type="italics"/>Ter tii.<emph.end type="italics"/> </s>
             
<s id="id.2.1.40.1.1.3.0"> <margin.target id="note67"></margin.target>29 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.41.1.0.0.0" type="main">        
<s id="id.2.1.41.1.1.1.0"> Huius autem effectus ratio ab Ari&longs;totele po&longs;ita, hic manife&longs;ta in <arrow.to.target n="note68"></arrow.to.target><lb/>
tueri pote&longs;t.  </s>
             
<s id="id.2.1.41.1.1.2.0"> &longs;it enim punctum N vbi CS EF &longs;e inuicem &longs;ecant.  </s>
             
<s id="id.2.1.41.1.1.3.0"> <lb/>
&amp; quoniam HE e&longs;t ip&longs;i HF &aelig;qualis; erit NE maior NF. li&shy;<lb/>
nea ergo CS, quam perpendiculum vocat, libram EF in partes di <lb/>
uidet in&aelig;quales.  </s>
             
<s id="id.2.1.41.1.1.4.0"> <expan abbr="c&ugrave;m">cum</expan> itaq; pars libr&aelig; NE &longs;it maior NF; atq; id, <lb/>
quod plus e&longs;t, nece&longs;&longs;e e&longs;t, deor&longs;um ferri: libra ergo EF ex parte E <lb/>
deor&longs;um mouebitur, donec in AB redeat. </s>
    </p>       
<p id="id.2.1.42.1.0.0.0" type="margin">        
<s id="id.2.1.42.1.1.1.0"> <margin.target id="note68"></margin.target><emph type="italics"/>Ari&longs;totelis ratio.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.43.1.0.0.0" type="main">        
<s id="id.2.1.43.1.1.1.0"> Ex iis pr&aelig;terea, qu&aelig; ha<lb/>
ctenus dicta &longs;unt inferre li<lb/>
cet, libram EF velocius ab <lb/>
eo &longs;itu in AB moueri; <expan abbr="vnd&egrave;">vnde</expan> <lb/>
linea EF in directum pro&shy;<lb/>
tracta in centrum mundi <lb/>
perueniat.  </s>
             
<s id="id.2.1.43.1.1.2.0"> vt &longs;it EFS recta <lb/>
linea.  </s>
             
<s id="id.2.1.43.1.1.3.0"> &amp; quoniam CD <lb/>
CH, &longs;unt inter &longs;e &longs;e &aelig;qua<lb/>
les.  </s>
             
<s id="id.2.1.43.1.1.4.0"> &longs;i igitur centro C, &longs;pa<lb/>
tioq; CD, circulus de&longs;cri&shy;<lb/>
batur DHM; erunt pun&shy;<lb/>
cta DH in circuli circum&shy;<lb/>
ferentia.  </s>
             
<s id="id.2.1.43.1.1.5.0"> Quoniam au&shy;<lb/>
tem CH ip&longs;i EF e&longs;t per&shy;<lb/>
pendicularis; continget li&shy;<lb/>
nea EHS circulum DHM <lb/>
in puncto H.  </s>
     
<s id="id.2.1.43.1.1.5.0.a"> pondus igi&shy;<lb/>
tur in H (&longs;icuti &longs;upra de&shy;<lb/>
mon&longs;trauimus) grauius <lb/>
<arrow.to.target n="fig43"></arrow.to.target><lb/>
erit, <expan abbr="qu&agrave;m">quam</expan> in alio &longs;itu circuli DHM.  </s>
     
<s id="id.2.1.43.1.1.5.0.b"> ergo magnitudo ex EF <lb/>
ponderibus, &amp; libra EF compo&longs;ita, cuius centrum grauitatis e&longs;t <lb/>
in H, in hoc &longs;itu magis grauitabit, <expan abbr="qu&agrave;m">quam</expan> in quocunq; alio &longs;itu 
<pb/>
circuli fuerit punctum H. <lb/>
ab hoc igitur &longs;itu velo&shy;<lb/>
cius, <expan abbr="qu&agrave;m">quam</expan> <expan abbr="&agrave;">a</expan> quocunq; <lb/>
alio mouebitur.  </s>
             
<s id="id.2.1.43.1.1.6.0"> &amp; &longs;i H <lb/>
propius fuerit ip&longs;i D mi <lb/>
nus grauitabit, minu&longs;q; <lb/>
ab eo &longs;itu mouebitur.  </s>
             
<s id="id.2.1.43.1.1.7.0"> <lb/>
&longs;emper enim de&longs;cen&longs;us <lb/>
obliquior e&longs;t, &amp; minus re<lb/>
ctus.  </s>
             
<s id="id.2.1.43.1.1.8.0"> libra ergo EF velo<lb/>
cius ab hoc &longs;itu mouebi&shy;<lb/>
tur, <expan abbr="qu&agrave;m">quam</expan> ab alio &longs;itu.  </s>
             
<s id="id.2.1.43.1.1.9.0"> &amp; <lb/>
&longs;i propius ad AB acce&shy;<lb/>
det, inde minus mouebi<lb/>
tur.  </s>
             
<s id="id.2.1.43.1.1.10.0"> Deinde <expan abbr="qu&ograve;">quo</expan> longius <lb/>
punctum H <expan abbr="&agrave;">a</expan> puncto C <lb/>
di&longs;tabit, velocius moue&shy;<lb/>
bitur; quod <expan abbr="n&otilde;">non</expan> <expan abbr="&longs;ol&utilde;">&longs;olum</expan> ex Ari<lb/>
&longs;totele in principio qu&aelig;&longs;t&shy;<lb/>
io num mechanicarum, &amp; <lb/>
<arrow.to.target n="fig44"></arrow.to.target><lb/>
ex &longs;uperius dictis patet; <expan abbr="ver&ugrave;m">verum</expan> etiam ex iis, qu&aelig; infra in &longs;exta <lb/>
propo&longs;itione dicemus, manife&longs;tum erit.  </s>
             
<s id="id.2.1.43.1.1.11.0"> libra igitur EF, <expan abbr="qu&ograve;">quo</expan> ma<lb/>
gis ab eius centro di&longs;tabit, adhuc velocius mouebitur.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig43" place="text">       </figure>       
<p id="id.2.1.43.1.2.1.0" type="caption">        
<s id="id.2.1.43.1.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig44" place="text">       </figure>       
<p id="id.2.1.43.1.2.3.0" type="caption">        
<s id="id.2.1.43.1.2.3.0.capt"> YYY </s>
    </p>       
<pb n="25"/>
       
<p id="id.2.1.43.3.0.0.0" type="main">        
<s id="id.2.1.43.3.1.1.0"> Sit deinde libra AB, <lb/>
cuius centrum C &longs;it infra li<lb/>
bram; &longs;intq; in AB pon<lb/>
dera &aelig;qualia; libraq; &longs;it <lb/>
mota in EF.  </s>
     
<s id="id.2.1.43.3.1.1.0.a"> Dico maio&shy;<lb/>
rem habere grauitatem <lb/>
pondus in F, <expan abbr="qu&agrave;m">quam</expan> pondus <lb/>
in E. atq; ideo libram EF <lb/>
deor&longs;um ex parte F moue&shy;<lb/>
ri.  </s>
             
<s id="id.2.1.43.3.1.2.0"> Producatur DC ex <lb/>
vtraq; parte v&longs;q; ad mun&shy;<lb/>
di centrum S, &amp; v&longs;q; ad <lb/>
O, lineaq; HS ducatur, <lb/>
cui <expan abbr="&agrave;">a</expan> punctis EF &aelig;quidi&shy;<lb/>
&longs;tantes ducantur GEk FL; <lb/>
connectanturq; CE CF: <lb/>
atq; centro C, &longs;patioq; CE <lb/>
circulus de&longs;cribatur AEO <lb/>
BF.  </s>
     
<s id="id.2.1.43.3.1.2.0.a"> &longs;imiliter demon&longs;tra&shy;<lb/>
bitur puncta ABEF in <lb/>
circuli circumferentia e&longs;&longs;e; <lb/>
de&longs;cen&longs;umq; libr&aelig; EF <expan abbr="vn&aacute;">vna</expan> <lb/>
cum ponderibus rectum &longs;e<lb/>
<expan abbr="cund&ugrave;m">cundum</expan> lineam HS fieri; <lb/>
ponderumq; in EF &longs;ecun <lb/>
<arrow.to.target n="fig45"></arrow.to.target><expan abbr="d&ugrave;m"><lb/>
dum</expan> lineas GK FL ip&longs;i HS &aelig;quidi&longs;tantes.  </s>
             
<s id="id.2.1.43.3.1.3.0"> Quoniam autem an<lb/>
gulus CFP &aelig;qualis e&longs;t angulo CEO: erit angulus HFP angulo <lb/>
HEO maior.  </s>
             
<s id="id.2.1.43.3.1.4.0"> angulus <expan abbr="ver&ograve;">vero</expan> HFL &aelig;qualis e&longs;t angulo HEG. <expan abbr="&agrave;">a</expan> <arrow.to.target n="note69"></arrow.to.target><lb/>
quibus igitur &longs;i demantur anguli HFP HEO, erit angulus <lb/>
LFP angulo GEO minor.  </s>
             
<s id="id.2.1.43.3.1.5.0"> quare de&longs;cen&longs;us ponderis in F rectior <lb/>
erit a&longs;cen&longs;u ponderis in E. ergo naturalis potentia ponderis in <lb/>
F re&longs;i&longs;tentiam violenti&aelig; ponderis in E &longs;uperabit.  </s>
             
<s id="id.2.1.43.3.1.6.0"> &amp; ideo ma&shy;<lb/>
iorem habebit grauitatem pondus in F, <expan abbr="qu&agrave;m">quam</expan> pondus in E.  </s>
     
<s id="id.2.1.43.3.1.6.0.a"> <lb/>
Pondus igitur in F deor&longs;um, pondus <expan abbr="ver&ograve;">vero</expan> in E &longs;ur&longs;um mo&shy;<lb/>
uebitur. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig45" place="text">       </figure>       
<p id="id.2.1.43.3.2.1.0" type="caption">        
<s id="id.2.1.43.3.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.44.1.0.0.0" type="margin">        
<s id="id.2.1.44.1.1.1.0"> <margin.target id="note69"></margin.target>29 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.45.1.0.0.0" type="main">        
<s id="id.2.1.45.1.1.1.0"> Ari&longs;totelis quoq; ratio hic per&longs;picua erit.  </s>
             
<s id="id.2.1.45.1.1.2.0"> &longs;it enim punctum <arrow.to.target n="note70"></arrow.to.target>
<pb/>
N vbi CO EF &longs;e inuicem <lb/>
&longs;ecant; erit NF maior <lb/>
NE.  </s>
     
<s id="id.2.1.45.1.1.2.0.a"> &amp; quoniam CO per <lb/>
pendiculum (&longs;ecund&ugrave;m <lb/>
ip&longs;um) libram EF in par <lb/>
tes in&aelig;quales diuidit, &amp; <lb/>
maior pars e&longs;t ver&longs;us F, hoc <lb/>
e&longs;t NF; libra EF ex par <lb/>
te F deor&longs;um mouebitur: <lb/>
<expan abbr="c&ugrave;mid">cumid</expan>, quod plus e&longs;t, deor<lb/>
&longs;um feratur. </s>
    </p>       
<p id="id.2.1.46.1.0.0.0" type="margin">        
<s id="id.2.1.46.1.1.1.0"> <margin.target id="note70"></margin.target><emph type="italics"/>Ari&longs;totelis ratio.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.47.1.0.0.0" type="main">        
<s id="id.2.1.47.1.1.1.0"> Similiter, <expan abbr="&eacute;x">ex</expan> dictis <lb/>
quoq; eliciemus libram EF <lb/>
centrum habens infra li&shy;<lb/>
bram, <expan abbr="qu&ograve;">quo</expan> magis <expan abbr="&agrave;">a</expan> &longs;itu <lb/>
AB di&longs;tabit, velocius mo <lb/>
ueri.  </s>
             
<s id="id.2.1.47.1.1.2.0"> centrum enim graui <lb/>
tatis H, <expan abbr="qu&ograve;">quo</expan> magis <expan abbr="&aacute;">a</expan> pun&shy;<lb/>
cto D di&longs;tat, <expan abbr="e&ograve;">eo</expan> volecius <lb/>
pondus ex EF ponderibus, <lb/>
libraq; EF compo&longs;itum <lb/>
mouebitur, donec angulus <lb/>
CHS rectus euadat.  </s>
             
<s id="id.2.1.47.1.1.3.0"> ad&shy;<lb/>
huc in&longs;uper velocius moue<lb/>
bitur, <expan abbr="qu&ograve;">quo</expan> libram <expan abbr="&agrave;">a</expan> centro <lb/>
C magis di&longs;tabit. <arrow.to.target n="fig46"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.47.2.0.0.0" type="main">        
<s id="id.2.1.47.2.1.1.0"> Ex ip&longs;orum quinetiam rationibus, ac fal&longs;is &longs;upo&longs;itionibus iam <lb/>
declaratos libr&aelig; effectus, ac motus deducere, ac manife&longs;tare libet; <lb/>
vt quanta &longs;it veritatis efficacia appareat, <expan abbr="quipp&egrave;">quippe</expan> ex fal&longs;is etiam <lb/>
eluce&longs;cere contendit.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig46" place="text">       </figure>       
<p id="id.2.1.47.2.2.1.0" type="caption">        
<s id="id.2.1.47.2.2.1.0.capt"> YYY </s>
    </p>       
<pb n="26"/>
       
<p id="id.2.1.47.4.0.0.0" type="main">        
<s id="id.2.1.47.4.1.1.0"> Exponantur eadem, &longs;ci <lb/>
licet &longs;it circulus AEBF; <lb/>
<expan abbr="libraqu&eacute;">libraque</expan> AB, cuius cen&shy;<lb/>
trum C &longs;it &longs;upra libram, <lb/>
moueatur in EF.  </s>
     
<s id="id.2.1.47.4.1.1.0.a"> dico <lb/>
pondus in E maiorem ibi <lb/>
habere grauitatem, <expan abbr="qu&agrave;m">quam</expan> <lb/>
pondus in F; libramq; EF <lb/>
in AB redire.  </s>
             
<s id="id.2.1.47.4.1.2.0"> Ducantur <lb/>
<expan abbr="&agrave;">a</expan> punctis EF ip&longs;i AB <lb/>
perpendiculares EL FM, <lb/>
qu&aelig; inter &longs;e &aelig;quidi&longs;tan&shy;<lb/>
tes  <arrow.to.target n="note71"></arrow.to.target><arrow.to.target n="fig47"></arrow.to.target> erunt; &longs;itq; punctum N, vbi AB EF &longs;e inuicem &longs;ecant.  </s>
             
<s id="id.2.1.47.4.1.3.0"> <lb/>
Quoniam igitur angulus FNM e&longs;t &aelig;qualis angulo ENL, &amp; an&shy;<lb/>
gulus  <arrow.to.target n="note72"></arrow.to.target> F MN rectus recto ELN &aelig;qualis, ac reliquus NFM reli&shy;<lb/>
quo  <arrow.to.target n="note73"></arrow.to.target> NEL e&longs;t etiam &aelig;qualis; erit triangulum NLE triangu<lb/>
lo NMF &longs;imile.  </s>
             
<s id="id.2.1.47.4.1.4.0"> vt igitur NE ad EL, ita NF ad FM; &amp; per <arrow.to.target n="note74"></arrow.to.target><lb/>
mutando vt EN ad NF, ita EL ad FM. &longs;ed <expan abbr="c&ugrave;m">cum</expan> &longs;it HE ip&longs;i <arrow.to.target n="note75"></arrow.to.target><lb/>
HF &aelig;qualis, erit EN maior NF; quare &amp; EL maior erit FM.  </s>
     
<s id="id.2.1.47.4.1.4.0.a"> <lb/>
&amp; quoniam dum pondus in E per circumferentiiam EA de&longs;cendit, <lb/>
pondus in F per circumferentiam FB ip&longs;i circumferenti&aelig; EA <lb/>
&aelig;qualem a&longs;cendit; de&longs;cen&longs;u&longs;q; ponderis in E de directo (vt ip&shy;<lb/>
&longs;i dicunt) capit EL: a&longs;cen&longs;us <expan abbr="ver&ograve;">vero</expan> ponderis in F de directo ca&shy;<lb/>
pit FM; minus de directo capiet a&longs;cen&longs;us ponderis in F, <expan abbr="qu&agrave;m">quam</expan> <lb/>
de&longs;cen&longs;us ponderis in E. maiorem igitur grauitatem habebit pon<lb/>
dus in E, <expan abbr="qu&agrave;m">quam</expan> pondus in F. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig47" place="text">       </figure>       
<p id="id.2.1.47.4.2.1.0" type="caption">        
<s id="id.2.1.47.4.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.48.1.0.0.0" type="margin">        
<s id="id.2.1.48.1.1.1.0"> <margin.target id="note71"></margin.target>28 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.48.1.1.2.0"> <margin.target id="note72"></margin.target>15 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.48.1.1.3.0"> <margin.target id="note73"></margin.target>29 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.48.1.1.4.0"> <margin.target id="note74"></margin.target>4 <emph type="italics"/>Sexti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.48.1.1.5.0"> <margin.target id="note75"></margin.target>16 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.49.1.0.0.0" type="main">        
<s id="id.2.1.49.1.1.1.0"> Producatur CD ex vtraq; parte in OP, qu&aelig; lineam EF in <lb/>
puncto S &longs;ecet.  </s>
             
<s id="id.2.1.49.1.1.2.0"> &amp; quoniam (vt aiunt) <expan abbr="qu&ograve;">quo</expan> magis pondus <expan abbr="&agrave;">a</expan> li&shy;<lb/>
nea directionis OP di&longs;tat, <expan abbr="e&ograve;">eo</expan> fit grauius; idcirco hoc quoq; me <lb/>
dio pondus in E maiorem habere grauitauitatem pondere in F o&shy;<lb/>
&longs;tendetur.  </s>
             
<s id="id.2.1.49.1.1.3.0"> Ducantur <expan abbr="&agrave;">a</expan> punctis EF ip&longs;i OP perpendiculares EQ <lb/>
FR. &longs;imili ratione o&longs;tendetur, triangulum QES triangulo RFS <lb/>
&longs;imile e&longs;&longs;e; lineamq; EQ ip&longs;a RF maiorem e&longs;&longs;e.  </s>
             
<s id="id.2.1.49.1.1.4.0"> pondus itaq; <lb/>
in E magis <expan abbr="&agrave;">a</expan> linea OP di&longs;tabit, <expan abbr="qu&agrave;m">quam</expan> pondus in F; ac propterea <lb/>
pondus in E maiorem habebit grauitatem pondere in F. ex quibus <lb/>
reditus libr&aelig; EF in AB manife&longs;tus apparet.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.49.3.0.0.0" type="main">        
<s id="id.2.1.49.3.1.1.0"> Si autem centrum libr&aelig; <lb/>
&longs;it infra libram, tunc pon&shy;<lb/>
dus depre&longs;&longs;um maiorem <lb/>
habere grauitatem eleuato <lb/>
ii&longs;dem mediis o&longs;tendetur.  </s>
             
<s id="id.2.1.49.3.1.2.0"> <lb/>
ducantur <expan abbr="&agrave;">a</expan> punctis EF ip&shy;<lb/>
&longs;i AB perpendiculares EL <lb/>
FM. &longs;imiliter demon&longs;tra<lb/>
bitur EL maiorem e&longs;&longs;e <lb/>
FM; &amp; ob id de&longs;cen&longs;us <lb/>
ponderis in F minus de di <lb/>
recto capiet, <expan abbr="qu&agrave;m">quam</expan> a&longs;cen&shy;<lb/>
<arrow.to.target n="fig48"></arrow.to.target><lb/>
&longs;us ponderis in E: quocirca re&longs;i&longs;tentia violenti&aelig; ponderis in E &longs;u<lb/>
perabit naturalem propen&longs;ionem ponderis in F. ergo pondus in E <lb/>
pondere in F grauius erit. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig48" place="text">       </figure>       
<p id="id.2.1.49.3.2.1.0" type="caption">        
<s id="id.2.1.49.3.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.49.4.0.0.0" type="main">        
<s id="id.2.1.49.4.1.1.0"> Producatur etiam CD ex vtraq; parte in OP; ip&longs;iq; <expan abbr="&agrave;">a</expan> punctis <lb/>
EF perpendiculares ducantur EQ FR. eodem pror&longs;us modo <lb/>
o&longs;tendetur, lineam EQ maiorem e&longs;&longs;e FR. pondus <expan abbr="ide&ograve;">ideo</expan> in E ma<lb/>
gis <expan abbr="&agrave;">a</expan> linea directionis OP di&longs;tabit, <expan abbr="qu&agrave;m">quam</expan> pondus in F. maio&shy;<lb/>
rem igitur grauitatem habebit pondus in E, <expan abbr="qu&agrave;m">quam</expan> pondus in F. <lb/>
ex quibus &longs;equitur, libram EF ex parte E deor&longs;um moueri. </s>
    </p>       
<p id="id.2.1.49.5.0.0.0" type="main">        
<s id="id.2.1.49.5.1.1.0"> Ari&longs;toteles itaq; has duas <expan abbr="tant&ugrave;m">tantum</expan> qu&aelig;&longs;tiones propo&longs;uit, ter&shy;<lb/>
tiamq; reliquit; &longs;cilicet <expan abbr="c&ugrave;m">cum</expan> centrum libr&aelig; in ip&longs;a e&longs;t libra: hanc <lb/>
autem ommi&longs;sit, vt notam, quemadmodum res valde notas pr&aelig;&shy;<lb/>
termittere &longs;olet.  </s>
             
<s id="id.2.1.49.5.1.2.0"> nam cui dubium, &longs;i pondus in eius centro gra<lb/>
uitatis &longs;u&longs;tineatur, quin maneat?  </s>
             
<s id="id.2.1.49.5.1.3.0"> Ea <expan abbr="ver&ograve;">vero</expan>, qu&aelig; ex ip&longs;ius &longs;enten<lb/>
tia attulimus, aliquis reprehendere po&longs;&longs;et, nos integram eius &longs;enten<lb/>
tiam <expan abbr="minim&egrave;">minime</expan> protuli&longs;&longs;e affimans.  </s>
             
<s id="id.2.1.49.5.1.4.0"> nam <expan abbr="c&ugrave;m">cum</expan> in &longs;ecunda parte &longs;e<lb/>
cund&aelig; qu&aelig;&longs;tionis proponit, cur libra, trutina deor&longs;um con&longs;tituta, <lb/>
quando deor&longs;um lato pondere qui&longs;piam id amouet, non a&longs;cen<lb/>
dit, &longs;ed manet?  </s>
             
<s id="id.2.1.49.5.1.5.0"> non a&longs;&longs;erit adhuc libram deor&longs;um moueri; &longs;ed <lb/>
manere.  </s>
             
<s id="id.2.1.49.5.1.6.0"> quod in vltima quoq; conclu&longs;ione colligi&longs;&longs;e videtur.  </s>
             
<s id="id.2.1.49.5.1.7.0"> Ve <lb/>
<expan abbr="r&ugrave;m">rum</expan> hoc non &longs;olum nobis non repugnat, &longs;ed &longs;i <expan abbr="rect&egrave;">recte</expan> intelligitur, <lb/>
<expan abbr="maxim&egrave;">maxime</expan> &longs;uffragatur.  </s>
    </p>       
<pb n="27"/>
       
<p id="id.2.1.49.7.0.0.0" type="main">        
<s id="id.2.1.49.7.1.1.0"> Sit enim libra AB <lb/>
horizonti &aelig;quidi&longs;tans, <lb/>
cuius centrum E &longs;it <lb/>
infra libram.  </s>
             
<s id="id.2.1.49.7.1.2.0"> quia ve <lb/>
<expan abbr="r&ograve;">ro</expan> Ari&longs;toteles libram, <lb/>
&longs;icuti actu e&longs;t, con&longs;ide<lb/>
rat; <expan abbr="ide&ograve;">ideo</expan> nece&longs;&longs;e e&longs;t <lb/>
trutinam, vel aliquid <lb/>
aliud infra centrum E <lb/>
collocare, vt EF <lb/>
(quod quidem truti&shy;<lb/>
na erit) ita vt centrum <lb/>
E &longs;u&longs;tineat.  </s>
             
<s id="id.2.1.49.7.1.3.0"> &longs;itq; per&shy;<lb/>
<arrow.to.target n="fig49"></arrow.to.target><lb/>
pendiculum ECD. &amp; vt libra AB ab hoc moueatur &longs;itu; dicit <lb/>
Ari&longs;toteles, ponatur pondus in B, quod <expan abbr="c&ugrave;m">cum</expan> &longs;it graue, libram ex <lb/>
parte B deor&longs;um mouebit; <expan abbr="put&aacute;">puta</expan> in G. ita vt propter impedimen<lb/>
tum deor&longs;um amplius moueri non poterit.  </s>
             
<s id="id.2.1.49.7.1.4.0"> non enim dicit Ari<lb/>
&longs;toteles, moueatur libra ex parte B deor&longs;um, quou&longs;q; libuerit; dein <lb/>
de relinquatur, vt nos diximus: &longs;ed pr&aelig;cipit, vt in ip&longs;o B po&shy;<lb/>
natur pondus, quod ex ip&longs;ius natura deor&longs;um &longs;emper mouebi&shy;<lb/>
tur; donec libra trutin&aelig;, &longs;iue alicui alii adh&aelig;reat.  </s>
             
<s id="id.2.1.49.7.1.5.0"> &amp; quando B erit <lb/>
in G, erit libra in GH; in quo &longs;itu, ablato pondere, manebit: <lb/>
<expan abbr="c&ugrave;m">cum</expan> maior pars libr&aelig; <expan abbr="&agrave;">a</expan> perpendiculo &longs;it ver&longs;us G, qu&aelig; e&longs;t DG, <lb/>
<expan abbr="qu&agrave;m">quam</expan> DH.  </s>
     
<s id="id.2.1.49.7.1.5.0.a"> nec deor&longs;um amplius mouebitur; nam libra, vel <lb/>
trutin&aelig;, vel alteri cuipiam, quod centrum libr&aelig; &longs;u&longs;tineat, incum<lb/>
bet.  </s>
             
<s id="id.2.1.49.7.1.6.0"> &longs;i enim huic non adh&aelig;reret, libra ex parte G deor&longs;um ex <lb/>
ip&longs;ius &longs;ententia moueretur; <expan abbr="c&ugrave;m">cum</expan> id, quod plus e&longs;t, &longs;cilicet DG, <lb/>
deor&longs;um ferri &longs;it nece&longs;&longs;e. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig49" place="text">       </figure>       
<p id="id.2.1.49.7.2.1.0" type="caption">        
<s id="id.2.1.49.7.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.49.8.0.0.0" type="main">        
<s id="id.2.1.49.8.1.1.0"> C&aelig;terum quis adhuc dicere poterit, &longs;i paruum imponatur pon<lb/>
dus in B, mouebitur quidem libra deor&longs;um, non autem v&longs;q; ad <lb/>
G. in <expan abbr="qu&ograve;">quo</expan> &longs;itu <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> Ari&longs;totelem, ablato pondere, mane&shy;<lb/>
re deberet.  </s>
             
<s id="id.2.1.49.8.1.2.0"> quod experimento patet; <expan abbr="c&ugrave;m">cum</expan> in vna <expan abbr="tant&ugrave;m">tantum</expan> libr&aelig; <lb/>
extremitate, impo&longs;ito onere, hocq; vel maiore, vel minore, libra <lb/>
plus, <expan abbr="minu&longs;u&egrave;">minu&longs;ue</expan> inclinetur.  </s>
             
<s id="id.2.1.49.8.1.3.0"> Quod e&longs;t quidem veri&longs;&longs;imum, centro &longs;upra <lb/>
libram, non autem infra, neq; in ip&longs;a libra collocato.  </s>
             
<s id="id.2.1.49.8.1.4.0"> Vt exempli <lb/>
gratia.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.49.10.0.0.0" type="main">        
<s id="id.2.1.49.10.1.1.0"> Sit libra horizonti &aelig;&shy;<lb/>
quidi&longs;tans AB, cuius cen<lb/>
trum C &longs;it &longs;upra libram, <lb/>
perpendiculumq; CD ho<lb/>
rizonti perpendiculare, <lb/>
quod ex parte D produca<lb/>
tur in H.  </s>
     
<s id="id.2.1.49.10.1.1.0.a"> Quoniam enim <lb/>
con&longs;iderata libr&aelig; grauita&shy;<lb/>
te, erit punctum D libr&aelig; <lb/>
centrum grauitatis.  </s>
             
<s id="id.2.1.49.10.1.2.0"> &longs;i ergo <lb/>
in B paruum imponatur <lb/>
pondus, cuius centrum <lb/>
<arrow.to.target n="fig50"></arrow.to.target><lb/>
grauitatis &longs;it in puncto B; magnitudinis ex libra AB, &amp; pondere <lb/>
in B compo&longs;it&aelig; non erit amplius centrum grauitatis D; &longs;ed erit in <lb/>
<arrow.to.target n="note76"></arrow.to.target> linea DB, vt in E: ita vt DE ad EB &longs;it, vt pondus in B ad gra&shy;<lb/>
uitatem libr&aelig; AB. Connectatur CE.  </s>
     
<s id="id.2.1.49.10.1.2.0.a"> Quoniam autem pun&shy;<lb/>
ctum Ce&longs;t immobile, dum libra mouetur, punctum E circuli cir<lb/>
cumferentiam EFG de&longs;cribet, cuius &longs;emidiameter CE, &amp; cen&shy;<lb/>
trum C. quia <expan abbr="ver&ograve;">vero</expan> CD horizonti e&longs;t perpendicularis, linea CE <lb/>
horizonti perpendicularis nequaquam erit.  </s>
             
<s id="id.2.1.49.10.1.3.0"> quare magnitudo ex <lb/>
AB, &amp; pondere in B compo&longs;ita <expan abbr="minim&egrave;">minime</expan> in hoc &longs;itu manebit; &longs;ed <lb/>
<arrow.to.target n="note77"></arrow.to.target> deor&longs;um <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> eius grauitatis centrum E per circumferen&shy;<lb/>
tiam EFG mouebitur; donec CE horizonti perpendicularis eua<lb/>
dat; hoc e&longs;t, donec CE in CDF perueniat.  </s>
             
<s id="id.2.1.49.10.1.4.0"> atq; tunc libra AB <lb/>
mota erit in kL, in quo &longs;itu libra <expan abbr="vn&aacute;">vna</expan> cum pondere manebit.  </s>
             
<s id="id.2.1.49.10.1.5.0"> nec <lb/>
deor&longs;um amplius mouebitur.  </s>
             
<s id="id.2.1.49.10.1.6.0"> Si <expan abbr="ver&ograve;">vero</expan> in B ponatur pondus graui&shy;<lb/>
us; centrum grauitatis totius magnitudinis erit ip&longs;i B propius, vt in <lb/>
M. &amp; tunc libra deor&longs;um, donec iuncta CM in linea CDH per <lb/>
ueniat, mouebitur.  </s>
             
<s id="id.2.1.49.10.1.7.0"> Ex maiore igitur, &amp; minore pondere in B po<lb/>
&longs;ito, libra plus, <expan abbr="minu&longs;u&egrave;">minu&longs;ue</expan> inclinabitur.  </s>
             
<s id="id.2.1.49.10.1.8.0"> ex quo &longs;equitur pondus B <lb/>
quarta circuli parte minorem &longs;emper circumferentiam de&longs;cribe&shy;<lb/>
re, <expan abbr="c&ugrave;m">cum</expan> angulus FCE &longs;it &longs;emper acutus.  </s>
             
<s id="id.2.1.49.10.1.9.0"> nunquam enim punctum <lb/>
B v&longs;q; ad lineam CH perueniet, <expan abbr="c&ugrave;m">cum</expan> centrum grauitatis ponde&shy;<lb/>
ris, &amp; libr&aelig; &longs;imul &longs;emper inter DB exi&longs;tat.  </s>
             
<s id="id.2.1.49.10.1.10.0"> <expan abbr="qu&ograve;">quo</expan> tamen pondus <lb/>
in B grauius fuerit, maiorem quoq; circumferentiam de&longs;cribet.  </s>
             
<s id="id.2.1.49.10.1.11.0"> <lb/>
<expan abbr="e&ograve;">eo</expan> enim magis punctum B ad lineam CH accedet.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig50" place="text">       </figure>       
<p id="id.2.1.49.10.2.1.0" type="caption">        
<s id="id.2.1.49.10.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.50.1.0.0.0" type="margin">        
<s id="id.2.1.50.1.1.1.0"> <margin.target id="note76"></margin.target>6 <emph type="italics"/>Primi Ar chim. de &aelig;quep.<emph.end type="italics"/>  </s>
             
<s id="id.2.1.50.1.1.2.0"> [de &aelig;quep.<emph.end type="italics"/>] </s>
             
<s id="id.2.1.50.1.1.3.0"> <margin.target id="note77"></margin.target>1. <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.51.1.0.0.0" type="main">        
<pb n="28"/>
       
<s id="id.2.1.51.1.2.1.0"> Habeat autem libra AB <lb/>
centrum C in ip&longs;a libra, atq; <lb/>
in eius medio: erit C libr&aelig; <lb/>
centrum quoq; grauitatis; <lb/>
<expan abbr="&agrave;">a</expan> quo ip&longs;i AB, horizontiq; <lb/>
perpendicularis ducatur FC <lb/>
G. ponatur deinde in B <lb/>
quoduis pondus; erit totius <lb/>
magnitudinis centrum gra&shy;<lb/>
uitatis <expan abbr="put&aacute;">puta</expan> in E; ita vt CE <lb/>
<arrow.to.target n="fig51"></arrow.to.target><lb/>
ad EB &longs;it, vt pondus in B ad libr&aelig; grauitatem.  </s>
             
<s id="id.2.1.51.1.2.2.0"> &amp; quoniam CE <lb/>
non e&longs;t horizonti perpendicularis, libra AB, atq; pondus in B <lb/>
in hoc &longs;itu nunquam manebunt; &longs;ed deor&longs;um ex parte B mouebun<lb/>
tur, donec CE horizonti fiat perpendicularis.  </s>
             
<s id="id.2.1.51.1.2.3.0"> hoc e&longs;t donec li&shy;<lb/>
bra AB in FG perueniat.  </s>
             
<s id="id.2.1.51.1.2.4.0"> ex quo patet, quolibet pondus in B <lb/>
circuli quartam &longs;emper de&longs;cribere. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig51" place="text">       </figure>       
<p id="id.2.1.51.1.3.1.0" type="caption">        
<s id="id.2.1.51.1.3.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.51.2.0.0.0" type="main">        
<s id="id.2.1.51.2.1.1.0"> Sit autem centrum Cin&shy;<lb/>
fra libram AB. &longs;itq; DCE <lb/>
perpendiculum.  </s>
             
<s id="id.2.1.51.2.1.2.0"> &longs;imiliter <lb/>
po&longs;ito in B pondere, cen&shy;<lb/>
trum grauitatis magnitudi<lb/>
nis ex AB libra, &amp; ponde<lb/>
re in B compo&longs;it&aelig; in linea <lb/>
DB erit; vt in F; ita vt DF <lb/>
ad FB &longs;it, vt pondus in B <lb/>
<arrow.to.target n="fig52"></arrow.to.target><lb/>
ad libr&aelig; pondus.  </s>
             
<s id="id.2.1.51.2.1.3.0"> Iungatur CF. &amp; quoniam CD horizonti e&longs;t <lb/>
perpendicularis; linea CF horizonti nequaquam perpendicula&shy;<lb/>
ris exi&longs;tet.  </s>
             
<s id="id.2.1.51.2.1.4.0"> quare magnitudo ex AB libra, ac pondere in B com<lb/>
po&longs;ita in hoc &longs;itu nunquam per&longs;i&longs;tet; &longs;ed deor&longs;um, ni&longs;i aliquid <lb/>
impediat, mouebitur; donec CF in DCE perueniat: in quo &longs;itu <lb/>
libra <expan abbr="vn&aacute;">vna</expan> cum pondere manebit.  </s>
             
<s id="id.2.1.51.2.1.5.0"> &amp; punctum B erit vt in G, atq; <lb/>
punctum A in H, libraq; GH non amplius centrum infra, &longs;ed &longs;u<lb/>
pra ip&longs;am habebit.  </s>
             
<s id="id.2.1.51.2.1.6.0"> quod idem &longs;emper eueniet; quamuis mini&shy;<lb/>
mum imponatur pondus in B. ergo priu&longs;quam B perueniat ad <lb/>
G; nece&longs;&longs;e e&longs;t libram, &longs;iue trutin&aelig; deor&longs;um po&longs;it&aelig;, vel alicui 
<pb/>
alteri, quod centrum C &longs;u&shy;<lb/>
&longs;tineat, occurrere; ibiq; ad&shy;<lb/>
h&aelig;rere.  </s>
             
<s id="id.2.1.51.2.1.7.0"> ex hoc &longs;equitur, pon<lb/>
dus in B vltra lineam Dk <lb/>
&longs;emper moueri; ac circuli <lb/>
quarta maiorem &longs;emper cir<lb/>
<expan abbr="cumfer&etilde;tiam">cumferentiam</expan> de&longs;cribere: e&longs;t <lb/>
enim angulus FCE &longs;emper <lb/>
obtu&longs;us, <expan abbr="c&ugrave;m">cum</expan> angulus DCF <lb/>
&longs;emper &longs;it acutus.  </s>
             
<s id="id.2.1.51.2.1.8.0"> <expan abbr="qu&ograve;">quo</expan> au&shy;<lb/>
<arrow.to.target n="fig53"></arrow.to.target><lb/>
tem pondus in B fuerit leuius, maiorem tamen adhuc circumfe&shy;<lb/>
rentiam de&longs;cribet.  </s>
             
<s id="id.2.1.51.2.1.9.0"> nam <expan abbr="qu&ograve;">quo</expan> pondus in G leuius fuerit, <expan abbr="e&ograve;">eo</expan> ma&shy;<lb/>
gis pondus in G eleuabitur; libraq; GH ad &longs;itum horizonti &aelig;qui<lb/>
di&longs;tantem propius accedet.  </s>
             
<s id="id.2.1.51.2.1.10.0"> qu&aelig; omnia ex iis, qu&aelig; &longs;upra dixi&shy;<lb/>
mus, manife&longs;ta &longs;unt. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig52" place="text">       </figure>       
<p id="id.2.1.51.2.2.1.0" type="caption">        
<s id="id.2.1.51.2.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig53" place="text">       </figure>       
<p id="id.2.1.51.2.2.3.0" type="caption">        
<s id="id.2.1.51.2.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.51.3.0.0.0" type="main">        
<s id="id.2.1.51.3.1.1.0"> His demon&longs;tratis.  </s>
             
<s id="id.2.1.51.3.1.2.0"> Manife&longs;tum e&longs;t, centrum libr&aelig; cau&longs;am e&longs;&longs;e <lb/>
diuer&longs;itatis effectuum in libra.  </s>
             
<s id="id.2.1.51.3.1.3.0"> atq; patet omnes Archimedis de <lb/>
&aelig;queponderantibus propo&longs;itiones ad hoc pertinentes in omni &longs;itu <lb/>
veras e&longs;&longs;e.  </s>
             
<s id="id.2.1.51.3.1.4.0"> hoc e&longs;t &longs;iue libra &longs;it horizonti &aelig;quidi&longs;tans, &longs;iue non: <lb/>
dummodo centrum libr&aelig; in ip&longs;a &longs;it libra; quemadmodum ip&longs;e <lb/>
con&longs;iderat.  </s>
             
<s id="id.2.1.51.3.1.5.0"> &amp; quamquam libra brachia habeat in&aelig;qualia, idem eue<lb/>
niet; eodemq; pro&longs;us modo o&longs;tendetur, centrum libr&aelig; diuer&longs;imo <lb/>
<expan abbr="d&egrave;">de</expan> collocatum varios producere effectus. </s>
    </p>       
<p id="id.2.1.51.4.0.0.0" type="main">        
<s id="id.2.1.51.4.1.1.0"> Sit enim libra AB hori&shy;<lb/>
zonti &aelig;quidi&longs;tans; &amp; in AB <lb/>
&longs;int pondera in&aelig;qualia, quo <lb/>
rum grauitatis centrum &longs;it <lb/>
C: &longs;u&longs;pendaturq; libra in <lb/>
eodem puncto C. &amp; mo&shy;<lb/>
ueatur libra in DE.  </s>
     
<s id="id.2.1.51.4.1.1.0.a"> mani <lb/>
<arrow.to.target n="note78"></arrow.to.target> fe&longs;tum e&longs;t libram non &longs;o&shy;<lb/>
lum in DE, &longs;ed in quouis <lb/>
alio &longs;itu manere. <arrow.to.target n="fig54"></arrow.to.target> </s>
    </p>       
<pb n="29"/>
       
<p id="id.2.1.51.6.0.0.0" type="main">        
<s id="id.2.1.51.6.1.1.0"> Sit autem centrum libr&aelig; <lb/>
AB &longs;upra C in F; &longs;itq; <lb/>
FC ip&longs;i AB, &amp; horizonti <lb/>
perpendicularis: &amp; &longs;i mo&shy;<lb/>
ueatur libra in DE, linea <lb/>
CF mota erit in FG; qu&aelig; <lb/>
<expan abbr="c&ugrave;m">cum</expan> non &longs;it horizonti per&shy;<lb/>
pendicularis, libra DE <arrow.to.target n="note79"></arrow.to.target><lb/>
deor&longs;um ex parte D moue<lb/>
bitur, donec FG in FC <lb/>
redeat: atq; tunc libra DE <lb/>
in AB erit, in <expan abbr="qu&ograve;">quo</expan> &longs;itu <lb/>
quoq; manebit. <arrow.to.target n="fig55"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.51.7.0.0.0" type="main">        
<s id="id.2.1.51.7.1.1.0"> Et &longs;i centrum libr&aelig; F <lb/>
&longs;it infra libram; &longs;itq; mota <lb/>
libra in DE; <expan abbr="prim&ugrave;m">primum</expan> qui <lb/>
dem manife&longs;tum e&longs;t li&shy;<lb/>
bram in AB manere; in <arrow.to.target n="note80"></arrow.to.target><lb/>
DE <expan abbr="ver&ograve;">vero</expan> deor&longs;um ex par <lb/>
te E moueri: <expan abbr="c&ugrave;m">cum</expan> linea <lb/>
FG non &longs;it horizonti per&shy;<lb/>
pendicularis. <arrow.to.target n="fig56"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.51.8.0.0.0" type="main">        
<s id="id.2.1.51.8.1.1.0"> Ex his determinatis &longs;i libra &longs;it <lb/>
arcuata, vel libr&aelig; brachia angulum <lb/>
con&longs;tituant; centrumq; diuer&longs;imo <lb/>
<expan abbr="d&egrave;">de</expan> collocetur (quamquam h&aelig;c pro<lb/>
<expan abbr="pri&egrave;">prie</expan> non &longs;it libra) varios tamen <lb/>
huius quoq; effectus o&longs;tendere pote<lb/>
rimus.  </s>
             
<s id="id.2.1.51.8.1.2.0"> Vt &longs;it libra ACB, cuius <lb/>
centrum, circa quod vertitur, &longs;it C. <lb/>
ductaq; AB, &longs;it arcus &longs;iue angulus <lb/>
<arrow.to.target n="fig57"></arrow.to.target><lb/>
ACB &longs;upra lineam AB; &amp; in AB grauitatis centra ponderum <lb/>
ponantur, qu&aelig; in hoc &longs;itu maneant.  </s>
             
<s id="id.2.1.51.8.1.3.0"> moueatur deinde libra ab 
<pb/>
hoc &longs;itu, <expan abbr="put&aacute;">puta</expan> in ECF.  </s>
     
<s id="id.2.1.51.8.1.3.0.a"> Dico li&shy;<lb/>
bram ECF in ACB redire.  </s>
             
<s id="id.2.1.51.8.1.4.0"> to&shy;<lb/>
tius magnitudinis centrum grauita<lb/>
tis inueniatur D. &amp; CD iunga&shy;<lb/>
tur.  </s>
             
<s id="id.2.1.51.8.1.5.0"> Quoniam enim pondera AB <lb/>
<arrow.to.target n="note81"></arrow.to.target> manent, linea CD horizonti per&shy;<lb/>
pendicularis erit.  </s>
             
<s id="id.2.1.51.8.1.6.0"> quando igitur <lb/>
libra erit in ECF, linea CD erit <lb/>
<expan abbr="put&aacute;">puta</expan> in CG; qu&aelig; <expan abbr="c&ugrave;m">cum</expan> non &longs;it ho<lb/>
<arrow.to.target n="fig58"></arrow.to.target><lb/>
rizonti perpendicularis; libra ECF in ACB redibit.  </s>
             
<s id="id.2.1.51.8.1.7.0"> quod idem <lb/>
eueniet, &longs;i centrum C &longs;upra libram con&longs;tituatur, vt in H. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig54" place="text">       </figure>       
<p id="id.2.1.51.8.2.1.0" type="caption">        
<s id="id.2.1.51.8.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig55" place="text">       </figure>       
<p id="id.2.1.51.8.2.3.0" type="caption">        
<s id="id.2.1.51.8.2.3.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig56" place="text">       </figure>       
<p id="id.2.1.51.8.2.5.0" type="caption">        
<s id="id.2.1.51.8.2.5.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig57" place="text">       </figure>       
<p id="id.2.1.51.8.2.7.0" type="caption">        
<s id="id.2.1.51.8.2.7.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig58" place="text">       </figure>       
<p id="id.2.1.51.8.2.9.0" type="caption">        
<s id="id.2.1.51.8.2.9.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.52.1.0.0.0" type="margin">        
<s id="id.2.1.52.1.1.1.0"> <margin.target id="note78"></margin.target><emph type="italics"/>Per def. <expan abbr="c&etilde;tri">centri</expan> grauitatis.<emph.end type="italics"/> </s>
             
<s id="id.2.1.52.1.1.2.0"> <margin.target id="note79"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.52.1.1.3.0"> <margin.target id="note80"></margin.target>1. <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.52.1.1.4.0"> <margin.target id="note81"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.53.1.0.0.0" type="main">        
<s id="id.2.1.53.1.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> arcus, &longs;iue angulus <lb/>
ACB, &longs;it infra lineam AB; eo <lb/>
dem modo libram ECF, cuius <lb/>
centrum, &longs;iue &longs;it in C, &longs;iue in H, <lb/>
deor&longs;um ex parte F moueri o&shy;<lb/>
&longs;tendemus. <arrow.to.target n="fig59"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.53.2.0.0.0" type="main">        
<s id="id.2.1.53.2.1.1.0"> Sit autem angulus ACB &longs;upra lineam AB; ac libr&aelig; centrum <lb/>
&longs;it H; lineaq; CH libram &longs;u&longs;tineat; &amp; moueatur libra in EKF: <lb/>
libra EkF in ACB redibit.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig59" place="text">       </figure>       
<p id="id.2.1.53.2.2.1.0" type="caption">        
<s id="id.2.1.53.2.2.1.0.capt"> YYY </s>
    </p>       
<pb n="30"/>
       
<p id="id.2.1.53.4.0.0.0" type="main">        
<s id="id.2.1.53.4.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> centrum libr&aelig; &longs;it D, quocunq; modo moueatur libra; <lb/>
vbirelinquetur, manebit. </s>
    </p>       
<p id="id.2.1.53.5.0.0.0" type="main">        
<s id="id.2.1.53.5.1.1.0"> Si deinde punctum H &longs;it infra lineam AB; tunc libra EkF <lb/>
deor&longs;um ex parte F mouebitur. </s>
    </p>       
<p id="id.2.1.53.6.0.0.0" type="main">        
<s id="id.2.1.53.6.1.1.0"> Similiq; pror&longs;us ratione, &longs;i an<lb/>
gulus ACB &longs;it infra lineam AB; <lb/>
&longs;itq; libr&aelig; centrum H; &longs;u&longs;tineaturq; <lb/>
libra linea CH; &longs;i libra ab hoc mo<lb/>
ueatur &longs;itu, deor&longs;um ex parte pon&shy;<lb/>
deris inferioris mouebitur.  </s>
             
<s id="id.2.1.53.6.1.2.0"> &amp; &longs;i cen<lb/>
trum libr&aelig; &longs;it D; vbi relinquetur, <lb/>
manebit.  </s>
             
<s id="id.2.1.53.6.1.3.0"> &longs;i <expan abbr="ver&ograve;">vero</expan> &longs;it in K; &longs;i ab eiu&longs; <lb/>
<arrow.to.target n="fig60"></arrow.to.target><lb/>
modi moueatur &longs;itu, in eundem pro&longs;us redibit.  </s>
             
<s id="id.2.1.53.6.1.4.0"> qu&aelig; omnia ex iis, <lb/>
qu&aelig; in principio diximus, &longs;unt manife&longs;ta.  </s>
             
<s id="id.2.1.53.6.1.5.0"> &longs;imiliter &longs;i centrum li<lb/>
br&aelig;, vel in altero brachiorum, vel intra, vel extra vtcunq; po<lb/>
natur; eadem inueniemus.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig60" place="text">       </figure>       
<p id="id.2.1.53.6.2.1.0" type="caption">        
<s id="id.2.1.53.6.2.1.0.capt"> YYY </s>
    </p>       
<pb/>
       
<p id="id.2.1.53.8.0.0.0" type="head">        
<s id="id.2.1.53.8.1.1.0"> PROPOSITIO. V. </s>
    </p>       
<p id="id.2.1.53.9.0.0.0" type="main">        
<s id="id.2.1.53.9.1.1.0"> Duo pondera in libra appen&longs;a, &longs;i libra inter <lb/>
h&aelig;c ita diuidatur, vt partes ponderibus per&shy;<lb/>
mutatim re&longs;pondeant; <expan abbr="t&agrave;m">tam</expan> in punctis appen&longs;is <lb/>
ponderabunt, <expan abbr="qu&agrave;m">quam</expan> &longs;i vtraq; ex diui&longs;ionis pun&shy;<lb/>
cto &longs;u&longs;pendantur. <arrow.to.target n="fig61"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.53.10.0.0.0" type="main">        
<s id="id.2.1.53.10.1.1.0"> Sit AB libra, cuius centrum C; &longs;intq; duo pondera EF ex pun<lb/>
ctis BG &longs;u&longs;pen&longs;a: diuidaturq; BG in H, ita vt BH ad HG <lb/>
eandem habeat proportionem, quam pondus E ad pondus F.  </s>
     
<s id="id.2.1.53.10.1.1.0.a"> <lb/>
Dico pondera EF <expan abbr="t&agrave;m">tam</expan> in BG ponderare, <expan abbr="qu&agrave;m">quam</expan> &longs;i vtraq; ex pun<lb/>
cto H &longs;u&longs;pendantur.  </s>
             
<s id="id.2.1.53.10.1.2.0"> [fiat AC ip&longs;i CH &aelig;qualis.]  </s>
             
<s id="id.2.1.53.10.1.3.0"> fiat AC ip&longs;i CH &aelig;qualis. &amp; vt AC ad <lb/>
CG, ita fiat pondus E ad pondus L. &longs;imiliter vt AC ad CB, <lb/>
ita fiat pondus F ad pondus M. ponderaq; LM ex puncto A &longs;u<lb/>
&longs;pendantur.  </s>
             
<s id="id.2.1.53.10.1.4.0"> Quoniam enim AC e&longs;t &aelig;qualis CH, erit BC ad <lb/>
CH vt pondus M ad pondus F.  </s>
     
<s id="id.2.1.53.10.1.4.0.a"> &amp; quoniam maior e&longs;t BC, <lb/>
<expan abbr="qu&agrave;m">quam</expan> CH; erit &amp; pondus M ip&longs;o F maius.  </s>
             
<s id="id.2.1.53.10.1.5.0"> diuidatur igitur pon<lb/>
dus M in duas partes QR, &longs;itq; pars Q ip&longs;i F &aelig;qualis; erit BC <lb/>
<arrow.to.target n="note82"></arrow.to.target> ad CH, vt RQ ad Q: &amp; diuidendo, vt BH ad HC, ita R ad <expan abbr="q.">que</expan> <lb/>
<arrow.to.target n="note83"></arrow.to.target> deinde conuertendo, vt CH ad HB, ita Q ad R.  </s>
     
<s id="id.2.1.53.10.1.5.0.a"> Pr&aelig;terea quo&shy;<lb/>
niam CH e&longs;t &aelig;qualis ip&longs;i CA, erit HC ad CG, vt pondus <lb/>
E ad pondus L: maior autem e&longs;t HC, <expan abbr="qu&agrave;m">quam</expan> CG; erit &amp; pon&shy;
<pb n="31"/>
dus E pondere L maius.  </s>
             
<s id="id.2.1.53.10.1.6.0"> diuidatur itaq; pondus E in duas partes <lb/>
NO ita, vt pars O &longs;it ip&longs;i L &aelig;qualis, erit HC ad CG, vt to&shy;<lb/>
tum NO ad O; &amp; diuidendo, vt HG ad GC, ita N ad O: <arrow.to.target n="note84"></arrow.to.target><lb/>
conuertendoq; vt CG ad GH, ita O ad N. &amp; iterum com&shy;<lb/>
ponendo, vt CH ad HG, ita ON ad N. vt autem GH <arrow.to.target n="note85"></arrow.to.target><lb/>
ad HB, ita e&longs;t F ad ON. quare ex &aelig;quali, vt CH ad HB, ita F <arrow.to.target n="note86"></arrow.to.target><lb/>
ad N. &longs;ed vt CH ad HB ita e&longs;t Q ad R: erit igitur Q ad R, vt <arrow.to.target n="note87"></arrow.to.target><lb/>
F ad N; &amp; permutando, vt Q ad F, ita R ad N. e&longs;t autem pars <arrow.to.target n="note88"></arrow.to.target><lb/>
Q ip&longs;i F &aelig;qualis; quare &amp; pars R ip&longs;i N &aelig;qualis erit.  </s>
             
<s id="id.2.1.53.10.1.7.0"> Itaq; <expan abbr="c&ugrave;m">cum</expan> <lb/>
pondus L &longs;it ip&longs;i O &aelig;quale, &amp; pondus F ip&longs;i Q etiam &aelig;quale, atq; <lb/>
pars R ip&longs;i N &aelig;qualis; erunt pondera LM ip&longs;is EF ponderibus <lb/>
&aelig;qualia.  </s>
             
<s id="id.2.1.53.10.1.8.0"> &amp; quoniam e&longs;t, vt AC ad CG, ita pondus E ad pon&shy;<lb/>
dus L; pondera EL &aelig;queponderabunt.  </s>
             
<s id="id.2.1.53.10.1.9.0"> &longs;imiliter quoniam e&longs;t, vt <arrow.to.target n="note89"></arrow.to.target><lb/>
AC ad CB, ita pundus F ad pondus M; pondera quoq; FM <lb/>
&aelig;queponderabunt.  </s>
             
<s id="id.2.1.53.10.1.10.0"> Pondera igitur LM ponderibus EF in BG <arrow.to.target n="note90"></arrow.to.target><lb/>
appen&longs;is &aelig;queponderabunt.  </s>
             
<s id="id.2.1.53.10.1.11.0"> <expan abbr="c&ugrave;m">cum</expan> autem di&longs;tantia CA &aelig;qualis &longs;it <lb/>
di&longs;tanti&aelig; CH; &longs;i igitur vtraq; pondera EF in H appendantur, <lb/>
pondera LM ip&longs;is EF ponderibus in H appen&longs;is &aelig;quepondera&shy;<lb/>
bunt.  </s>
             
<s id="id.2.1.53.10.1.12.0"> &longs;ed LM ip&longs;is EF in GB quoq; &aelig;queponderant: <expan abbr="&aelig;qu&egrave;">&aelig;que</expan> <arrow.to.target n="note91"></arrow.to.target><lb/>
igitur grauia erunt pondera EF in GB, vt in H appen&longs;a.  </s>
             
<s id="id.2.1.53.10.1.13.0"> <expan abbr="t&agrave;m">tam</expan> igi<lb/>
tur ponderabunt in BG, <expan abbr="qu&agrave;m">quam</expan> in H appen&longs;a. <arrow.to.target n="fig62"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.53.11.0.0.0" type="main">        
<s id="id.2.1.53.11.1.1.0"> Sint autem pondera EF in CB appen&longs;a; &longs;itq; C libr&aelig; centrum; <lb/>
&amp; diuidatur CB in H, ita vt CH ad HB &longs;it, vt pondus in F ad <lb/>
E.  </s>
     
<s id="id.2.1.53.11.1.1.0.a"> Dico pondera EF <expan abbr="t&agrave;m">tam</expan> in CB ponderare, <expan abbr="qu&agrave;m">quam</expan> in puncto H.  </s>
     
<s id="id.2.1.53.11.1.1.0.b"> <lb/>
fiat CA ip&longs;i CH &aelig;qualis, &amp; vt CA ad CB, ita fiat pondus F ad <lb/>
aliud D, quod appendatur in A.  </s>
     
<s id="id.2.1.53.11.1.1.0.c"> Quoniam enim CH e&longs;t &aelig;qua&shy;
<pb/>
<arrow.to.target n="fig63"></arrow.to.target><lb/>
lis CA, erit CH ad CB, vt F ad D; &amp; maior quidem e&longs;t CB, <lb/>
<expan abbr="qu&agrave;m">quam</expan> CH; idcirco D pondere F maius erit.  </s>
             
<s id="id.2.1.53.11.1.2.0"> Diuidatur ergo D <lb/>
in duas partes Gk, &longs;itq; G ip&longs;i F &aelig;qualis; erit vt BC ad CH, <lb/>
vt Gk ad G; &amp; diuidendo, vt BH ad HC, ita K ad G; &amp; conuer <lb/>
<arrow.to.target n="note92"></arrow.to.target> tendo, vt CH ad HB, ita G ad k.  </s>
             
<s id="id.2.1.53.11.1.3.0"> Vt autem CH ad HB, ita e&longs;t <lb/>
<arrow.to.target n="note93"></arrow.to.target> F ad E. vt igitur G ad k, ita e&longs;t F ad E; &amp; permutando vt G <lb/>
<arrow.to.target n="note94"></arrow.to.target> ad F, ita k ad E. &longs;unt autem GF &aelig;qualia; erunt &amp; kE inter &longs;e <lb/>
&longs;e &aelig;qualia.  </s>
             
<s id="id.2.1.53.11.1.4.0"> <expan abbr="c&ugrave;m">cum</expan> itaq; pars G &longs;it ip&longs;i F &aelig;qualis, &amp; K ip&longs;i E; erit <lb/>
totum C k ip&longs;is EF ponderibus &aelig;quale.  </s>
             
<s id="id.2.1.53.11.1.5.0"> &amp; quoniam AC e&longs;t ip&shy;<lb/>
&longs;i CH &aelig;qualis; &longs;i igitur pondera EF ex puncto H &longs;u&longs;pendantur, <lb/>
pondus D ip&longs;is EF in H appen&longs;is &aelig;queponderabit.  </s>
             
<s id="id.2.1.53.11.1.6.0"> &longs;ed &amp; ip&longs;is <lb/>
&aelig;queponderat in CB, hoc e&longs;t F in B, &amp; E in C; <expan abbr="c&ugrave;m">cum</expan> &longs;it vt AC <lb/>
ad CB, ita F ad. D. pondus enim E ex centro libr&aelig; C &longs;u&longs;pen&shy;<lb/>
&longs;um non efficit, vt libra in alterutram moueatur partem. </s>
             
<s id="id.2.1.53.11.1.7.0"> [D. pondus enim E ex centro libr&aelig; C &longs;u&longs;pen&shy;<lb/>
&longs;um non efficit, vt libra in alterutram moueatur partem.]  </s>
             
<s id="id.2.1.53.11.1.8.0"> <expan abbr="t&agrave;m">tam</expan> igi&shy;<lb/>
tur grauia erunt pondera EF in CB, <expan abbr="qu&agrave;m">quam</expan> in H appen&longs;a. 
<pb n="32"/>
<arrow.to.target n="fig64"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.53.12.0.0.0" type="main">        
<s id="id.2.1.53.12.1.1.0"> Sit deniq; libra AB, &amp; ex punctis AB &longs;u&longs;pen&longs;a &longs;int pondera <lb/>
EF; &longs;itq; centrum libr&aelig; C intra pondera; diuidaturq; AB in <lb/>
D, ita vt AD ad DB &longs;it, vt pondus F ad pondus E.  </s>
     
<s id="id.2.1.53.12.1.1.0.a"> Dico pon<lb/>
dera EF <expan abbr="t&agrave;m">tam</expan> in AB ponderare, <expan abbr="qu&aacute;m">quam</expan> &longs;i vtraq; ex puncto D &longs;u&longs;pen<lb/>
dantur.  </s>
             
<s id="id.2.1.53.12.1.2.0"> fiat CG &aelig;qualis ip&longs;i CD; &amp; vt DC ad CA, ita fiat <lb/>
pondus E ad aliud H; quod appendatur in D. vt autem GC ad <lb/>
CB, ita fiat pondus F ad aliud K; appendaturq; k in G.  </s>
     
<s id="id.2.1.53.12.1.2.0.a"> <expan abbr="Quoni&atilde;">Quoniam</expan> enim <lb/>
e&longs;t, vt BC ad CG, hoc e&longs;t ad CD, ita pondus k ad F; erit K ma <lb/>
ius pondere F. quare diuidatur pondus k in L, &amp; MN; fiatq; <lb/>
pars L ip&longs;i F &aelig;qualis; erit vt BC ad CD, vt totum LMN ad <lb/>
L; &amp; diuidendo, vt BD ad DC, ita pars MN ad partem L. vt <arrow.to.target n="note95"></arrow.to.target><lb/>
igitur BD ad DC, ita pars MN ad F. vt autem AD ad DB, <lb/>
ita F ad E: quare ex &aelig;quali, vt AD ad DC, ita MN ad E. <expan abbr="c&ugrave;m">cum</expan> <arrow.to.target n="note96"></arrow.to.target><expan abbr="ver&ograve;"><lb/>
vero</expan> AD &longs;it ip&longs;a CD maior; erit &amp; pars MN pondere E <lb/>
maior: diuidatur ergo MN in duas partes MN, &longs;itq; M &aelig;qua <lb/>
lis ip&longs;i E. erit vt AD ad DC, vt NM ad M; &amp; diuidendo, vt <arrow.to.target n="note97"></arrow.to.target><lb/>
AC ad CD, ita N ad M: conuertendoq; vt DC ad CA, ita M <lb/>
ad N. vt autem DC ad CA, ita e&longs;t E ad H; erit igitur M ad N <arrow.to.target n="note98"></arrow.to.target><lb/>
vt E ad H; &amp; permutando, vt M ad E, ita N ad H. &longs;ed ME <arrow.to.target n="note99"></arrow.to.target><lb/>
&longs;unt inter &longs;e &aelig;qualia, erunt NH inter &longs;e&longs;e quoq; &aelig;qualia.  </s>
             
<s id="id.2.1.53.12.1.3.0"> &amp; quo&shy;<lb/>
niam ita e&longs;t AC ad CD, vt H ad E: pondera HE &aelig;queponde&shy;<lb/>
rabunt.  <arrow.to.target n="note100"></arrow.to.target>  </s>
             
<s id="id.2.1.53.12.1.4.0"> &longs;imiliter quoniam e&longs;t vt GC ad CB, ita F ad k, ponde&shy;
<pb/>
<arrow.to.target n="fig65"></arrow.to.target><lb/>
<arrow.to.target n="note101"></arrow.to.target> ra etiam kF &aelig;queponderabunt.  </s>
             
<s id="id.2.1.53.12.1.5.0"> pondera igitur Ek HF in li&shy;<lb/>
bra AB, cuius centrum C, &aelig;queponderabunt.  </s>
             
<s id="id.2.1.53.12.1.6.0"> <expan abbr="c&ugrave;m">cum</expan> autem GC <lb/>
ip&longs;i CD &longs;it &aelig;qualis, &amp; pondus H &longs;it ip&longs;i N &aelig;quale; pondera NH <lb/>
&aelig;queponderabunt.  </s>
             
<s id="id.2.1.53.12.1.7.0"> &amp; quoniam omnia &aelig;queponderant, demptis <lb/>
<arrow.to.target n="note102"></arrow.to.target> HN ponderibus, qu&aelig; &aelig;queponderant, reliqua &aelig;queponderabunt; <lb/>
hoc e&longs;t pondera EF &amp; pondus LM ex centro libr&aelig; C &longs;u&longs;pen&longs;a.  </s>
             
<s id="id.2.1.53.12.1.8.0"> <lb/>
quia <expan abbr="ver&ograve;">vero</expan> pars L ip&longs;i F e&longs;t &aelig;qualis, &amp; pars M ip&longs;i E &aelig;qualis; erit <lb/>
totum LM ip&longs;is FE ponderibus &longs;imul &longs;umptis &aelig;quale.  </s>
             
<s id="id.2.1.53.12.1.9.0"> &amp; <expan abbr="c&ugrave;m">cum</expan> <lb/>
&longs;it CG ip&longs;i CD &aelig;qualis, &longs;i igitur pondera EF ex puncto D &longs;u&longs;pen&shy;<lb/>
dantur, pondera EF in D appen&longs;a ip&longs;i LM &aelig;queponderabunt.  </s>
             
<s id="id.2.1.53.12.1.10.0"> quare <lb/>
LM <expan abbr="t&agrave;m">tam</expan> ip&longs;is EF in AB appen&longs;is &aelig;queponderat, <expan abbr="qu&agrave;m">quam</expan> in pun<lb/>
cto D appen&longs;is.  </s>
             
<s id="id.2.1.53.12.1.11.0"> libra enim &longs;emper eodem modo manet.  </s>
             
<s id="id.2.1.53.12.1.12.0"> Ponde&shy;<lb/>
<arrow.to.target n="note103"></arrow.to.target> ra ergo EF <expan abbr="t&agrave;m">tam</expan> in AB ponderabunt, <expan abbr="qu&agrave;m">quam</expan> in puncto D. quod <lb/>
demon&longs;tre oportebat. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig61" place="text">       </figure>       
<p id="id.2.1.53.12.2.1.0" type="caption">        
<s id="id.2.1.53.12.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig62" place="text">       </figure>       
<p id="id.2.1.53.12.2.3.0" type="caption">        
<s id="id.2.1.53.12.2.3.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig63" place="text">       </figure>       
<p id="id.2.1.53.12.2.5.0" type="caption">        
<s id="id.2.1.53.12.2.5.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig64" place="text">       </figure>       
<p id="id.2.1.53.12.2.7.0" type="caption">        
<s id="id.2.1.53.12.2.7.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig65" place="text">       </figure>       
<p id="id.2.1.53.12.2.9.0" type="caption">        
<s id="id.2.1.53.12.2.9.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.54.1.0.0.0" type="margin">        
<s id="id.2.1.54.1.1.1.0"> <margin.target id="note82"></margin.target>17 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.54.1.1.2.0"> <margin.target id="note83"></margin.target><emph type="italics"/>Cor.<emph.end type="italics"/>4 <emph type="italics"/>quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.54.1.1.3.0"> <margin.target id="note84"></margin.target>17 <emph type="italics"/>Quinti.  </s>
             
<s id="id.2.1.54.1.1.4.0"> Cor.<emph.end type="italics"/>4 <emph type="italics"/>quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.54.1.1.5.0"> <margin.target id="note85"></margin.target>18 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.54.1.1.6.0"> <margin.target id="note86"></margin.target>23 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.54.1.1.7.0"> <margin.target id="note87"></margin.target>11 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.54.1.1.8.0"> <margin.target id="note88"></margin.target>16 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.54.1.1.9.0"> <margin.target id="note89"></margin.target>6 <emph type="italics"/>Primi Archim. de &aelig;quep.<emph.end type="italics"/> </s>
             
<s id="id.2.1.54.1.1.10.0"> [de &aelig;quep.<emph.end type="italics"/>] </s>
             
<s id="id.2.1.54.1.1.11.0"> <margin.target id="note90"></margin.target>2 <emph type="italics"/>Com. not. huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.54.1.1.12.0"> [not.]  </s>
             
<s id="id.2.1.54.1.1.13.0"> [huius.<emph.end type="italics"/>] </s>
             
<s id="id.2.1.54.1.1.14.0"> <margin.target id="note91"></margin.target>3 <emph type="italics"/>Com. not. huius.<emph.end type="italics"/>  </s>
             
<s id="id.2.1.54.1.1.15.0"> [huius.<emph.end type="italics"/>] </s>
             
<s id="id.2.1.54.1.1.16.0"> <margin.target id="note92"></margin.target>17 <emph type="italics"/>Quinti.  </s>
             
<s id="id.2.1.54.1.1.17.0"> Cor.<emph.end type="italics"/>4 <emph type="italics"/>quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.54.1.1.18.0"> <margin.target id="note93"></margin.target>11 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.54.1.1.19.0"> <margin.target id="note94"></margin.target>16 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.54.1.1.20.0"> <margin.target id="note95"></margin.target>17 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.54.1.1.21.0"> <margin.target id="note96"></margin.target>23 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.54.1.1.22.0"> <margin.target id="note97"></margin.target>17 <emph type="italics"/>Quinti.  </s>
             
<s id="id.2.1.54.1.1.23.0"> Cor.<emph.end type="italics"/>4 <emph type="italics"/>quinti<emph.end type="italics"/> </s>
             
<s id="id.2.1.54.1.1.24.0"> <margin.target id="note98"></margin.target>11 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.54.1.1.25.0"> <margin.target id="note99"></margin.target>16 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.54.1.1.26.0"> <margin.target id="note100"></margin.target>6 <emph type="italics"/>Primi Ar chim. de &aelig;quep.<emph.end type="italics"/> </s>
             
<s id="id.2.1.54.1.1.27.0"> [de &aelig;quep.<emph.end type="italics"/>] </s>
             
<s id="id.2.1.54.1.1.28.0"> <margin.target id="note101"></margin.target>2 <emph type="italics"/>Com.not. huius.<emph.end type="italics"/>  </s>
             
<s id="id.2.1.54.1.1.29.0"> [huius.<emph.end type="italics"/>] </s>
             
<s id="id.2.1.54.1.1.30.0"> <margin.target id="note102"></margin.target>1 <emph type="italics"/>Com.not. huius.<emph.end type="italics"/>  </s>
             
<s id="id.2.1.54.1.1.31.0"> [huius.<emph.end type="italics"/>] </s>
             
<s id="id.2.1.54.1.1.32.0"> <margin.target id="note103"></margin.target>3 <emph type="italics"/>Com.not. huius.<emph.end type="italics"/>  </s>
             
<s id="id.2.1.54.1.1.33.0"> [huius.<emph.end type="italics"/>] </s>
    </p>       
<p id="id.2.1.55.1.0.0.0" type="main">        
<s id="id.2.1.55.1.1.1.0"> H&aelig;c autem omnia (mechanic&egrave; tamen ma&shy;<lb/>
gis) aliter o&longs;tendemus. 
<pb n="33"/>
<arrow.to.target n="fig66"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.55.2.0.0.0" type="main">        
<s id="id.2.1.55.2.1.1.0"> Sit libra AB, cuius centrum C; &longs;intq; vt in primo ca&longs;u duo pon<lb/>
dera EF ex punctis BG &longs;u&longs;pen&longs;a: &longs;itq; GH ad HB, vt pondus <lb/>
F ad pondus E.  </s>
     
<s id="id.2.1.55.2.1.1.0.a"> Dico pondera EF <expan abbr="t&agrave;m">tam</expan> in GB ponderare, <expan abbr="qu&agrave;m">quam</expan> <lb/>
&longs;i vtraq; ex diui&longs;ionis puncto H &longs;u&longs;pendantur.  </s>
             
<s id="id.2.1.55.2.1.2.0"> Con&longs;truantur ea <lb/>
dem, hoc e&longs;t fiat AC ip&longs;i CH &aelig;qualis, &amp; ex puncto A duo ap&shy;<lb/>
pendantur pondera LM, ita vt pondus E ad pondus L, &longs;it vt <lb/>
CA ad CG; vt autem CB ad CA, ita &longs;it pondus M ad pondus <lb/>
F.  </s>
     
<s id="id.2.1.55.2.1.2.0.a"> pondera LM ip&longs;is EF in GB appen&longs;is (vt &longs;upra dictum e&longs;t) <lb/>
&aelig;queponderabunt.  </s>
             
<s id="id.2.1.55.2.1.3.0"> Sint deinde puncta NO centra grauitatis pon<lb/>
derum EF; connectanturq; GN BO; iungaturq; NO, qu&aelig; tan&shy;<lb/>
quam libra erit; qu&aelig; etiam efficiat lineas GN BO inter &longs;e &longs;e &aelig;qui&shy;<lb/>
di&longs;tantes e&longs;&longs;e; <expan abbr="&agrave;">a</expan> punctoq; H horizonti perpendicularis ducatur <lb/>
HP, qu&aelig; NO &longs;ecet in P, atq; ip&longs;is GN BO &longs;it &aelig;quidi&longs;tans. <lb/>
deniq; connectatur GO, qu&aelig; HP &longs;ecet in R. </s>
             
<s id="id.2.1.55.2.1.4.0"> [<lb/>
deniq; connectatur GO, qu&aelig; HP &longs;ecet in R.] Quoniam igitur <lb/>
HR e&longs;t lateri BO trianguli GBO &aelig;quidi&longs;tans; erit GH ad HB, <lb/>
vt GR ad RO. &longs;imiliter quoniam RP e&longs;t lateri GN trianguli <arrow.to.target n="note104"></arrow.to.target><lb/>
OGN &aelig;quidi&longs;tans; erit GR ad RO, vt NP ad PO. quare <lb/>
vt GH ad HB, ita e&longs;t NP ad PO. vt autem GH ad HB, ita <arrow.to.target n="note105"></arrow.to.target><lb/>
e&longs;t pondus F ad pondus E; vt igitur NP ad PO, ita e&longs;t pondus <lb/>
F ad pondus E.  </s>
     
<s id="id.2.1.55.2.1.4.0.a"> punctum ergo P centrum erit grauitatis magni&shy;<lb/>
tudinis ex vtri&longs;q; EF ponderibus compo&longs;it&aelig;.  </s>
             
<s id="id.2.1.55.2.1.5.0"> Intelligantur itaq; <arrow.to.target n="note106"></arrow.to.target><lb/>
pondera EF ita e&longs;&longs;e <expan abbr="&agrave;">a</expan> libra NO connexa, ac &longs;i vna <expan abbr="tant&ugrave;m">tantum</expan> e&longs;&longs;et <lb/>
magnitudo ex vtri&longs;q; EF compo&longs;ita, in puncti&longs;q; BG appen&longs;a.  </s>
             
<s id="id.2.1.55.2.1.6.0"> &longs;i <lb/>
igitur ponderum &longs;u&longs;pen&longs;iones BG &longs;oluantur, manebunt pondera <arrow.to.target n="note107"></arrow.to.target><lb/>
EF ex HP &longs;u&longs;pen&longs;a; &longs;icuti in GB prius manebant.  </s>
             
<s id="id.2.1.55.2.1.7.0"> pondera <expan abbr="ver&ograve;">vero</expan> EF <lb/>
in GB appen&longs;a ip&longs;is LM ponderibus &aelig;queponderant, &amp; pondera 
<pb/>
<arrow.to.target n="fig67"></arrow.to.target><lb/>
EF ex puncto H &longs;u&longs;pen&longs;a, eandem habent con&longs;titutionem ad li&shy;<lb/>
bram AB, quam in BG appen&longs;a: eadem ergo pondera EF ex <lb/>
H &longs;u&longs;pen&longs;a ei&longs;dem ponderibus LM &aelig;queponderabunt.  </s>
             
<s id="id.2.1.55.2.1.8.0"> <expan abbr="&aelig;qu&egrave;">&aelig;que</expan> igi&shy;<lb/>
tur &longs;unt grauia pondera EF in GB, vt in H appen&longs;a. <arrow.to.target n="fig68"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.55.3.0.0.0" type="main">        
<s id="id.2.1.55.3.1.1.0"> Similiter demon&longs;trabitur, pondera EF in quibu&longs;cunq; aliis pun<lb/>
ctis appen&longs;a <expan abbr="t&agrave;m">tam</expan> <expan abbr="p&otilde;derare">ponderare</expan>, <expan abbr="qu&agrave;m">quam</expan> &longs;i vt raq; ex diui&longs;ionis puncto H &longs;u<lb/>
&longs;pendantur.  </s>
             
<s id="id.2.1.55.3.1.2.0"> &longs;i enim (vt &longs;upra docuimus) in libra pondera inue&shy;<lb/>
niantur, quibus pondera EF &aelig;queponderent; eadem pondera EF <lb/>
ex H &longs;u&longs;pen&longs;a ei&longs;dem inuentis ponderibus &aelig;queponderabunt; <expan abbr="c&ugrave;m">cum</expan> <lb/>
punctum P &longs;it &longs;emper eorum centrum grauitatis; &amp; HP horizon <lb/>
ri perpendicularis.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig66" place="text">       </figure>       
<p id="id.2.1.55.3.2.1.0" type="caption">        
<s id="id.2.1.55.3.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig67" place="text">       </figure>       
<p id="id.2.1.55.3.2.3.0" type="caption">        
<s id="id.2.1.55.3.2.3.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig68" place="text">       </figure>       
<p id="id.2.1.55.3.2.5.0" type="caption">        
<s id="id.2.1.55.3.2.5.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.56.1.0.0.0" type="margin">        
<s id="id.2.1.56.1.1.1.0"> <margin.target id="note104"></margin.target>2 <emph type="italics"/>Sexti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.56.1.1.2.0"> <margin.target id="note105"></margin.target>11 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.56.1.1.3.0"> <margin.target id="note106"></margin.target>6 <emph type="italics"/>Primi Ar chim. de &aelig;quep.<emph.end type="italics"/> </s>
             
<s id="id.2.1.56.1.1.4.0"> [de &aelig;quep.<emph.end type="italics"/>] </s>
             
<s id="id.2.1.56.1.1.5.0"> <margin.target id="note107"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.57.1.0.0.0" type="head">        
<pb n="34"/>
       
<s id="id.2.1.57.1.2.1.0"> PROPOSITIO. VI. </s>
    </p>       
<p id="id.2.1.57.2.0.0.0" type="main">        
<s id="id.2.1.57.2.1.1.0"> Pondera &aelig;qualia in libra appen&longs;a eam in gra<lb/>
uitate proportionem habent; quam di&longs;tanti&aelig;, ex <lb/>
quibus appenduntur. <arrow.to.target n="fig69"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.57.3.0.0.0" type="main">        
<s id="id.2.1.57.3.1.1.0"> Sit libra BAC &longs;u&longs;pen&longs;a ex puncto A; &amp; &longs;ecetur AC vtcunq; <lb/>
in D: ex punctis autem DC appendantur &aelig;qualia pondera EF. <lb/>
 </s>
     
<s id="id.2.1.57.3.1.1.0.a"> Dico pondus F ad pondus E eam in grauitate proportionem ha&shy;<lb/>
bere, quam habet di&longs;tantia CA ad di&longs;tantiam AD.  </s>
     
<s id="id.2.1.57.3.1.1.0.b"> fiat enim vt <lb/>
CA ad AD, ita pondus F ad aliud pondus, quod &longs;it G.  </s>
     
<s id="id.2.1.57.3.1.1.0.c"> Dico pri <lb/>
<expan abbr="m&uacute;m">mum</expan> pondera GF ex puncto C &longs;u&longs;pen&longs;a <expan abbr="tant&ugrave;m">tantum</expan> ponderare, quan<lb/>
<expan abbr="t&ugrave;m">tum</expan> pondera EF ex punctis DC.  </s>
     
<s id="id.2.1.57.3.1.1.0.d"> Secetur DC bifariam in H, &amp; <lb/>
ex H appendantur vtraq; pondera EF. ponderabunt EF &longs;imul <lb/>
&longs;umpta in eo &longs;itu, <expan abbr="quant&ugrave;m">quantum</expan> ponderant in DC. ponatur BA <arrow.to.target n="note108"></arrow.to.target><lb/>
&aelig;qualis AH, &longs;eceturq; BA in K, ita vt &longs;it KA &aelig;qualis AD: <lb/>
deinde ex puncto B appendatur pondus L duplum ponderis F, <lb/>
hoc e&longs;t &aelig;quale duobus ponderibus EF, quod quidem &aelig;queponde<lb/>
rabit ponderibus EF in H appen&longs;is, hoc e&longs;t appen&longs;is in DC.  </s>
     
<s id="id.2.1.57.3.1.1.0.e"> <expan abbr="Quoni&atilde;">Quoniam</expan> <lb/>
igitur, vt CA ad AD, ita e&longs;t pondus F ad pondus G; erit compo<lb/>
nendo vt CA AD ad AD, hoc e&longs;t vt Ck ad AD, ita ponde&shy;<lb/>
ra  <arrow.to.target n="note109"></arrow.to.target> FG ad pondus G. &longs;ed <expan abbr="c&ugrave;m">cum</expan> &longs;it, vt CA ad AD, ita F pon&shy;<lb/>
dus ad pondus G; erit conuertendo, vt DA ad AC, ita pondus <arrow.to.target n="note110"></arrow.to.target><lb/>
G ad pondus F; &amp; con&longs;equentium dupla, vt DA ad duplam ip&longs;ius <lb/>
AC, ita pondus G ad duplum ponderis F, hoc e&longs;t ad pondus <lb/>
L.  </s>
     
<s id="id.2.1.57.3.1.1.0.f"> Quare vt Ck ad DA, ita pondera EF ad pondus G; &amp; vt 
<pb/>
<arrow.to.target n="fig70"></arrow.to.target><lb/>
<arrow.to.target n="note111"></arrow.to.target> AD ad <expan abbr="dupl&atilde;">duplam</expan> ip&longs;ius AC, ita pondus G ad pondus L; ergo ex &aelig;quali, <lb/>
vt Ck ad <expan abbr="dupl&atilde;">duplam</expan> ip&longs;ius AC, ita pondera FG ad pondus L. &longs;ed vt Ck <lb/>
ad duplam AC, ita dimidia CK, videlicet AH, hoc e&longs;t BA, ad <lb/>
AC.  </s>
     
<s id="id.2.1.57.3.1.1.0.g"> Vt igitur BA ad AC, ita FG pondera ad pondus L.  </s>
     
<s id="id.2.1.57.3.1.1.0.h"> Qua <lb/>
re ex &longs;exta eiu&longs;dem primi Archimedis, duo pondera FG ex pun<lb/>
cto C &longs;u&longs;pen&longs;a <expan abbr="tant&ugrave;m">tantum</expan> ponderabunt, <expan abbr="quant&ugrave;m">quantum</expan> pondus L ex B; <lb/>
hoc e&longs;t <expan abbr="quant&ugrave;m">quantum</expan> pondera EF ex punctis DC &longs;u&longs;pen&longs;a.  </s>
             
<s id="id.2.1.57.3.1.2.0"> Itaq; quo<lb/>
niam pondera FG <expan abbr="tant&ugrave;m">tantum</expan> ponderant, quantum pondera EF; &longs;u&shy;<lb/>
blato communi pondere F, <expan abbr="t&agrave;m">tam</expan> ponderabit pondus G in C ap&shy;<lb/>
pen&longs;um, <expan abbr="qu&agrave;m">quam</expan> pondus E in D.  </s>
     
<s id="id.2.1.57.3.1.2.0.a"> ac propterea pondus F ad pon&shy;<lb/>
<arrow.to.target n="note112"></arrow.to.target> dus E eam in grauitate proportionem habet, quam habet ad pon<lb/>
dus G. &longs;ed pondus F ad G erat, vt CA ad AD:. ergo &amp; F pon&shy;<lb/>
dus ad pondus E eam in grauitate proportionem habebit, quam ha<lb/>
bet CA ad AD. quod demon&longs;trare oportebat. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig69" place="text">       </figure>       
<p id="id.2.1.57.3.2.1.0" type="caption">        
<s id="id.2.1.57.3.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig70" place="text">       </figure>       
<p id="id.2.1.57.3.2.3.0" type="caption">        
<s id="id.2.1.57.3.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.58.1.0.0.0" type="margin">        
<s id="id.2.1.58.1.1.1.0"> <margin.target id="note108"></margin.target>5 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.58.1.1.2.0"> <margin.target id="note109"></margin.target>18 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.58.1.1.3.0"> <margin.target id="note110"></margin.target><emph type="italics"/>Cor.<emph.end type="italics"/> 4 <emph type="italics"/>quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.58.1.1.4.0"> <margin.target id="note111"></margin.target>22 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.58.1.1.5.0"> <margin.target id="note112"></margin.target>7 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.59.1.0.0.0" type="main">        
<s id="id.2.1.59.1.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> in libra <lb/>
BAC pondera EF <lb/>
&aelig;qualia ex punctis <lb/>
BC &longs;u&longs;pendantur; &longs;i&shy;<lb/>
militer dico pondus <lb/>
E ad pondus F eam <lb/>
<arrow.to.target n="fig71"></arrow.to.target><lb/>
in grauitate proportionem habere, <expan abbr="qu&agrave;m">quam</expan> habet di&longs;tantia CA ad di<lb/>
&longs;tantiam AB.  </s>
     
<s id="id.2.1.59.1.1.1.0.a"> fiat AD ip&longs;i AB &aelig;qualis, &amp; ex puncto D &longs;u&longs;pen&shy;<lb/>
datur pondus G &aelig;quale ponderi F; quod etiam ip&longs;i E erit &aelig;quale.  </s>
             
<s id="id.2.1.59.1.1.2.0"> <lb/>
&amp; quoniam AD e&longs;t &aelig;qualis ip&longs;i AB; pondera FG &aelig;queponde<lb/>
rabunt, eandemq; habebunt grauitatem.  </s>
             
<s id="id.2.1.59.1.1.3.0"> <expan abbr="c&ugrave;m">cum</expan> autem grauitas pon<lb/>
deris E ad grauitatem ponderis G &longs;it, vt CA ad AD; erit graui<lb/>
tas ponderis E ad grauitatem ponderis F, vt CA ad AD, hoc e&longs;t <lb/>
CA ad AB. quod erat quoq; o&longs;tendendum.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig71" place="text">       </figure>       
<p id="id.2.1.59.1.2.1.0" type="caption">        
<s id="id.2.1.59.1.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.59.2.0.0.0" type="head">        
<pb n="35"/>
       
<s id="id.2.1.59.3.1.1.0"> ALITER. </s>
    </p>       
<p id="id.2.1.59.4.0.0.0" type="main">        
<s id="id.2.1.59.4.1.1.0"> Sit libra BAC, cu&shy;<lb/>
ius centrum A; in pun&shy;<lb/>
ctis <expan abbr="ver&ograve;">vero</expan> BC pondera <lb/>
appendantur &aelig;qualia G <lb/>
F: &longs;itq; <expan abbr="prim&ugrave;m">primum</expan> cen&shy;<lb/>
trum A vtcunque inter <lb/>
BC.  </s>
     
<s id="id.2.1.59.4.1.1.0.a"> Dico pondus F ad <lb/>
pondus G eam in graui<lb/>
<arrow.to.target n="fig72"></arrow.to.target><lb/>
tate proportionem habere, quam habet di&longs;tantia CA ad di&longs;tan&shy;<lb/>
tiam AB.  </s>
     
<s id="id.2.1.59.4.1.1.0.b"> fiat vt BA ad AC, ita pondus F ad aliud H, quod ap<lb/>
pendatur in B: pondera HF ex A &aelig;queponderabunt.  </s>
             
<s id="id.2.1.59.4.1.2.0"> &longs;ed <expan abbr="c&ugrave;m">cum</expan> <arrow.to.target n="note113"></arrow.to.target><lb/>
pondera FG &longs;int &aelig;qualia, habebit pondus H ad pondus G ean&shy;<lb/>
dem proportionem, quam habet ad F. vt igitur CA ad AB, ita <arrow.to.target n="note114"></arrow.to.target><lb/>
e&longs;t H ad G. vt autem H ad G, ita e&longs;t grauitas ip&longs;ius H ad graui<lb/>
tatem ip&longs;ius G; <expan abbr="c&ugrave;m">cum</expan> in eodem puncto B &longs;int appen&longs;a.  </s>
             
<s id="id.2.1.59.4.1.3.0"> quare vt CA <lb/>
ad AB, ita grauitas ponderis H ad grauitatem ponderis G. <expan abbr="c&ugrave;m">cum</expan> au<lb/>
tem grauitas ponderis F in C appen&longs;i &longs;it &aelig;qualis grauitati ponderis <lb/>
H in B; erit grauitas ponderis F ad grauitatem ponderis G, vt CA <lb/>
ad AB, videlicet vt di&longs;tantia ad di&longs;tantiam. quod demon&longs;trare <lb/>
oportebat. </s>
             
<s id="id.2.1.59.4.1.4.0"> [quod demon&longs;trare <lb/>
oportebat.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig72" place="text">       </figure>       
<p id="id.2.1.59.4.2.1.0" type="caption">        
<s id="id.2.1.59.4.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.60.1.0.0.0" type="margin">        
<s id="id.2.1.60.1.1.1.0"> <margin.target id="note113"></margin.target>6 <emph type="italics"/>Primi Ar chim. de &aelig;quep.<emph.end type="italics"/>  </s>
             
<s id="id.2.1.60.1.1.2.0"> [de &aelig;quep.<emph.end type="italics"/>] </s>
             
<s id="id.2.1.60.1.1.3.0"> <margin.target id="note114"></margin.target>7 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.61.1.0.0.0" type="main">        
<s id="id.2.1.61.1.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> libra B <lb/>
AC &longs;ecetur vtcunq; <lb/>
in D, &amp; in DC ap&shy;<lb/>
pendantur pondera <lb/>
&aelig;qualia EF.  </s>
     
<s id="id.2.1.61.1.1.1.0.a"> Dico <lb/>
&longs;imiliter ita e&longs;&longs;e gra&shy;<lb/>
<arrow.to.target n="fig73"></arrow.to.target><lb/>
uitatem ponderis F ad grauitatem ponderis E, vt di&longs;tantia CA ad <lb/>
di&longs;tantiam AD.  </s>
     
<s id="id.2.1.61.1.1.1.0.b"> fiat AB &aelig;qualis ip&longs;i AD, &amp; in B appendatur <lb/>
pondus G &aelig;quale ponderi E, &amp; ponderi F.  </s>
     
<s id="id.2.1.61.1.1.1.0.c"> Quoniam enim AB e&longs;t <lb/>
&aelig;qualis AD; pondera GE &aelig;queponderabunt.  </s>
             
<s id="id.2.1.61.1.1.2.0"> &longs;ed <expan abbr="c&ugrave;m">cum</expan> grauitas <lb/>
ponderis F ad grauitatem ponderis G &longs;it, vt CA ad AB, &amp; graui<lb/>
tas ponderis E &longs;it &aelig;qualis grauitati ponderis G; erit grauitas pon-<lb/>
deris F ad grauitatem ponderis E, vt CA ad AB, hoc e&longs;t vt CA <lb/>
ad AD. quod demon&longs;trare oportebat.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig73" place="text">       </figure>       
<p id="id.2.1.61.1.2.1.0" type="caption">        
<s id="id.2.1.61.1.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.61.2.0.0.0" type="head">        
<pb/>
       
<s id="id.2.1.61.3.1.1.0"> COROLLARIVM. </s>
    </p>       
<p id="id.2.1.61.4.0.0.0" type="main">        
<s id="id.2.1.61.4.1.1.0"> Ex hoc manife&longs;tum e&longs;t, <expan abbr="qu&ograve;">quo</expan> pondus <expan abbr="&agrave;">a</expan> centro <lb/>
libr&aelig; magis di&longs;tat, <expan abbr="e&ograve;">eo</expan> grauius e&longs;&longs;e; &amp; per con&longs;e&shy;<lb/>
quens velocius moueri. </s>
    </p>       
<p id="id.2.1.61.5.0.0.0" type="main">        
<s id="id.2.1.61.5.1.1.0"> <arrow.to.target n="note115"></arrow.to.target> Hinc pr&aelig;terea &longs;tater&aelig; quoq; ratio <expan abbr="facil&egrave;">facile</expan> o&longs;ten <lb/>
detur. </s>
    </p>       
<p id="id.2.1.62.1.0.0.0" type="margin">        
<s id="id.2.1.62.1.1.1.0"> <margin.target id="note115"></margin.target><emph type="italics"/>Stater&aelig; ratio.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.63.1.0.0.0" type="main">        
<s id="id.2.1.63.1.1.1.0"> Sit enim &longs;tate<lb/>
r&aelig; &longs;capus AB, cu<lb/>
ius trutina &longs;it in <lb/>
C; &longs;itq; &longs;tater&aelig; <lb/>
appendiculum E. <lb/>
appendatur in A <lb/>
pondus D, quod <lb/>
&aelig;queponderet ap<lb/>
pendiculo E in F <lb/>
<arrow.to.target n="fig74"></arrow.to.target><lb/>
appen&longs;o.  </s>
             
<s id="id.2.1.63.1.1.2.0"> aliud quoq; appendatur pondus G in A, quod etiam <lb/>
appendiculo E in B appen&longs;o &aelig;queponderet.  </s>
             
<s id="id.2.1.63.1.1.3.0"> Dico grauitatem <lb/>
ponderis D ad grauitatem ponderis G ita e&longs;&longs;e, vt CF ad CB.  </s>
     
<s id="id.2.1.63.1.1.3.0.a"> <lb/>
Quoniam enim grauitas ponderis D e&longs;t &aelig;qualis grauitati ponde&shy;<lb/>
ris E in F appen&longs;i, &amp; grauitas ponderis G e&longs;t &aelig;qualis grauitati pon<lb/>
deris E in B; erit grauitas ponderis D ad grauitatem ponderis E in <lb/>
F, vt grauitas ponderis G ad grauitatem ponderis E in B: &amp; permu<lb/>
<arrow.to.target n="note116"></arrow.to.target> tando, vt grauitas ponderis D ad grauitatem ponderis G, ita graui<lb/>
tas ip&longs;ius E in F, ad grauitatem ip&longs;ius E in B; grauitas autem pon <lb/>
<arrow.to.target n="note117"></arrow.to.target> deris E in F ad grauitatem ponderis E in B e&longs;t, vt CF ad CB; vt <lb/>
igitur grauitas ponderis D ad grauitatem ponderis G, ita e&longs;t CF <lb/>
ad CB &longs;i ergo pars &longs;capi CB in partes diuidatur &aelig;quales, &longs;olo <lb/>
pondere E, &amp; propius, &amp; longius <expan abbr="&agrave;">a</expan> puncto C po&longs;ito; ponderum <lb/>
grauitates, qu&aelig; ex puncto A &longs;u&longs;penduntur inter &longs;e &longs;e not&aelig; erunt.  </s>
             
<s id="id.2.1.63.1.1.4.0"> 
<pb n="36"/>
Vt &longs;i di&longs;tantia CB tripla &longs;it di&longs;tanti&aelig; CF, erit quoq; grauitas ip&shy;<lb/>
&longs;ius G grauitatis ip&longs;ius D tripla, quod demon&longs;trare oportebat. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig74" place="text">       </figure>       
<p id="id.2.1.63.1.2.1.0" type="caption">        
<s id="id.2.1.63.1.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.64.1.0.0.0" type="margin">        
<s id="id.2.1.64.1.1.1.0"> <margin.target id="note116"></margin.target>16 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.64.1.1.2.0"> <margin.target id="note117"></margin.target>6 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.65.1.0.0.0" type="main">        
<s id="id.2.1.65.1.1.1.0"> Alio quoq; modo &longs;tatera vti po&longs;&longs;umus, vt <lb/>
ponderum grauitates not&aelig; reddantur. </s>
    </p>       
<p id="id.2.1.65.2.0.0.0" type="main">        
<s id="id.2.1.65.2.1.1.0"> Sit &longs;capus AB, cuius tru&shy;<lb/>
tina &longs;it in C; &longs;itq; &longs;tater&aelig; ap<lb/>
pendiculum E, quod appen&shy;<lb/>
datur in A; <expan abbr="&longs;intqu&eacute;">&longs;intque</expan> pon&shy;<lb/>
dera DG in&aelig;qualia, quorum <lb/>
inter &longs;e &longs;e grauitatum propor&shy;<lb/>
tiones qu&aelig;rimus: appenda&shy;<lb/>
tur pondus D in B, ita vt ip&longs;i <lb/>
<arrow.to.target n="fig75"></arrow.to.target><lb/>
E &aelig;queponderet.  </s>
             
<s id="id.2.1.65.2.1.2.0"> &longs;imiliter pondus G appendatur in F, quod ei&shy;<lb/>
dem ponderi E &aelig;queponderet.  </s>
             
<s id="id.2.1.65.2.1.3.0"> dico D ad G ita e&longs;&longs;e, vt CF ad <lb/>
CB.  </s>
     
<s id="id.2.1.65.2.1.3.0.a"> Quoniam enim pondera DE &aelig;queponderant, erit D ad E, <arrow.to.target n="note118"></arrow.to.target><lb/>
vt CA ad CB. <expan abbr="c&ugrave;m">cum</expan> autem pondera quoque GE &aelig;quepon&shy;<lb/>
derent, erit pondus E ad pondus G, vt FC ad CA; quare ex &aelig;qua <lb/>
li pondus D ad pondus G ita erit, vt CF ad CB. quod o&longs;tende <arrow.to.target n="note119"></arrow.to.target><lb/>
re quoq; oportebat.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig75" place="text">       </figure>       
<p id="id.2.1.65.2.2.1.0" type="caption">        
<s id="id.2.1.65.2.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.66.1.0.0.0" type="margin">        
<s id="id.2.1.66.1.1.1.0"> <margin.target id="note118"></margin.target>6 <emph type="italics"/>Primi Ar chim. de &aelig;quep.<emph.end type="italics"/> </s>
             
<s id="id.2.1.66.1.1.2.0"> [de &aelig;quep.<emph.end type="italics"/>] </s>
             
<s id="id.2.1.66.1.1.3.0"> <margin.target id="note119"></margin.target>23 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.67.1.0.0.0" type="head">        
<pb/>
       
<s id="id.2.1.67.1.2.1.0"> PROPOSITIO VII. </s>
     <lb/>
       
<s id="id.2.1.67.1.4.1.0"> PROBLEMA. </s>
    </p>       
<p id="id.2.1.67.2.0.0.0" type="main">        
<s id="id.2.1.67.2.1.1.0"> Quotcunque datis in libra ponderibus <lb/>
vbicunque appen&longs;is, centrum libr&aelig; inuenire, <lb/>
ex quo &longs;i &longs;u&longs;pendatur libra, data pondera ma&shy;<lb/>
neant. <arrow.to.target n="fig76"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.67.3.0.0.0" type="main">        
<s id="id.2.1.67.3.1.1.0"> Sit libra AB, &longs;intq; data quotcunque pondera CDEFG. <lb/>
accipiantur in libra vtcunque puncta AHkLB, ex quibus <lb/>
data pondera &longs;pu&longs;pendantur.  </s>
             
<s id="id.2.1.67.3.1.2.0"> Centrum libr&aelig; inuenire oportet, <lb/>
ex quo &longs;i fiat &longs;u&longs;pen&longs;io, data pondera maneant.  </s>
             
<s id="id.2.1.67.3.1.3.0"> Diuidatur 
<pb n="37"/>
<arrow.to.target n="fig77"></arrow.to.target><lb/>
AH in M, ita vt HM ad MA, &longs;it vt grauitas ponderis <lb/>
C ad grauitatem ponderis D.  </s>
     
<s id="id.2.1.67.3.1.3.0.a"> deinde diuidatur BL in N, ita <lb/>
vt LN ad NB, &longs;it vt grauitas ponderis G ad grauitatem pon<lb/>
deris F. diuidaturq; MN in O, ita vt MO ad ON &longs;it, vt <lb/>
grauitas ponderum FG ad grauitatem ponderum CD.  </s>
     
<s id="id.2.1.67.3.1.3.0.b"> <expan abbr="tandem&shy;qu&eacute;">tandem&shy;<lb/>
que</expan> diuidatur kO in P, ita vt kP ad PO, &longs;it vt grauitas pon<lb/>
derum CDFG ad grauitatem ponderis E.  </s>
     
<s id="id.2.1.67.3.1.3.0.c"> Quoniam igitur pon <lb/>
dera CDFG <expan abbr="t&agrave;m">tam</expan> ponderant in O, <expan abbr="qu&agrave;m">quam</expan> CD in M, &amp; FG in N; <arrow.to.target n="note120"></arrow.to.target><lb/>
&aelig;queponderabunt pondera CD in M, &amp; FG in N, &amp; pondus E <lb/>
in K, &longs;i ex puncto P &longs;u&longs;pendantur.  </s>
             
<s id="id.2.1.67.3.1.4.0"> <expan abbr="c&ugrave;m">cum</expan> <expan abbr="ver&ograve;">vero</expan> pondera CD tan<lb/>
<expan abbr="t&ugrave;m">tum</expan> ponderent in M, <expan abbr="quant&ugrave;m">quantum</expan> in AH, &amp; FG in N, <expan abbr="quant&ugrave;m">quantum</expan> <lb/>
in LB; pondera CDFG ex AHLB punctis &longs;u&longs;pen&longs;a, &amp; pon&shy;<lb/>
dus E ex k, &longs;i ex P &longs;u&longs;pendantur, &aelig;queponderabunt, atq; mane&shy;<lb/>
bunt.  </s>
             
<s id="id.2.1.67.3.1.5.0"> Inuentum e&longs;t ergo centrum libr&aelig; P, ex quo data pondera <lb/>
manent. quod facere oportebat. </s>
             
<s id="id.2.1.67.3.1.6.0"> [quod facere oportebat.]  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig76" place="text">       </figure>       
<p id="id.2.1.67.3.2.1.0" type="caption">        
<s id="id.2.1.67.3.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig77" place="text">       </figure>       
<p id="id.2.1.67.3.2.3.0" type="caption">        
<s id="id.2.1.67.3.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.68.1.0.0.0" type="margin">        
<s id="id.2.1.68.1.1.1.0"> <margin.target id="note120"></margin.target>5 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.69.1.0.0.0" type="head">        
<pb/>
       
<s id="id.2.1.69.1.2.1.0"> COROLLARIVM. </s>
    </p>       
<p id="id.2.1.69.2.0.0.0" type="main">        
<s id="id.2.1.69.2.1.1.0"> Ex hoc manife&longs;tum e&longs;t, &longs;i ponderum CDEFG <lb/>
centra grauitatis e&longs;&longs;ent in AHKLB punctis; e&longs;&shy;<lb/>
&longs;et punctum P magnitudinis ex omnibus CD <lb/>
EFG ponderibus compo&longs;it&aelig; centrum graui&shy;<lb/>
tatis. <arrow.to.target n="fig78"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.69.3.0.0.0" type="main">        
<s id="id.2.1.69.3.1.1.0"> Hoc enim ex definitione centri grauitatis patet, <expan abbr="c&ugrave;m">cum</expan> ponde&shy;<lb/>
ra, &longs;i ex puncto P &longs;u&longs;pendantur, maneant.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig78" place="text">       </figure>       
<p id="id.2.1.69.3.2.1.0" type="caption">        
<s id="id.2.1.69.3.2.1.0.capt"> YYY </s>
    </p>       

      </chap>
      <pb n="38"/>
      <chap>
<p id="id.2.1.69.4.0.0.0" type="head">        
<s id="id.2.1.69.5.1.1.0"> DE VECTE. </s>
     <lb/>
<s id="id.2.1.69.5.3.1.0"> LEMMA. </s>
    </p>       
<p id="id.2.1.69.6.0.0.0" type="main">        
<s id="id.2.1.69.6.1.1.0"> Sint quatuor magnitudines A <lb/>
BCD; &longs;itq; A maior B, &amp; C ma<lb/>
ior D.  </s>
     
<s id="id.2.1.69.6.1.1.0.a"> Dico A ad D maiorem <lb/>
habere proportionem; <expan abbr="qu&agrave;m">quam</expan> <lb/>
habet B ad C. </s>
    </p>       
<p id="id.2.1.69.7.0.0.0" type="main">        
<s id="id.2.1.69.7.1.1.0"> Quoniam enim A ad C maiorem habet pro&shy;<lb/>
portionem, <expan abbr="qu&agrave;m">quam</expan> B ad C; &amp; A ad D maio&shy;<lb/>
rem  <arrow.to.target n="note121"></arrow.to.target> quoq; habet proportionem, quam habet <lb/>
ad C: A igitur ad D maiorem habebit, quam B <lb/>
ad C. quod demon&longs;trare oportebat.  </s>
     <lb/>
      </p>       
<p id="id.2.1.70.1.0.0.0" type="margin">        
<s id="id.2.1.70.1.1.1.0"> <margin.target id="note121"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.71.1.0.0.0" type="main">        

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure place="text">       </figure>       
<p id="id.2.1.71.1.1.1.0" type="caption">        
<s id="id.2.1.71.1.1.1.0.capt"> YYY </s>
     <lb/>
       
<s id="id.2.1.71.1.3.1.0"> PROPOSITIO I. </s>
    </p>       
<p id="id.2.1.71.2.0.0.0" type="main">        
<s id="id.2.1.71.2.1.1.0"> Potentia &longs;u&longs;tinens pondus vecti appen&longs;um; <lb/>
eandem ad ip&longs;um pondus proportionem habe&shy;<lb/>
bit, quam vectis di&longs;tantia inter fulcimentum, ac <lb/>
ponderis &longs;u&longs;pen&longs;ionem ad di&longs;tantiam <expan abbr="&agrave;">a</expan> fulcimen<lb/>
to ad potentiam interiectam. 
<pb/>
<arrow.to.target n="fig79"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.71.3.0.0.0" type="main">        
<s id="id.2.1.71.3.1.1.0"> Sit vectis AB, cuius fulcimentum C; &longs;itq; pondus D ex A &longs;u&shy;<lb/>
&longs;pen&longs;um AH, ita vt AH &longs;it &longs;emper horizonti perpendicularis: <lb/>
&longs;itq; potentia &longs;u&longs;tinens pondus in B.  </s>
     
<s id="id.2.1.71.3.1.1.0.a"> Dico potentiam in B ad pon<lb/>
dus D ita e&longs;&longs;e, vt CA ad CB.  </s>
     
<s id="id.2.1.71.3.1.1.0.b"> fiat vt BC ad CA, ita pondus D <lb/>
<arrow.to.target n="note122"></arrow.to.target> ad aliud pondus E, <expan abbr="quipp&egrave;">quippe</expan> quod &longs;i in B appendatur; ip&longs;i D &aelig;que <lb/>
ponderabit, exi&longs;tente C amborum grauitatis centro.  </s>
             
<s id="id.2.1.71.3.1.2.0"> quare poten<lb/>
tia &aelig;qualis ip&longs;i E ibidem con&longs;tituta ip&longs;i D &aelig;queponderabit, vecte <lb/>
AB, eius fulcimento in C collocato, hoc e&longs;t prohibebit, ne pon<lb/>
dus D deor&longs;um vergat, quemadmodum prohibet pondus E.  </s>
     
<s id="id.2.1.71.3.1.2.0.a"> Po<lb/>
<arrow.to.target n="note123"></arrow.to.target> tentia <expan abbr="ver&ograve;">vero</expan> in B ad pondus D eandem habet proportionem, quam <lb/>
pondus E ad idem pondus D: ergo potentia in B ad pondus D <lb/>
erit, vt CA ad CB; hoc e&longs;t vectis di&longs;tantia <expan abbr="&agrave;">a</expan> fulcimento ad pon<lb/>
deris &longs;u&longs;pendium ad di&longs;tantiam <expan abbr="&agrave;">a</expan> fulcimento ad potentiam. quod <lb/>
demon&longs;trare oportebat. </s>
             
<s id="id.2.1.71.3.1.3.0"> [quod <lb/>
demon&longs;trare oportebat.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig79" place="text">       </figure>       
<p id="id.2.1.71.3.2.1.0" type="caption">        
<s id="id.2.1.71.3.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.72.1.0.0.0" type="margin">        
<s id="id.2.1.72.1.1.1.0"> <margin.target id="note122"></margin.target>6 <emph type="italics"/>Primi Ar chim. de &aelig;quep.<emph.end type="italics"/> </s>
             
<s id="id.2.1.72.1.1.2.0"> [de &aelig;quep.<emph.end type="italics"/>] </s>
             
<s id="id.2.1.72.1.1.3.0"> <margin.target id="note123"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 7 <emph type="italics"/>quinti.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.73.1.0.0.0" type="main">        
<s id="id.2.1.73.1.1.1.0"> Hinc <expan abbr="facil&egrave;">facile</expan> o&longs;tendi pote&longs;t, fulcimentum <expan abbr="qu&ograve;">quo</expan> <lb/>
ponderi fuerit propius, minorem ad idem pon&shy;<lb/>
dus &longs;u&longs;tinendum requiri potentiam. </s>
    </p>       
<p id="id.2.1.73.2.0.0.0" type="main">        
<s id="id.2.1.73.2.1.1.0"> Ii&longs;dem po&longs;i&shy;<lb/>
tis, &longs;it fulcimen <lb/>
tum in F ip&longs;i A <lb/>
propius, <expan abbr="qu&agrave;m">quam</expan> <lb/>
C; fiatq; vt BF <lb/>
ad FA, ita pon<lb/>
dus D ad aliud <lb/>
<arrow.to.target n="fig80"></arrow.to.target><lb/>
G, quod &longs;i appendatur in B, pondera DG ex fulcimento E <lb/>
<arrow.to.target n="note124"></arrow.to.target> &aelig;queponderabunt.  </s>
             
<s id="id.2.1.73.2.1.2.0"> quoniam autem BF maior e&longs;t BC, &amp; CA <lb/>
<arrow.to.target n="note125"></arrow.to.target> maior AC; maior erit proportio BF ad FA, <expan abbr="qu&agrave;m">quam</expan> BC ad CA: 
<pb n="39"/>
&amp; ideo maior quoq; erit proportio ponderis D ad pondus G, <lb/>
<expan abbr="qu&agrave;m">quam</expan> idem D ad E: pondus igitur G minus erit pondere E. <expan abbr="c&ugrave;m">cum</expan> <arrow.to.target n="note126"></arrow.to.target><lb/>
autem potentia in B ip&longs;i G &aelig;qualis ponderi D &aelig;queponderet, mi&shy;<lb/>
nor potentia, <expan abbr="qu&agrave;m">quam</expan> ea, qu&aelig; ponderi E e&longs;t &aelig;qualis, pondus D &longs;u<lb/>
&longs;tinebit; exi&longs;tente vecte AB, eius <expan abbr="ver&ograve;">vero</expan> fulcimento vbi F, <expan abbr="qu&agrave;m">quam</expan> &longs;i <lb/>
fuerit vbi C. &longs;imiliter quoq; o&longs;tendetur, <expan abbr="qu&ograve;">quo</expan> propius erit fulci&shy;<lb/>
mentum ponderi D, adhuc &longs;emper minorem requiri potentiam <lb/>
ad &longs;u&longs;tinendum pondus D. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig80" place="text">       </figure>       
<p id="id.2.1.73.2.2.1.0" type="caption">        
<s id="id.2.1.73.2.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.74.1.0.0.0" type="margin">        
<s id="id.2.1.74.1.1.1.0"> <margin.target id="note124"></margin.target><emph type="italics"/>Ex eadem Sexta.<emph.end type="italics"/> </s>
             
<s id="id.2.1.74.1.1.2.0"> <margin.target id="note125"></margin.target><emph type="italics"/>Lemma.<emph.end type="italics"/> </s>
             
<s id="id.2.1.74.1.1.3.0"> <margin.target id="note126"></margin.target>10 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.75.1.0.0.0" type="head">        
<s id="id.2.1.75.1.1.1.0"> COROLLARIVM. </s>
    </p>       
<p id="id.2.1.75.2.0.0.0" type="main">        
<s id="id.2.1.75.2.1.1.0"> Vnde <expan abbr="pal&agrave;m">palam</expan> colligere licet, exi&longs;tente AF ip&longs;a <lb/>
FB minore, minorem quoq; requiri potentiam <lb/>
in ip&longs;o B pondere D &longs;u&longs;tinendo.  </s>
             
<s id="id.2.1.75.2.1.2.0"> &aelig;quali <expan abbr="ver&ograve;">vero</expan> <lb/>
&aelig;qualem. maiore <expan abbr="ver&ograve;">vero</expan> maiorem.  </s>
             
<s id="id.2.1.75.2.1.3.0"> [maiore <expan abbr="ver&ograve;">vero</expan> maiorem.]  </s>
    </p>       
<p id="id.2.1.75.3.0.0.0" type="head">        
<s id="id.2.1.75.3.1.1.0"> PROPOSITIO II. </s>
    </p>       
<p id="id.2.1.75.4.0.0.0" type="main">        
<s id="id.2.1.75.4.1.1.0"> Alio modo vecte vti po&longs;sumus. </s>
    </p>       
<p id="id.2.1.75.5.0.0.0" type="main">        
<s id="id.2.1.75.5.1.1.0"> Sit vectis AB, cuius <lb/>
fulcimentum &longs;it B, &amp; <lb/>
pondus C vtcunq; in <lb/>
D inter AB appen&shy;<lb/>
&longs;um; &longs;itq; potentia in <lb/>
A &longs;u&longs;tinens pondus C.  </s>
     
<s id="id.2.1.75.5.1.1.0.a"> <lb/>
Dico vt BD ad BA, <lb/>
<arrow.to.target n="fig81"></arrow.to.target><lb/>
ita e&longs;&longs;e potentiam in A ad pondus C. appendatur in A pondus <lb/>
E &aelig;quale ip&longs;i C; &amp; vt AB ad BD, ita fiat pondus E ad aliud F. <lb/>
&amp; quoniam pondera CE &longs;unt inter &longs;e &longs;e &aelig;qualia, erit pondus C <lb/>
ad pondus F, vt AB ad BD. appendatur quoq; pondus F in A. <lb/>
&amp; quoniam pondus E ad pondus F e&longs;t, vt grauitas ip&longs;ius E ad gra&shy;<lb/>
uitatem  <arrow.to.target n="note127"></arrow.to.target> ip&longs;ius F; &amp; pondus E ad F e&longs;t, vt AB ad BD; vt igitur <lb/>
grauitas ponderis E ad grauitatem ponderis F, ita e&longs;t AB ab BD. <lb/>
vt autem AB ad BD, ita e&longs;t grauitas ponderis E ad grauitatem <arrow.to.target n="note128"></arrow.to.target>
<pb/>
ponderis C: quare gra<lb/>
uitas ponderis E ad <lb/>
grauitatem ponderis <lb/>
F ita erit, vt grauitas <lb/>
ponderis E ad gra&shy;<lb/>
uitatem ponderis C.  </s>
     
<s id="id.2.1.75.5.1.1.0.b"> <lb/>
Pondera igitur CF <lb/>
<arrow.to.target n="fig82"></arrow.to.target><lb/>
<arrow.to.target n="note129"></arrow.to.target> eandem habent grauitatem.  </s>
             
<s id="id.2.1.75.5.1.2.0"> Ponatur itaq; potentia in A &longs;u&longs;tinens <lb/>
pondus F; erit potentia in A &aelig;qualis ip&longs;i ponderi F.  </s>
     
<s id="id.2.1.75.5.1.2.0.a"> &amp; quoniam <lb/>
pondus F in A appen&longs;um <expan abbr="&aelig;qu&egrave;">&aelig;que</expan> graue e&longs;t, vt pondus C in D ap&shy;<lb/>
pen&longs;um; eandem proportionem habebit potentia in A ad grauita&shy;<lb/>
<arrow.to.target n="note130"></arrow.to.target> tem ponderis F in A appen&longs;i, quam habet ad grauitatem ponde&shy;<lb/>
ris C in D appen&longs;i.  </s>
             
<s id="id.2.1.75.5.1.3.0"> Potentia <expan abbr="ver&ograve;">vero</expan> in A ip&longs;i F &aelig;qualis &longs;u&longs;tinet <lb/>
pondus F, ergo potentia in A pondus quoq; C &longs;u&longs;tinebit.  </s>
             
<s id="id.2.1.75.5.1.4.0"> Itaq; <lb/>
<expan abbr="c&ugrave;m">cum</expan> potentia in A &longs;it &aelig;qualis ponderi F, &amp; pondus C ad pon&shy;<lb/>
dus F &longs;it, vt AB ad BD; erit pondus C ad potentiam in A, vt <lb/>
<arrow.to.target n="note131"></arrow.to.target> AB ad BD. &amp; <expan abbr="&egrave;">e</expan> conuer&longs;o, vt BD ad BA, ita potentia in A ad <lb/>
pondus C. potentia ergo ad pondus ita erit, vt di&longs;tantia fulci&shy;<lb/>
mento, ac ponderis &longs;u&longs;pen&longs;ioni intercepta ad di&longs;tantiam <expan abbr="&agrave;">a</expan> fulci <lb/>
mento ad potentiam. quod oportebat demon&longs;trare. </s>
             
<s id="id.2.1.75.5.1.5.0"> [quod oportebat demon&longs;trare.]  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig81" place="text">       </figure>       
<p id="id.2.1.75.5.2.1.0" type="caption">        
<s id="id.2.1.75.5.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig82" place="text">       </figure>       
<p id="id.2.1.75.5.2.3.0" type="caption">        
<s id="id.2.1.75.5.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.76.1.0.0.0" type="margin">        
<s id="id.2.1.76.1.1.1.0"> <margin.target id="note127"></margin.target><emph type="italics"/>In &longs;exta huius de libra Ex<emph.end type="italics"/> 11 <emph type="italics"/>quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.76.1.1.2.0"> <margin.target id="note128"></margin.target>6 <emph type="italics"/>Huius. de libra.<emph.end type="italics"/> </s>
             
<s id="id.2.1.76.1.1.3.0"> [de libra.<emph.end type="italics"/>] </s>
             
<s id="id.2.1.76.1.1.4.0"> <margin.target id="note129"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 9 <emph type="italics"/>quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.76.1.1.5.0"> <margin.target id="note130"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 7 <emph type="italics"/>quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.76.1.1.6.0"> <margin.target id="note131"></margin.target><emph type="italics"/>Cor.<emph.end type="italics"/> 4 <emph type="italics"/>quinti.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.77.1.0.0.0" type="head">        
<s id="id.2.1.77.1.1.1.0"> ALITER. </s>
     <lb/>
       

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure place="text">       </figure>       
<p id="id.2.1.77.1.3.1.0" type="caption">        
<s id="id.2.1.77.1.3.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.77.2.0.0.0" type="main">        
<s id="id.2.1.77.2.1.1.0"> Sit vectis AB, cuius fulcimentum &longs;it B, &amp; pondus E ex puncto <lb/>
C &longs;u&longs;pen&longs;um; &longs;itq; vis in A &longs;u&longs;tinens pondus E.  </s>
     
<s id="id.2.1.77.2.1.1.0.a"> Dico vt BC ad BA, <lb/>
ita e&longs;&longs;e potentiam in A ad pondus E.  </s>
     
<s id="id.2.1.77.2.1.1.0.b"> Producatur AB in C, &amp; <lb/>
fiat BD &aelig;qualis BC; &amp; ex puncto D appendatur pondus F &aelig;qua <lb/>
le ponderi E; itemq; ex puncto A &longs;u&longs;pendatur pondus G ita, vt <lb/>
pondus F ad pondus G eandem habeat proportionem, quam AB 
<pb n="40"/>
ad BA. pondera FG &aelig;queponderabunt.  </s>
             
<s id="id.2.1.77.2.1.2.0"> <expan abbr="c&ugrave;m">cum</expan> autem &longs;it CB &aelig;qua <lb/>
lis BD, pondera quoq; FE &aelig;qualia &aelig;queponderabunt.  </s>
             
<s id="id.2.1.77.2.1.3.0"> pondera <lb/>
<expan abbr="ver&ograve;">vero</expan> FEG in libra, &longs;eu vecte DBA appen&longs;a, cuius fulcimentum <lb/>
e&longs;t B, non &aelig;queponderabunt; &longs;ed ex parte A deor&longs;um tendent.  </s>
             
<s id="id.2.1.77.2.1.4.0"> po<lb/>
natur itaq; in A tanta vis, vt pondera FEG &aelig;queponderent; erit <lb/>
potentia in A &aelig;qualis ponderi G. pondera enim FE <expan abbr="&aelig;queponder&atilde;t">&aelig;queponderant</expan>, <lb/>
&amp; vis in A nihil aliud efficere debet, ni&longs;i &longs;u&longs;tinere <expan abbr="p&otilde;dus">pondus</expan> G, ne de&longs;cen<lb/>
dat.  </s>
             
<s id="id.2.1.77.2.1.5.0"> &amp; quoniam pondera FEG, &amp; potentia in A &aelig;queponderant, <lb/>
demptis igitur FG ponderibus, qu&aelig; &aelig;queponderant, reliqua &aelig;que <lb/>
ponderabunt; &longs;cilicet potentia in A ponderi E, hoc e&longs;t potentia <lb/>
in A pondus E &longs;u&longs;tinebit, ita vt vectis AB maneat, vt prius erat.  </s>
             
<s id="id.2.1.77.2.1.6.0"> <lb/>
<expan abbr="C&ugrave;m">Cum</expan> autem potentia in A &longs;it &aelig;qualis ponderi G, &amp; pondus E pon<lb/>
deri F &aelig;quale; habebit potentia in A ad pondus E eandem pro&shy;<lb/>
portionem, quam habet BD, hoc e&longs;t BC ad BA. quod demon&shy;<lb/>
&longs;trare oportebat.  </s>
    </p>       
<p id="id.2.1.77.3.0.0.0" type="head">        
<s id="id.2.1.77.3.1.1.0"> COROLLARIVM I. </s>
    </p>       
<p id="id.2.1.77.4.0.0.0" type="main">        
<s id="id.2.1.77.4.1.1.0"> Ex hoc etiam (vt prius) manife&longs;tum e&longs;&longs;e po&shy;<lb/>
te&longs;t, &longs;i ponatur pondus E propius fulcimento B, <lb/>
vt in H; minorem potentiam in A &longs;u&longs;tinere po&longs;&shy;<lb/>
&longs;e ip&longs;um pondus. </s>
    </p>       
<p id="id.2.1.77.5.0.0.0" type="main">        
<s id="id.2.1.77.5.1.1.0"> Minorem enim proportionem habet HB ad BA, quam CB ad <arrow.to.target n="note132"></arrow.to.target><lb/>
BA. &amp; <expan abbr="qu&ograve;">quo</expan> propius pondus erit fulcimento, adhuc &longs;emper mino <lb/>
rem po&longs;&longs;e potentiam &longs;u&longs;tinere pondus E &longs;imiliter o&longs;tendetur.  </s>
    </p>       
<p id="id.2.1.78.1.0.0.0" type="margin">        
<s id="id.2.1.78.1.1.1.0"> <margin.target id="note132"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.79.1.0.0.0" type="head">        
<s id="id.2.1.79.1.1.1.0"> COROLLARIVM II. </s>
    </p>       
<p id="id.2.1.79.2.0.0.0" type="main">        
<s id="id.2.1.79.2.1.1.0"> Sequitur etiam potentiam in A &longs;emper mino <lb/>
rem e&longs;&longs;e pondere E. </s>
    </p>       
<p id="id.2.1.79.3.0.0.0" type="main">        
<s id="id.2.1.79.3.1.1.0"> Sumatur enim inter AB quoduis punctum C, &longs;emper BC <lb/>
minor erit BA. </s>
    </p>       
<pb/>
       
<p id="id.2.1.79.5.0.0.0" type="head">        
<s id="id.2.1.79.5.1.1.0"> COROLLARIVM III. </s>
    </p>       
<p id="id.2.1.79.6.0.0.0" type="main">        
<s id="id.2.1.79.6.1.1.0"> Ex hoc quoq; elici pote&longs;t, &longs;i du&aelig; fuerint poten<lb/>
ti&aelig;, vna in A, altera in B, &amp; vtraq; &longs;u&longs;tentet <lb/>
pondus E; potentiam in A ad potentiam in B e&longs;&shy;<lb/>
&longs;e, vt BC ad CA. </s>
    </p>       
<p id="id.2.1.79.7.0.0.0" type="main">        
<s id="id.2.1.79.7.1.1.0"> Vectis enim BA fungi&shy;<lb/>
tur officio duorum <expan abbr="vecti&utilde;">vectium</expan>; <lb/>
&amp; AB &longs;unt tanquam duo <lb/>
fulcimenta, hoc e&longs;t quan&shy;<lb/>
do AB e&longs;t vectis, &amp; poten<lb/>
tia &longs;u&longs;tinens in A; erit eius <lb/>
<arrow.to.target n="fig83"></arrow.to.target><lb/>
fulcimentum B.  </s>
     
<s id="id.2.1.79.7.1.1.0.a"> Quando <expan abbr="ver&ograve;">vero</expan> BA e&longs;t vectis, &amp; potentia in B; <lb/>
erit A fulcimentum: &amp; pondus &longs;emper ex puncto C remanet &longs;u&shy;<lb/>
&longs;pen&longs;um.  </s>
             
<s id="id.2.1.79.7.1.2.0"> &amp; quoniam potentia in A ad pondus E e&longs;t, vt BC ad <lb/>
BA; vt autem pondus E ad potentiam, qu&aelig; e&longs;t in B, ita e&longs;t <lb/>
<arrow.to.target n="note133"></arrow.to.target> BA ad AC; erit ex &aelig;quali, potentia in A ad potentiam in B, vt <lb/>
BC ad CA. &amp; hoc modo <expan abbr="facil&egrave;">facile</expan> etiam proportionem, qu&aelig; in <lb/>
Qu&aelig;&longs;tionibus Mechanicis qu&aelig;&longs;tione vige&longs;ima nona ab Ari&longs;totele <lb/>
ponitur, noui&longs;&longs;e poterimus.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig83" place="text">       </figure>       
<p id="id.2.1.79.7.2.1.0" type="caption">        
<s id="id.2.1.79.7.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.80.1.0.0.0" type="margin">        
<s id="id.2.1.80.1.1.1.0"> <margin.target id="note133"></margin.target>22 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.81.1.0.0.0" type="head">        
<s id="id.2.1.81.1.1.1.0"> COROLLARIVM IIII. </s>
    </p>       
<p id="id.2.1.81.2.0.0.0" type="main">        
<s id="id.2.1.81.2.1.1.0"> E&longs;t etiam manife&longs;tum, vtra&longs;q; potentias in A, <lb/>
&amp; B &longs;imul &longs;umptas &aelig;quales e&longs;&longs;e ponderi E. </s>
    </p>       
<p id="id.2.1.81.3.0.0.0" type="main">        
<s id="id.2.1.81.3.1.1.0"> Pondus enim E ad potentiam in A e&longs;t, vt BA ad BC; &amp; idem <lb/>
pondus E ad potentiam in B e&longs;t, vt BA ad AC; quare pondus <lb/>
E ad vtra&longs;q; potentias in A, &amp; B &longs;imul &longs;umptas e&longs;t, vt AB ad BC <lb/>
CA &longs;imul, hoc e&longs;t ad BA. pondus igitur E vtri&longs;q; potentiis &longs;imul <lb/>
&longs;umptis &aelig;quale erit.  </s>
    </p>       
<p id="id.2.1.81.4.0.0.0" type="head">        
<pb n="41"/>
       
<s id="id.2.1.81.5.1.1.0"> PROPOSITIO III. </s>
    </p>       
<p id="id.2.1.81.6.0.0.0" type="main">        
<s id="id.2.1.81.6.1.1.0"> Alio quoq; modo vecte vti po&longs;sumus. </s>
    </p>       
<p id="id.2.1.81.7.0.0.0" type="main">        
<s id="id.2.1.81.7.1.1.0"> Sit Vectis AB, <lb/>
cuius fulcimentum <lb/>
B; &longs;itq; ex puncto <lb/>
A pondus C appen&shy;<lb/>
&longs;um; &longs;itq; potentia <lb/>
in D vtcunq; inter <lb/>
AB &longs;u&longs;tinens pon&shy;<lb/>
dus C.  </s>
     
<s id="id.2.1.81.7.1.1.0.a"> Dico vt AB <lb/>
<arrow.to.target n="fig84"></arrow.to.target><lb/>
ad BD, ita e&longs;&longs;e potentiam in D ad pondus C.  </s>
     
<s id="id.2.1.81.7.1.1.0.b"> Appendatur ex <lb/>
puncto D pondus E &aelig;quale ip&longs;i C; &amp; vt BD ad BA, ita fiat pon<lb/>
dus E ad aliud F. &amp; <expan abbr="c&ugrave;m">cum</expan> pondera CE &longs;int inter &longs;e &longs;e &aelig;qualia; erit <lb/>
pondus C ad pondus F, vt BD ad BA.  </s>
     
<s id="id.2.1.81.7.1.1.0.c"> appendatur pondus <lb/>
F quoq; in D.  </s>
     
<s id="id.2.1.81.7.1.1.0.d"> &amp; quoniam pondus E ad ip&longs;um F e&longs;t, vt grauitas <lb/>
ponderis E ad grauitatem ponderis F; &amp; pondus E ad pondus F <arrow.to.target n="note134"></arrow.to.target><lb/>
e&longs;t, vt BD ad BA: vt igitur grauitas ponderis E ad grauitatem <lb/>
ponderis F, ita e&longs;t BD ad BA. vt autem BD ad BA, ita e&longs;t gra <arrow.to.target n="note135"></arrow.to.target><lb/>
uitas ponderis E ad grauitatem ponderis C; quare grauitas ponde&shy;<lb/>
ris E ad grauitatem ponderis F eandem habet proportionem, <lb/>
quam habet ad grauitatem ponderis C. pondera ergo CF eandem <arrow.to.target n="note136"></arrow.to.target><lb/>
habent grauitatem.  </s>
             
<s id="id.2.1.81.7.1.2.0"> &longs;it igitur potentia in D &longs;u&longs;tinens pondus F, <lb/>
erit potentia in D ip&longs;i ponderi F &aelig;qualis.  </s>
             
<s id="id.2.1.81.7.1.3.0"> &amp; quoniam pondus F <lb/>
in D <expan abbr="&aelig;qu&egrave;">&aelig;que</expan> graue e&longs;t, vt pondus C in A; habebit potentia in D <lb/>
eandem proportionem ad grauitatem ponderis F, quam habet ad <arrow.to.target n="note137"></arrow.to.target><lb/>
grauitatem ponderis C.  </s>
     
<s id="id.2.1.81.7.1.3.0.a"> &longs;ed potentia in D pondus F &longs;u&longs;tinet; po&shy;<lb/>
tentia igitur in D pondus quoq; C &longs;u&longs;tinebit: &amp; pondus C ad po&shy;<lb/>
tentiam in D ita erit, vt pondus C ad pondus F; &amp; C ad F e&longs;t, vt <lb/>
BD ad BA; erit igitur pondus C ad potentiam in D, vt BD ad <lb/>
BA: &amp; conuertendo, vt AB ad BD, ita potentia in D ad pondus <lb/>
C.  </s>
     
<s id="id.2.1.81.7.1.3.0.b"> potentia ergo ad pondus e&longs;t, vt di&longs;tantia <expan abbr="&agrave;">a</expan> fulcimento ad pon<lb/>
deris &longs;u&longs;pendium ad di&longs;tantiam <expan abbr="&agrave;">a</expan> fulcimento ad potentiam. quod <lb/>
demon&longs;trare oportebat. </s>
             
<s id="id.2.1.81.7.1.4.0"> [quod <lb/>
demon&longs;trare oportebat.]  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig84" place="text">       </figure>       
<p id="id.2.1.81.7.2.1.0" type="caption">        
<s id="id.2.1.81.7.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.82.1.0.0.0" type="margin">        
<s id="id.2.1.82.1.1.1.0"> <margin.target id="note134"></margin.target><emph type="italics"/>In &longs;exta huius de libra.<emph.end type="italics"/> </s>
             
<s id="id.2.1.82.1.1.2.0"> <margin.target id="note135"></margin.target>6 <emph type="italics"/>Huius de libra.<emph.end type="italics"/> </s>
             
<s id="id.2.1.82.1.1.3.0"> <margin.target id="note136"></margin.target>9 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.82.1.1.4.0"> <margin.target id="note137"></margin.target>7 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.83.1.0.0.0" type="head">        
<pb/>
       
<s id="id.2.1.83.1.2.1.0"> ALITER. </s>
     <lb/>
       

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure place="text">       </figure>       
<p id="id.2.1.83.1.4.1.0" type="caption">        
<s id="id.2.1.83.1.4.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.83.2.0.0.0" type="main">        
<s id="id.2.1.83.2.1.1.0"> Sit vectis AB, cuius fulcimentum B; &amp; ex puncto A &longs;it pon&shy;<lb/>
dus C &longs;u&longs;pen&longs;um; &longs;itq; potentia in D &longs;u&longs;tinens pondus C.  </s>
     
<s id="id.2.1.83.2.1.1.0.a"> Dico <lb/>
vt AB ad BD, ita e&longs;&longs;e potentiam in D ad pondus C.  </s>
     
<s id="id.2.1.83.2.1.1.0.b"> Produca<lb/>
tur AB in E, fiatq; BE &aelig;qualis ip&longs;i BA; &amp; ex puncto E appen<lb/>
datur pondus F &aelig;quale ponderi C; &amp; vt BD ad BE, ita fiat pon<lb/>
dus F ad aliud G, quod ex puncto D &longs;u&longs;pendatur.  </s>
             
<s id="id.2.1.83.2.1.2.0"> pondera FG <lb/>
&aelig;queponderabunt.  </s>
             
<s id="id.2.1.83.2.1.3.0"> &amp; quoniam AB e&longs;t &aelig;qualis BE, &amp; pondera <lb/>
FC &aelig;qualia; &longs;imiliter pondera FC &aelig;queponderabunt.  </s>
             
<s id="id.2.1.83.2.1.4.0"> Pondera <lb/>
<expan abbr="ver&ograve;">vero</expan> FGC &longs;u&longs;pen&longs;a in vecte EBA, cuius fulcimentum e&longs;t B, non <lb/>
&aelig;queponderabunt; &longs;ed ex parte A deor&longs;um tendent.  </s>
             
<s id="id.2.1.83.2.1.5.0"> Ponatur igi<lb/>
tur in D tanta vis, vt pondera FGC &aelig;queponderent; erit po&shy;<lb/>
tentia in D &aelig;qualis ponderi G: pondera enim FC &aelig;queponde&shy;<lb/>
rant, &amp; potentia in D nil aliud efficere debet, ni&longs;i &longs;u&longs;tinere pon&shy;<lb/>
dus G ne de&longs;cendat.  </s>
             
<s id="id.2.1.83.2.1.6.0"> &amp; quoniam pondera FGC, &amp; potentia in <lb/>
D &aelig;queponderant, demptis igitur FG ponderibus, qu&aelig; &aelig;quepon<lb/>
derant; reliqua &aelig;queponderabunt, &longs;cilicet potentia in D ponderi C. <lb/>
hoc e&longs;t potentia in D pondus C &longs;u&longs;tinebit, ita vt vectis AB ma&shy;<lb/>
neat, vt prius.  </s>
             
<s id="id.2.1.83.2.1.7.0"> &amp; <expan abbr="c&ugrave;m">cum</expan> potentia in D &longs;it &aelig;qualis ponderi G, &amp; pon&shy;<lb/>
dus C &aelig;quale ponderi F; habebit potentia in D ad pondus C ean<lb/>
dem proportionem, quam EB, hoc e&longs;t AB ad BD. quod de&shy;<lb/>
mon&longs;trare oportebat.  </s>
    </p>       
<p id="id.2.1.83.3.0.0.0" type="head">        
<s id="id.2.1.83.3.1.1.0"> COROLLARIVM I. </s>
    </p>       
<p id="id.2.1.83.4.0.0.0" type="main">        
<s id="id.2.1.83.4.1.1.0"> Ex hoc etiam <expan abbr="p&agrave;tet">patet</expan>, vt prius, &longs;i coftituatur pon<lb/>
dus fulcimento B propius, vt in H; <expan abbr="&agrave;">a</expan> minori po&shy;<lb/>
tentia pondus ip&longs;um &longs;ub&longs;tineri debere.  </s>
    </p>       
<pb n="42"/>
       
<p id="id.2.1.83.6.0.0.0" type="main">        
<s id="id.2.1.83.6.1.1.0"> Minorem enim proportionem habet HB ad BD, <expan abbr="qu&agrave;m">quam</expan> AB ad <arrow.to.target n="note138"></arrow.to.target><lb/>
BD. &amp; <expan abbr="qu&ograve;">quo</expan> propius erit fulcimento, adhuc &longs;emper minorem re&shy;<lb/>
quiri potentiam.  </s>
    </p>       
<p id="id.2.1.84.1.0.0.0" type="margin">        
<s id="id.2.1.84.1.1.1.0"> <margin.target id="note138"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.85.1.0.0.0" type="head">        
<s id="id.2.1.85.1.1.1.0"> COROLLARIVM II. </s>
    </p>       
<p id="id.2.1.85.2.0.0.0" type="main">        
<s id="id.2.1.85.2.1.1.0"> Manife&longs;tum quoq; e&longs;t, potentiam in D &longs;emper <lb/>
maiorem e&longs;&longs;e pondere C. </s>
    </p>       
<p id="id.2.1.85.3.0.0.0" type="main">        
<s id="id.2.1.85.3.1.1.0"> Si enim inter AB &longs;umatur quoduis punctum D, &longs;emper AB <lb/>
maior erit BD. </s>
    </p>       
<p id="id.2.1.85.4.0.0.0" type="main">        
<s id="id.2.1.85.4.1.1.0"> Et aduertendum e&longs;t ha&longs;ce, quas attulimus demon&longs;trationes <lb/>
non &longs;olum vectibus horizonti &aelig;quidi&longs;tantibus, <expan abbr="ver&ugrave;m">verum</expan> etiam ve&shy;<lb/>
ctibus horizonti inclinatis ad h&aelig;c omnia o&longs;tendenda <expan abbr="commod&egrave;">commode</expan> <lb/>
aptari po&longs;&longs;e.  </s>
             
<s id="id.2.1.85.4.1.2.0"> quod ex iis, qu&aelig; de libra diximus, patet.  </s>
    </p>       
<p id="id.2.1.85.5.0.0.0" type="head">        
<s id="id.2.1.85.5.1.1.0"> PROPOSITIO IIII. </s>
    </p>       
<p id="id.2.1.85.6.0.0.0" type="main">        
<s id="id.2.1.85.6.1.1.0"> Si potentia pondus in vecte appen&longs;um mo&shy;<lb/>
ueat; erit &longs;patium potenti&aelig; mot&aelig; ad &longs;patium <lb/>
moti ponderis, vt di&longs;tantia <expan abbr="&agrave;">a</expan> fulcimento ad po&shy;<lb/>
tentiam ad di&longs;tantiam ab eodem ad ponderis &longs;u<lb/>
&longs;pen&longs;ionem.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.85.8.0.0.0" type="main">        
<s id="id.2.1.85.8.1.1.0"> Sit vectis AB, cuius ful&shy;<lb/>
cimentum C; &amp; ex puncto B <lb/>
&longs;it pondus D &longs;u&longs;pen&longs;um; &longs;itq; <lb/>
potentia in A mouens pon&shy;<lb/>
dus D vecte AB.  </s>
     
<s id="id.2.1.85.8.1.1.0.a"> Dico &longs;pa&shy;<lb/>
tium potenti&aelig; in A ad &longs;pa&shy;<lb/>
tium ponderis ita e&longs;&longs;e, vt CA <lb/>
ad CB.  </s>
     
<s id="id.2.1.85.8.1.1.0.b"> Moueatur vectis AB, <lb/>
&amp; vt pondus D &longs;ur&longs;um mo&shy;<lb/>
ueatur, oportet B &longs;ur&longs;um mo <lb/>
ueri, A <expan abbr="ver&ograve;">vero</expan> deor&longs;um.  </s>
             
<s id="id.2.1.85.8.1.2.0"> &amp; quo&shy;<lb/>
niam C e&longs;t punctum immobi<lb/>
le; idcirco dum A, &amp; B mo&shy;<lb/>
uentur, <expan abbr="circulor&utilde;">circulorum</expan> circumferen<lb/>
tias de&longs;cribent.  </s>
             
<s id="id.2.1.85.8.1.3.0"> Moueatur igi&shy;<lb/>
tur AB in EF; erunt AE <lb/>
<arrow.to.target n="fig85"></arrow.to.target><lb/>
BF circulorum circumferenti&aelig;, quorum &longs;emidiametri &longs;unt CA <lb/>
CB. tota compleatur circumferentia AGE, &amp; tota BHF; &longs;intq; <lb/>
KH puncta, vbi AB, &amp; EF circulum BHF &longs;ecant.  </s>
             
<s id="id.2.1.85.8.1.4.0"> Quoniam e&shy;<lb/>
<arrow.to.target n="note139"></arrow.to.target> nim angulus BCF e&longs;t &aelig;qualis angulo HCk; erit circumferentia <lb/>
<arrow.to.target n="note140"></arrow.to.target> kH circumferenti&aelig; BF &aelig;qualis.  </s>
             
<s id="id.2.1.85.8.1.5.0"> <expan abbr="c&ugrave;m">cum</expan> autem circumferenti&aelig; AE <lb/>
kH &longs;int &longs;ub eodem angulo ACE, &amp; circumferentia AE ad to&shy;<lb/>
tam circumferentiam AGE &longs;it, vt angulus ACE ad quatuor re&shy;<lb/>
ctos; vt autem idem angulus HCk ad quatuor rectos, ita quoq; <lb/>
e&longs;t circumferentia HK ad totam circumferentiam HBK; erit cir<lb/>
cumferentia AE ad totam circumferentiam AGE, vt circumfe&shy;<lb/>
<arrow.to.target n="note141"></arrow.to.target> rentia kH ad totam kFH. &amp; permutando, vt circumferentia <lb/>
AE ad circumferentiam kH, hoc e&longs;t BF, ita tota circumferen&shy;<lb/>
tia AGE ad totam circumferentiam BHF.  </s>
     
<s id="id.2.1.85.8.1.5.0.a"> tota <expan abbr="ver&ograve;">vero</expan> circumfe<lb/>
rentia AGE ita &longs;e habet ad totam BHF, vt diameter circuli AEG <lb/>
<arrow.to.target n="note142"></arrow.to.target> ad diametrum circuli BHF.  </s>
     
<s id="id.2.1.85.8.1.5.0.b"> Vt igitur circumferentia AE ad cir<lb/>
<arrow.to.target n="note143"></arrow.to.target> cumferentiam BF, ita diameter circuli AGE ad diametrum cir <lb/>
culi BHF: vt autem diameter ad diametrum, ita &longs;emidiameter <lb/>
ad &longs;emidiametrum, hoc e&longs;t CA ad CB: quare vt circumferen&shy;<lb/>
tia AE ad circumferentiam BF, ita CA ad CF. circumferentia <lb/>
<expan abbr="ver&ograve;">vero</expan> AE &longs;patium e&longs;t potenti&aelig; mot&aelig;, &amp; circumferentia BF e&longs;t 
<pb n="43"/>
&aelig;qualis &longs;patio ponderis D moti.  </s>
             
<s id="id.2.1.85.8.1.6.0"> &longs;patium enim motus ponderis <lb/>
D &longs;emper &aelig;quale e&longs;t &longs;patio motus puncti B, <expan abbr="c&ugrave;m">cum</expan> in B &longs;it appen<lb/>
&longs;um: &longs;patium ergo potenti&aelig; mot&aelig; ad &longs;patium moti ponderis e&longs;t, <lb/>
vt CA ad CB; hoc e&longs;t vt di&longs;tantia <expan abbr="&agrave;">a</expan> fulcimento ad potentiam <lb/>
ad di&longs;tantiam ab eodem ad ponderis &longs;u&longs;pen&longs;ionem. quod demon <lb/>
&longs;trare oportebat. </s>
             
<s id="id.2.1.85.8.1.7.0"> [quod demon<lb/>
&longs;trare oportebat.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig85" place="text">       </figure>       
<p id="id.2.1.85.8.2.1.0" type="caption">        
<s id="id.2.1.85.8.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.86.1.0.0.0" type="margin">        
<s id="id.2.1.86.1.1.1.0"> <margin.target id="note139"></margin.target>15 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.86.1.1.2.0"> <margin.target id="note140"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 26 <emph type="italics"/>tertii.<emph.end type="italics"/> </s>
             
<s id="id.2.1.86.1.1.3.0"> <margin.target id="note141"></margin.target>16 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.86.1.1.4.0"> <margin.target id="note142"></margin.target>23 <emph type="italics"/>Octaui Pappi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.86.1.1.5.0"> <margin.target id="note143"></margin.target>11 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.87.1.0.0.0" type="main">        
<s id="id.2.1.87.1.1.1.0"> Sit autem vectis AB, cu&shy;<lb/>
ius fulcimentum B; <expan abbr="potentia&shy;qu&eacute;">potentia&shy;<lb/>
que</expan> mouens in A; &amp; pondus <lb/>
in C.  </s>
     
<s id="id.2.1.87.1.1.1.0.a"> dico &longs;patium potenti&aelig; <lb/>
translat&aelig; ad &longs;patium transla<lb/>
ti ponderis ita e&longs;&longs;e, vt BA ad <lb/>
BC.  </s>
     
<s id="id.2.1.87.1.1.1.0.b"> Moueatur vectis, &amp; vt <lb/>
pondus sursum attollatur, ne&shy;<lb/>
ce&longs;&longs;e e&longs;t puncta C A &longs;ur&longs;um <lb/>
moueri.  </s>
             
<s id="id.2.1.87.1.1.2.0"> Moueatur igitur A <lb/>
&longs;ur&longs;um v&longs;q; ad D; &longs;itq; ve&shy;<lb/>
ctis motus BD. eodemq; <lb/>
modo (vt prius dictum e&longs;t) <lb/>
o&longs;tendemus puncta CA cir&shy;<lb/>
culorum circumferentias de&shy;<lb/>
<arrow.to.target n="fig86"></arrow.to.target><lb/>
&longs;cribere, <expan abbr="quor&utilde;">quorum</expan> &longs;emidiametri &longs;unt BA BC. &longs;imiliterq; o&longs;tendemus <lb/>
ita e&longs;&longs;e AD ad CE, vt &longs;emidiameter AB ad &longs;emidiametrum BC. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig86" place="text">       </figure>       
<p id="id.2.1.87.1.2.1.0" type="caption">        
<s id="id.2.1.87.1.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.87.2.0.0.0" type="main">        
<s id="id.2.1.87.2.1.1.0"> Eademq; ratione, &longs;i potentia e&longs;&longs;et in C, &amp; pondus in A, <lb/>
o&longs;tendetur ita e&longs;&longs;e CE ad AD, vt BC ad BA; hoc e&longs;t di&longs;tan<lb/>
tia <expan abbr="&agrave;">a</expan> fulcimento ad potentiam ad di&longs;tantiam ab eodem ad ponde<lb/>
ris &longs;u&longs;pen&longs;ionem. quod oportebat demon&longs;trare. </s>
             
<s id="id.2.1.87.2.1.2.0"> [quod oportebat demon&longs;trare.]  </s>
    </p>       
<p id="id.2.1.87.3.0.0.0" type="head">        
<s id="id.2.1.87.3.1.1.0"> COROLLARIVM. </s>
    </p>       
<p id="id.2.1.87.4.0.0.0" type="main">        
<s id="id.2.1.87.4.1.1.0"> Ex his manife&longs;tum e&longs;t maiorem habere pro&shy;<lb/>
portionem &longs;patium potenti&aelig; mouentis ad &longs;pa&shy;<lb/>
tium ponderis moti, <expan abbr="qu&agrave;m">quam</expan> pondus ad eandem <lb/>
potentiam. </s>
    </p>       
<p id="id.2.1.87.5.0.0.0" type="main">        
<s id="id.2.1.87.5.1.1.0"> Spatium enim potenti&aelig; ad &longs;patium ponderis eandem habet, 
<pb/>
quam pondus ad potentiam pondus &longs;u&longs;tinentem; potentia <expan abbr="ve&shy;r&ograve;">ve&shy;<lb/>
ro</expan> &longs;u&longs;tinens minor e&longs;t potentia mouente, quare minorem habebit <lb/>
<arrow.to.target n="note144"></arrow.to.target> proportionem pondus ad potentiam ip&longs;um mouentem, <expan abbr="qu&agrave;m">quam</expan> ad <lb/>
potentiam ip&longs;um &longs;u&longs;tinentem.  </s>
             
<s id="id.2.1.87.5.1.2.0"> &longs;patium igitur potenti&aelig; mouentis <lb/>
ad &longs;patium ponderis maiorem habebit proportionem, <expan abbr="qu&agrave;m">quam</expan> pon&shy;<lb/>
dus ad eandem potentiam.  </s>
    </p>       
<p id="id.2.1.88.1.0.0.0" type="margin">        
<s id="id.2.1.88.1.1.1.0"> <margin.target id="note144"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.89.1.0.0.0" type="head">        
<s id="id.2.1.89.1.1.1.0"> PROPOSITIO V. </s>
    </p>       
<p id="id.2.1.89.2.0.0.0" type="main">        
<s id="id.2.1.89.2.1.1.0"> Potentia quomodocunq; vecte pondus &longs;u&longs;ti&shy;<lb/>
nens ad ip&longs;um pondus eandem habebit propor&shy;<lb/>
tionem, quam di&longs;tantia <expan abbr="&agrave;">a</expan> fulcimento ad punctum, <lb/>
vbi <expan abbr="&agrave;">a</expan> centro grauitatis ponderis horizonti ducta <lb/>
perpendicularis vectem &longs;ecat, intercepta, ad <lb/>
di&longs;tantiam inter fulcimentum, &amp; potentiam. </s>
    </p>       
<p id="id.2.1.89.3.0.0.0" type="main">        
<s id="id.2.1.89.3.1.1.0"> Sit vectis AB <lb/>
horizonti &aelig;qui&shy;<lb/>
di&longs;tans, cuius ful<lb/>
cimentum N; &longs;it <lb/>
deinde pondus <lb/>
AC, cuius cen&shy;<lb/>
trum grauitatis <lb/>
&longs;it D, quod pri <lb/>
<expan abbr="m&ugrave;m">mum</expan> &longs;it infra ve<lb/>
ctem; pondus ve <lb/>
<expan abbr="r&ograve;">ro</expan> &longs;it ex punctis <lb/>
AO &longs;u&longs;pen&longs;um; <lb/>
<arrow.to.target n="fig87"></arrow.to.target><lb/>
&amp; <expan abbr="&agrave;">a</expan> puncto D horizonti, &amp; ip&longs;i AB perpendicularis ducatur DE.  </s>
     
<s id="id.2.1.89.3.1.1.0.a"> <lb/>
&longs;i <expan abbr="ver&ograve;">vero</expan> alii &longs;int quoq; vectes AF AG, quorum fulcimenta &longs;int <lb/>
HK; pondu&longs;q; AC in vecte AG ex punctis AQ &longs;it appen&longs;um; <lb/>
in vecte autem AF in punctis AP: lineaq; DE producta &longs;ecet <lb/>
AF in L, &amp; AG in M.  </s>
     
<s id="id.2.1.89.3.1.1.0.b"> dico potentiam in F pondus AC &longs;u&longs;tinen<lb/>
tem ad ip&longs;um pondus eam habere proportionem, quam habet kL 
<pb n="44"/>
ad kF; &amp; potentiam in B ad pondus eam habere, quam NE ad <lb/>
NB; &amp; potentiam in G ad pondus eam, quam HM ad HG.  </s>
     
<s id="id.2.1.89.3.1.1.0.c"> <lb/>
Quoniam enim DL horizonti e&longs;t perpendicularis, pondus AC <lb/>
vbicunq; in linea DL fuerit appen&longs;um, eodem modo, quo reperi&shy;<lb/>
tur, manebit.  </s>
             
<s id="id.2.1.89.3.1.2.0"> quare in vecte AB &longs;i &longs;u&longs;pen&longs;iones, qu&aelig; &longs;unt ad AO <lb/>
&longs;oluantur, pondus AC in E appen&longs;um eodem modo manebit, &longs;i&shy;<lb/>
cutinunc manet; hoc e&longs;t &longs;ublato puncto A, &amp; linea QO, codem <lb/>
modo pondus in E appen&longs;um manebit, vt ab ip&longs;is AO pun&shy;<lb/>
ctis &longs;u&longs;tinebatur; ex commentario Federici Commandini in &longs;extam <lb/>
Archimedis <expan abbr="propo&longs;ion&etilde;">propo&longs;ionem</expan> de quadratura parabol&aelig;, &amp; ex prima huius <lb/>
de libra.  </s>
             
<s id="id.2.1.89.3.1.3.0"> Itaq; quoniam pondus AC eandem ad libram habet con&longs;ti<lb/>
tutionem, &longs;iue in AO &longs;u&longs;tineatur, &longs;iue ex puncto E &longs;it appen&longs;um; <lb/>
eadem potentia in B idem pondus AC, &longs;iue in E, &longs;iue in AO <lb/>
&longs;u&longs;pen&longs;um &longs;u&longs;tinebit.  </s>
             
<s id="id.2.1.89.3.1.4.0"> potentia <expan abbr="ver&ograve;">vero</expan> in B &longs;u&longs;tinens pondus AC <lb/>
in E appen&longs;um ad ip&longs;um pondus ita &longs;e habet, vt NE ad NB; po&shy;<lb/>
tentia  <arrow.to.target n="note145"></arrow.to.target> igitur in B &longs;u&longs;tinens pondus AC ex punctis AO &longs;u&longs;pen<lb/>
&longs;um ad ip&longs;um pondus ita erit, vt NE ad NB.  </s>
     
<s id="id.2.1.89.3.1.4.0.a"> Non aliter o&longs;ten <lb/>
detur pondus AC ex puncto L &longs;u&longs;pen&longs;um manere, &longs;icuti <expan abbr="&agrave;">a</expan> pun<lb/>
ctis AP &longs;u&longs;tinetur; potentiamq; in F ad ip&longs;um pondus ita e&longs;&longs;e, vt kL <lb/>
ad KF.  </s>
     
<s id="id.2.1.89.3.1.4.0.b"> In vecte <expan abbr="ver&ograve;">vero</expan> AG pondus AC in M appen&longs;um ita mane <lb/>
re, vt <expan abbr="&agrave;">a</expan> punctis AQ &longs;u&longs;tinetur; potentiamq; in G ad pondus <lb/>
AC ita e&longs;&longs;e, vt HM ad HG; hoc e&longs;t vt di&longs;tantia <expan abbr="&agrave;">a</expan> fulcimento <lb/>
ad punctum, vbi <expan abbr="&agrave;">a</expan> centro grauitatis ponderis horizonti ducta <lb/>
perpendicularis vectem &longs;ecat, ad di&longs;tantiam <expan abbr="&agrave;">a</expan> fulcimento ad poten<lb/>
tiam. quod demon&longs;trare oportebat. </s>
             
<s id="id.2.1.89.3.1.5.0"> [quod demon&longs;trare oportebat.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig87" place="text">       </figure>       
<p id="id.2.1.89.3.2.1.0" type="caption">        
<s id="id.2.1.89.3.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.90.1.0.0.0" type="margin">        
<s id="id.2.1.90.1.1.1.0"> <margin.target id="note145"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.91.1.0.0.0" type="main">        
<s id="id.2.1.91.1.1.1.0"> Si autem FBG e&longs;&longs;ent vectium fulcimenta, potenti&aelig;q; e&longs;&longs;ent <lb/>
in KNH pondus &longs;u&longs;tinentes, &longs;imili modo o&longs;tendetur ita e&longs;&longs;e po<lb/>
tentiam in H ad pondus, vt GM ad GH; &amp; potentiam in N ad <lb/>
pondus, vt BE ad BN; ac potentiam in k ad pondus, vt FL <lb/>
ad Fk.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.91.3.0.0.0" type="main">        
<s id="id.2.1.91.3.1.1.0"> Et &longs;i vectes AB <lb/>
AF AG habeant <lb/>
fulcimenta in A, <lb/>
&amp; pondus &longs;it NO; <lb/>
deinde ab eius <lb/>
centro grauitatis <lb/>
D ducatur ip&longs;i A <lb/>
B, &amp; horizonti <lb/>
<expan abbr="perp&etilde;dicularis">perpendicularis</expan> D <lb/>
MEL; &longs;intq; po<lb/>
tenti&aelig; in FBG: <lb/>
&longs;imiliter o&longs;tende&shy;<lb/>
tur ita e&longs;&longs;e poten&shy;<lb/>
<arrow.to.target n="fig88"></arrow.to.target><lb/>
tiam in G pondus NO &longs;u&longs;tinentem ad ip&longs;um pondus, vt AM <lb/>
ad AG; ac potentiam in B, vt AE ad AB; &amp; potentiam in F, <lb/>
vt AL ad AF. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig88" place="text">       </figure>       
<p id="id.2.1.91.3.2.1.0" type="caption">        
<s id="id.2.1.91.3.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.91.4.0.0.0" type="main">        
<s id="id.2.1.91.4.1.1.0"> Sit deinde <lb/>
vectis AB ho<lb/>
rizonti &aelig;qui&shy;<lb/>
di&longs;tans, cuius <lb/>
fulcimentum <lb/>
D; &amp; &longs;it BE <lb/>
pondus, cuius <lb/>
centrum ??? graui<lb/>
tatis &longs;it F &longs;u&shy;<lb/>
pra vectem: <expan abbr="&agrave;">a</expan> <lb/>
punctoq; F ho <lb/>
rizonti, &amp; ip&longs;i <lb/>
AB ducatur <lb/>
<arrow.to.target n="fig89"></arrow.to.target><lb/>
FH; pondu&longs;q; <expan abbr="&agrave;">a</expan> puncto B, &amp; PQ &longs;u&longs;tineatur.  </s>
             
<s id="id.2.1.91.4.1.2.0"> Sint deinde alii ve&shy;<lb/>
ctes BL BM, quorum fulcimenta &longs;int NO; lineaq; FH producta &longs;e&shy;<lb/>
cet BM in k, &amp; BL in G; pondus autem in vecte BL in pun&shy;<lb/>
ctis BP &longs;u&longs;tineatur; in vecte autem BM <expan abbr="&agrave;">a</expan> puncto B, &amp; PR.  </s>
     
<s id="id.2.1.91.4.1.2.0.a"> Di&shy;<lb/>
co potentiam in L pondus BE vecte BL &longs;u&longs;tinentem ad ip&longs;um <lb/>
pondus eam habere proportionem, quam NG ad NL; &amp; po&shy;
<pb n="45"/>
tentiam in A ad pondus eam habere, quam DH ad DA; poten<lb/>
tiamq; in M ad pondus eam, quam Ok ad OM.  </s>
     
<s id="id.2.1.91.4.1.2.0.b"> Quoniam e&shy;<lb/>
nim <expan abbr="&agrave;">a</expan> centro grauitatis F ducta e&longs;t kF horizonti perpendicularis, <lb/>
ex quocunq; puncto line&aelig; kF &longs;u&longs;tineatur pondus, manebit; vt <arrow.to.target n="note146"></arrow.to.target><lb/>
nunc &longs;e habet.  </s>
             
<s id="id.2.1.91.4.1.3.0"> &longs;i igitur &longs;u&longs;tineatur in H, manebit vt prius; &longs;cili&shy;<lb/>
cet &longs;ublato puncto B, &amp; PQ, qu&aelig; pondus &longs;u&longs;tinent, pondus BE <lb/>
manebit, &longs;icuti ab ip&longs;is &longs;u&longs;tinebatur.  </s>
             
<s id="id.2.1.91.4.1.4.0"> quare in vecte AB graue&longs;cet <lb/>
in H, &amp; ad vectem eandem habebit con&longs;titutionem, quam prius; <lb/>
idcirco erit, ac &longs;i in H e&longs;&longs;et appen&longs;um.  </s>
             
<s id="id.2.1.91.4.1.5.0"> eadem igitur potentia <expan abbr="&igrave;dem">idem</expan> <lb/>
pondus BE, &longs;iue in H, &longs;iue in B, &amp; Q &longs;uffultum, &longs;u&longs;tinebit.  </s>
             
<s id="id.2.1.91.4.1.6.0"> Potentia ve <arrow.to.target n="note147"></arrow.to.target><expan abbr="r&ograve;"><lb/>
ro</expan> in A &longs;u&longs;tinens pondus BE vecte AB in H appen&longs;um ad ip&longs;um <lb/>
pondus eandem habet proportionem, quam DH ad DA; eadem <lb/>
ergo potentia in A &longs;u&longs;tinens pondus BE in punctis BQ &longs;u&longs;tenta <lb/>
tum ad ip&longs;um pondus erit, vt DH ad DA.  </s>
     
<s id="id.2.1.91.4.1.6.0.a"> Similiter o&longs;tende&shy;<lb/>
tur pondus BE &longs;i in G &longs;u&longs;tineatur, manere; &longs;icuti <expan abbr="&agrave;">a</expan> punctis BP <lb/>
&longs;u&longs;tinebatur: &amp; in puncto k, vt <expan abbr="&agrave;">a</expan> punctis BR. quare potentia in <lb/>
L &longs;u&longs;tinens pondus BE ad ip&longs;um pondus ita erit, vt NG ad NL. <lb/>
potentia <expan abbr="ver&ograve;">vero</expan> in M ad pondus, vt OK ad OM; hoc e&longs;t vt di&longs;tan<lb/>
tia <expan abbr="&agrave;">a</expan> fulcimento ad punctum, vbi <expan abbr="&agrave;">a</expan> centro grauitatis ponderis ho<lb/>
rizonti ducta perpendicularis vectem &longs;ecat, ad di&longs;tantiam <expan abbr="&agrave;">a</expan> fulci&shy;<lb/>
mento ad potentiam. quod demon&longs;trare quoq; oportebat. </s>
             
<s id="id.2.1.91.4.1.7.0"> [quod demon&longs;trare quoq; oportebat.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig89" place="text">       </figure>       
<p id="id.2.1.91.4.2.1.0" type="caption">        
<s id="id.2.1.91.4.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.92.1.0.0.0" type="margin">        
<s id="id.2.1.92.1.1.1.0"> <margin.target id="note146"></margin.target>1 <emph type="italics"/>Huius de libra.<emph.end type="italics"/> </s>
             
<s id="id.2.1.92.1.1.2.0"> <margin.target id="note147"></margin.target>1 <emph type="italics"/>Huius&lt;*&gt;<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.93.1.0.0.0" type="main">        
<s id="id.2.1.93.1.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> LAM e&longs;&longs;ent fulcimenta, &amp; potenti&aelig; in NDO; &longs;imi <lb/>
liter o&longs;tendetur ita e&longs;&longs;e potentiam in N ad pondus, vt LG ad L <lb/>
N; &amp; potentiam in D, vt AH ad AD; &amp; potentiam in O, vt <lb/>
Mk ad MO. 
<pb/>
 </s>
    </p>       
<p id="id.2.1.93.2.0.0.0" type="main">        
<s id="id.2.1.93.2.1.1.0"> Et &longs;i vectes BA <lb/>
BL BM habeant <lb/>
fulcimenta in B, &amp; <lb/>
pondus &longs;upra <expan abbr="vect&etilde;">vectem</expan> <lb/>
&longs;it NO; &amp; ab eius <lb/>
centro grauitatis F <lb/>
ducatur ip&longs;i AB, &amp; <lb/>
horizonti perpendi<lb/>
cularis FDEG; &longs;int <lb/>
<expan abbr="qu&eacute;">que</expan> potenti&aelig; in L <lb/>
AM; &longs;imiliter o&shy;<lb/>
&longs;tendetur ita e&longs;&longs;e po<lb/>
tentiam in L pon&shy;<lb/>
<arrow.to.target n="fig90"></arrow.to.target><lb/>
dus &longs;u&longs;tinentem ad ip&longs;um pondus, vt BD ad BL; &amp; potentiam <lb/>
in A ad pondus, vt BE ad BA, atq; potentiam in M, vt BG <lb/>
ad BM. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig90" place="text">       </figure>       
<p id="id.2.1.93.2.2.1.0" type="caption">        
<s id="id.2.1.93.2.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.93.3.0.0.0" type="main">        
<s id="id.2.1.93.3.1.1.0"> Sit deniq; <lb/>
vectis AB ho<lb/>
rizonti &aelig;qui&shy;<lb/>
di&longs;tans, cuius <lb/>
fulcimentum <lb/>
C, &amp; pondus <lb/>
DE habeat <expan abbr="c&etilde;">cen</expan><lb/>
trum grauita&shy;<lb/>
tis F in ip&longs;o <lb/>
vecte AB; <lb/>
&longs;intq; deniq; <lb/>
alii vectes G <lb/>
H kL, quo&shy;<lb/>
<arrow.to.target n="fig91"></arrow.to.target><lb/>
rum fulcimenta &longs;int MN; pondusq; in vecte GH &longs;u&longs;tineatur <expan abbr="&agrave;">a</expan> <lb/>
punctis GO; in vecte autem AB <expan abbr="&agrave;">a</expan> punctis AP; &amp; in uecte KL <lb/>
<expan abbr="&agrave;">a</expan> punctis KQ; &amp; centrum grauitatis F &longs;it quoq; in utroq; uecte <lb/>
GH kL; &longs;intq; potenti&aelig; in HBL.  </s>
     
<s id="id.2.1.93.3.1.1.0.a"> Dico potentiam in H ad <lb/>
pondus ita e&longs;&longs;e, ut NF ad NH; &amp; potentiam in B ad pondus, ut <lb/>
CF ad CB; ac potentiam in L ad pondus, ut MF ad ML.  </s>
     
<s id="id.2.1.93.3.1.1.0.b"> Quo&shy;<lb/>
niam enim F centrum e&longs;t grauitatis ponderis DE, &longs;i igitur in F 
<pb n="46"/>
&longs;u&longs;tineatur, pondus DE manebit &longs;icut prius, per deffinitionem cen<lb/>
tri grauitatis; eritq; ac&longs;iin F e&longs;&longs;et appen&longs;um; atq; in vecte eodem <lb/>
modo manebit, &longs;iue <expan abbr="&agrave;">a</expan> punctis AP, &longs;iue <expan abbr="&agrave;">a</expan> puncto F &longs;u&longs;tineatur.  </s>
             
<s id="id.2.1.93.3.1.2.0"> <lb/>
quod idem in vectibus GH kL eueniet; &longs;cilicet pondus eodem mo <lb/>
do manere, &longs;iue in F, &longs;iue in GO, vel in kQ &longs;u&longs;tineatur.  </s>
             
<s id="id.2.1.93.3.1.3.0"> eadem <lb/>
igitur potentia in B idem pondus DE, vel in F, vel in AP appen&longs;um <lb/>
&longs;u&longs;tinebit: &amp; quando appen&longs;um e&longs;t in F ad ip&longs;um pon&shy;<lb/>
dus e&longs;t, vt CF ad CB, ergo potentia &longs;u&longs;tinens pondus DE in <lb/>
AP appen&longs;um ad ip&longs;um pondus erit, vt CF ad CB. eodemq; mo <lb/>
do potentia in H ad pondus in GO appen&longs;um ita erit, vt NF ad <lb/>
NH. potentiaq; in L ad pondus in kQ appen&longs;um erit, vt MF <lb/>
ad ML. quod o&longs;tendere quoq; oportebat. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig91" place="text">       </figure>       
<p id="id.2.1.93.3.2.1.0" type="caption">        
<s id="id.2.1.93.3.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.93.4.0.0.0" type="main">        
<s id="id.2.1.93.4.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> HBL e&longs;&longs;ent fulcimenta, &amp; potenti&aelig; e&longs;&longs;ent in NCM; &longs;i&shy;<lb/>
militer o&longs;tendetur potentiam in N ad pondus ita e&longs;&longs;e, vt HF ad <lb/>
HN; &amp; potentiam in C, vt BF ad BC, &amp; potentiam in M, vt <lb/>
LF ad LM. </s>
    </p>       
<p id="id.2.1.93.5.0.0.0" type="main">        
<s id="id.2.1.93.5.1.1.0"> Et &longs;i vectes BA <lb/>
BC BD <expan abbr="habe&atilde;t">habeant</expan> ful<lb/>
cimenta in B, &longs;intq; <lb/>
pondera in EF GH <lb/>
kL, ita vt eorum <lb/>
centra MNO gra&shy;<lb/>
uitatis &longs;int in vecti<lb/>
bus; &longs;intq; poten&shy;<lb/>
ti&aelig; in CAD: &longs;imi <lb/>
liter o&longs;tendetur po<lb/>
tentiam in C ad <lb/>
pondus EF ita e&longs;&longs;e, <lb/>
<arrow.to.target n="fig92"></arrow.to.target><lb/>
vt BM ad BC, &amp; potentiam in A ad pondus GH, vt BN ad <lb/>
BA, potentiamq; in D ad pondus KL, vt BO ad BD. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig92" place="text">       </figure>       
<p id="id.2.1.93.5.2.1.0" type="caption">        
<s id="id.2.1.93.5.2.1.0.capt"> YYY </s>
    </p>       
<pb/>
       
<p id="id.2.1.93.7.0.0.0" type="head">        
<s id="id.2.1.93.7.1.1.0"> PROPOSITIO VI. </s>
    </p>       
<p id="id.2.1.93.8.0.0.0" type="main">        
<s id="id.2.1.93.8.1.1.0"> Sit AB recta linea, cui ad angulos &longs;it rectos <lb/>
AD, qu&aelig; ex parte A producatur vtcunq; v&longs;q; <lb/>
ad C; connectaturq; CB, qu&aelig; ex parte B quoq; <lb/>
producatur v&longs;q; ad E. ducantur deinde <expan abbr="&agrave;">a</expan> pun&shy;<lb/>
cto B vtcunq; inter AB BE line&aelig; BF BG ip&longs;i <lb/>
AB &aelig;quales; <expan abbr="&agrave;">a</expan> puncti&longs;q; FG ip&longs;is perpendicula&shy;<lb/>
res ducantur FH GK, qu&aelig; &amp; inter &longs;e &longs;e, &amp; ip&longs;i <lb/>
AD con&longs;tituantur &aelig;&shy;<lb/>
quales, ac &longs;i BA AD <lb/>
mot&aelig; &longs;int in BF FH, <lb/>
&amp; in BG GK; con&shy;<lb/>
nectanturq; CH CK, <lb/>
qu&aelig; lineas BF BG <lb/>
in punctis MN &longs;e&shy;<lb/>
cent.  </s>
             
<s id="id.2.1.93.8.1.2.0"> Dico BN mi&shy;<lb/>
norem e&longs;&longs;e BM, &amp; <lb/>
BM ip&longs;a BA. <lb/>
<arrow.to.target n="fig93"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.93.9.0.0.0" type="main">        
<s id="id.2.1.93.9.1.1.0"> Connectantur BD BH <lb/>
BK. &amp; quoniam du&aelig; line&aelig; <lb/>
DA AB duabus HF FB <lb/>
&longs;unt &aelig;quales, &amp; angulus <lb/>
DAB rectus recto HFB e&longs;t <lb/>
<arrow.to.target n="note148"></arrow.to.target> etiam &aelig;qualis; erunt reliqui <lb/>
anguli reliquis angulis &aelig;qua&shy;<lb/>
les, &amp; HB ip&longs;i DB &aelig;qualis.  </s>
             
<s id="id.2.1.93.9.1.2.0"> <lb/>
&longs;imiliter o&longs;tendetur triangu&shy;<lb/>
lum BkG triangulo BHF &aelig;qualem e&longs;&longs;e.  </s>
             
<s id="id.2.1.93.9.1.3.0"> quare centro B, inter&shy;
<pb n="47"/>
uallo quidem vna ip&longs;arum circulus de&longs;cribatur DH kE, qui li&shy;<lb/>
neas CH CK &longs;ecet in punctis OP; connectanturq; OB PB.  </s>
     
<s id="id.2.1.93.9.1.3.0.a"> <lb/>
Quoniam igitur punctum k propius e&longs;t ip&longs;i E, <expan abbr="qu&agrave;m">quam</expan> H; erit linea <arrow.to.target n="note149"></arrow.to.target><lb/>
Ck maior ip&longs;a CH, &amp; CP ip&longs;a CO minor: ergo PK ip&longs;a OH <lb/>
maior erit.  </s>
             
<s id="id.2.1.93.9.1.4.0"> Quoniam autem triangulum BkP &aelig;quicrure latera <lb/>
Bk BP lateribus BH BO trianguli BHO &aelig;quicruris &aelig;qualia ha<lb/>
bet, ba&longs;im <expan abbr="ver&ograve;">vero</expan> KP ba&longs;i HO maiorem, erit angulus kBP an&shy;<lb/>
gulo  <arrow.to.target n="note150"></arrow.to.target> HBO maior.  </s>
             
<s id="id.2.1.93.9.1.5.0"> ergo reliqui ad ba&longs;im anguli, hoc e&longs;t kPB <lb/>
PkB &longs;imul &longs;umpti, qui inter &longs;e &longs;unt &aelig;quales, reliquis ad ba&longs;im an&shy;<lb/>
gulis, <expan abbr="nemp&egrave;">nempe</expan> OHB HOB, qui etiam inter &longs;e &longs;unt &aelig;quales, mino&shy;<lb/>
res  <arrow.to.target n="note151"></arrow.to.target> erunt: <expan abbr="c&ugrave;m">cum</expan> omnes anguli cuiu&longs;cunq; trianguli duobus &longs;int rectis <lb/>
&aelig;quales.  </s>
             
<s id="id.2.1.93.9.1.6.0"> quare &amp; horum dimidii, &longs;cilicet NkB minor MHB.  </s>
     
<s id="id.2.1.93.9.1.6.0.a"> <lb/>
<expan abbr="C&ugrave;m">Cum</expan> autem angulus BkG &aelig;qualis &longs;it angulo BHF, erit NkG <lb/>
ip&longs;o MHF maior.  </s>
             
<s id="id.2.1.93.9.1.7.0"> &longs;i igitur <expan abbr="&agrave;">a</expan> puncto k con&longs;tituatur angulus GKQ <lb/>
ip&longs;i FHM &aelig;qualis, fiet triangulum GkQ triangulo FHM &aelig;qua <lb/>
le; nam duo anguli ad FH vnius duobus ad Gk alterius &longs;unt <lb/>
&aelig;quales, &amp; latus FH lateri Gk e&longs;t &aelig;quale, erit GQ ip&longs;i FM &aelig;&shy;<lb/>
quale.  <arrow.to.target n="note152"></arrow.to.target>  </s>
             
<s id="id.2.1.93.9.1.8.0"> ergo GN maior erit ip&longs;a FM.  </s>
     
<s id="id.2.1.93.9.1.8.0.a"> <expan abbr="C&ugrave;m">Cum</expan> itaq; BG ip&longs;i BF &longs;it &aelig;qua <lb/>
lis, erit BN minor ip&longs;a BM.  </s>
     
<s id="id.2.1.93.9.1.8.0.b"> <expan abbr="Qu&ograve;d">Quod</expan> autem BM &longs;it ip&longs;a BA minor, <lb/>
e&longs;t manife&longs;tum; <expan abbr="c&ugrave;m">cum</expan> BM ip&longs;a BF, qu&aelig; ip&longs;i BA e&longs;t &aelig;qualis, &longs;it <lb/>
minor. quod demon&longs;trare oportebat. </s>
             
<s id="id.2.1.93.9.1.9.0"> [quod demon&longs;trare oportebat.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig93" place="text">       </figure>       
<p id="id.2.1.93.9.2.1.0" type="caption">        
<s id="id.2.1.93.9.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.94.1.0.0.0" type="margin">        
<s id="id.2.1.94.1.1.1.0"> <margin.target id="note148"></margin.target>4 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.94.1.1.2.0"> <margin.target id="note149"></margin.target>8 <emph type="italics"/>Tertii.<emph.end type="italics"/> </s>
             
<s id="id.2.1.94.1.1.3.0"> <margin.target id="note150"></margin.target>25 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.94.1.1.4.0"> <margin.target id="note151"></margin.target>5 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.94.1.1.5.0"> <margin.target id="note152"></margin.target>26 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.95.1.0.0.0" type="main">        
<s id="id.2.1.95.1.1.1.0"> In&longs;uper &longs;i intra BG BE alia vtcunq; ducatur linea ip&longs;i BG &aelig;&shy;<lb/>
qualis; fiatq; operatio, quemadmodum &longs;upra dictum e&longs;t; &longs;imili&shy;<lb/>
ter o&longs;tendetur lineam BR minorem e&longs;&longs;e BN. &amp; <expan abbr="qu&ograve;">quo</expan> propius fue<lb/>
rit ip&longs;i BE, adhuc minorem &longs;emper e&longs;&longs;e.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.95.3.0.0.0" type="main">        
<s id="id.2.1.95.3.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> &aelig;qualia triangula BFH BGK &longs;int <lb/>
deor&longs;um inter BC BA con&longs;tituta; connectan&shy;<lb/>
turq; HC KC, qu&aelig; lineas BF BG ex parte <lb/>
FG productas in punctis MN &longs;ecent erit BN <lb/>
maior BM, &amp; BM ip&longs;a BA. </s>
    </p>       
<p id="id.2.1.95.4.0.0.0" type="main">        
<s id="id.2.1.95.4.1.1.0"> Nam producatur CH <lb/>
Ck v&longs;q; ad circumferentiam <lb/>
in OP, Connectanturq; BO <lb/>
BP; &longs;imili modo o&longs;tende&shy;<lb/>
tur lineam Pk maiorem e&longs; <lb/>
&longs;e OH, angulumq; PkB mi<lb/>
norem e&longs;&longs;e angulo OHB.  </s>
     
<s id="id.2.1.95.4.1.1.0.a"> &amp; <lb/>
quoniam angulus BHF e&longs;t <lb/>
&aelig;qualis angulo BkG; erit to<lb/>
tus PKG angulus angulo <lb/>
OHF minor: quare reliquus <lb/>
GKN reliquo FHM maior <lb/>
erit.  </s>
             
<s id="id.2.1.95.4.1.2.0"> &longs;i it aq; con&longs;tituatur angu<lb/>
lus GkQ ip&longs;i FHM &aelig;qua <lb/>
lis, linea KQ ip&longs;am GN ita <lb/>
&longs;ecabit, vt GQ ip&longs;i FM &aelig;qua <lb/>
lis euadat: quare maior.  </s>
             
<s id="id.2.1.95.4.1.3.0"> erit <lb/>
GN, <expan abbr="qu&agrave;m">quam</expan> FM; quibus &longs;i <lb/>
&aelig;quales adiiciantur BF BG, <lb/>
erit BN ip&longs;a BM maior.  </s>
             
<s id="id.2.1.95.4.1.4.0"> &amp; <lb/>
<expan abbr="c&ugrave;m">cum</expan> BM &longs;it ip&longs;a FB maior, <lb/>
erit quoq; ip&longs;a BA maior.  </s>
             
<s id="id.2.1.95.4.1.5.0"> &longs;i <lb/>
militer o&longs;tendetur, <expan abbr="qu&ograve;">quo</expan> pro <lb/>
pius fuerit BG ip&longs;i BC, li&shy;<lb/>
neam BN &longs;emper maiorem <lb/>
e&longs;&longs;e. <arrow.to.target n="fig94"></arrow.to.target> </s>
     
<pb n="48"/>
       
<s id="id.2.1.95.4.3.1.0"> PROPOSITIO VII. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig94" place="text">       </figure>       
<p id="id.2.1.95.4.4.1.0" type="caption">        
<s id="id.2.1.95.4.4.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.95.5.0.0.0" type="main">        
<s id="id.2.1.95.5.1.1.0"> Sit recta linea AB, <expan abbr="cu&igrave;">cui</expan> perpendicularis exi&shy;<lb/>
&longs;tat AD, qu&aelig; ex parte D producatur vtcunq; v&longs;q; <lb/>
ad C; connectaturq; CB, qu&aelig; producatur e&shy;<lb/>
tiam v&longs;q, ad E; &amp; inter AB BE line&aelig; &longs;imiliter <lb/>
vtcunq; ducantur BF BG ip&longs;i AB &aelig;quales; <expan abbr="&agrave;">a</expan> <lb/>
punctisq; FG line&aelig; FH GK ip&longs;i AB &aelig;quales, <lb/>
ip&longs;is <expan abbr="ver&ograve;">vero</expan> BF BG <expan abbr="per&shy;p&etilde;diculares">per&shy;<lb/>
pendiculares</expan> ducantur; <lb/>
ac &longs;i BA AD mot&aelig; <lb/>
&longs;int in BF FH BG <lb/>
GK: Connectanturq; <lb/>
CH CK, qu&aelig; lineas <lb/>
BF BG productas &longs;e&shy;<lb/>
cent in punctis MN.  </s>
     
<s id="id.2.1.95.5.1.1.0.a"> <lb/>
Dico BN maiorem e&longs; <lb/>
&longs;e BM, &amp; BM ip&longs;a BA. <lb/>
<arrow.to.target n="fig95"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.95.6.0.0.0" type="main">        
<s id="id.2.1.95.6.1.1.0"> Connectantur BD BH Bk, <lb/>
&amp; centro B, interuallo quidem <lb/>
BD, circulus de&longs;cribatur.  </s>
             
<s id="id.2.1.95.6.1.2.0"> &longs;imi <lb/>
liter vt in pr&aelig;cedenti demon&shy;<lb/>
&longs;trabimus puncta kHDOP in <lb/>
circuli circumferentia e&longs;&longs;e, trian<lb/>
gulaq; ABD FBH GBk in&shy;<lb/>
ter &longs;e &longs;e &aelig;qualia e&longs;&longs;e, atq; lineam <lb/>
Pk maiorem OH, angulumq; <lb/>
PKB minorem e&longs;&longs;e angulo O <lb/>
HB.  </s>
     
<s id="id.2.1.95.6.1.2.0.a"> Quoniam igitur angulus BHF &aelig;qualis e&longs;t angulo BkG, 
<pb/>
erit totus angulus PkG angu&shy;<lb/>
lo OHF minor: quare reliquus <lb/>
GkN reliquo FHM maior <lb/>
erit.  </s>
             
<s id="id.2.1.95.6.1.3.0"> &longs;i igitur fiat angulus GK <lb/>
Q ip&longs;i FHM &aelig;qualis, erit trian<lb/>
gulum GKQ triangulo FHM <lb/>
&aelig;quale, &amp; latus GQ lateri FM <lb/>
&aelig;quale; ergo maior erit GN ip<lb/>
&longs;a FM; ac propterea BN ma&shy;<lb/>
ior erit BM.  </s>
     
<s id="id.2.1.95.6.1.3.0.a"> BM autem ma&shy;<lb/>
ior erit BA; nam BM maior e&longs;t <lb/>
ip&longs;a BF. quod demon&longs;trare <lb/>
oportebat. <arrow.to.target n="fig96"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.95.7.0.0.0" type="main">        
<s id="id.2.1.95.7.1.1.0"> Eodemq; pror&longs;us modo, quo <lb/>
propius fuerit BG ip&longs;i BE, li&shy;<lb/>
neam BN &longs;emper maiorem e&longs;&longs;e <lb/>
o&longs;tendetur. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig95" place="text">       </figure>       
<p id="id.2.1.95.7.2.1.0" type="caption">        
<s id="id.2.1.95.7.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig96" place="text">       </figure>       
<p id="id.2.1.95.7.2.3.0" type="caption">        
<s id="id.2.1.95.7.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.95.8.0.0.0" type="main">        
<s id="id.2.1.95.8.1.1.0"> Si autem triangula BFH BGK deor&longs;um in&shy;<lb/>
ter AB BC con&longs;tituantur, ducanturq; CHO <lb/>
CKP, qu&aelig; lineas BF BG &longs;ecent in punctis M <lb/>
N; erit linea BN minor ip&longs;a BM, &amp; BM <lb/>
ip&longs;a BA. </s>
    </p>       
<pb n="49"/>
       
<p id="id.2.1.95.10.0.0.0" type="main">        
<s id="id.2.1.95.10.1.1.0"> Connectantur enim BO BP, <lb/>
&longs;imiliter o&longs;tendetur angulum <lb/>
PKB minorem e&longs;&longs;e OHB.  </s>
     
<s id="id.2.1.95.10.1.1.0.a"> &amp; <lb/>
quoniam angulus FHB &aelig;qua&shy;<lb/>
lis e&longs;t angulo GkB; erit angu<lb/>
lus GkN angulo FHM ma&shy;<lb/>
ior: quare &amp; linea GN ma&shy;<lb/>
ior erit ip&longs;a FM. ideoq; linea <lb/>
nea BN minor erit linea BM.  </s>
     
<s id="id.2.1.95.10.1.1.0.b"> <lb/>
<expan abbr="C&ugrave;m">Cum</expan> autem maior &longs;it BF ip&longs;a <lb/>
BM; erit BM ip&longs;a BA minor.  </s>
             
<s id="id.2.1.95.10.1.2.0"> Si&shy;<lb/>
miliq; modo o&longs;tendetur, <expan abbr="qu&ograve;">quo</expan> <lb/>
propius fuerit BG ip&longs;i BC, li&shy;<lb/>
neam BN &longs;emper minorem <lb/>
e&longs;&longs;e.  </s>
     <lb/>
       

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure place="text">       </figure>       
<p id="id.2.1.95.10.2.1.0" type="caption">        
<s id="id.2.1.95.10.2.1.0.capt"> YYY </s>
     <lb/>
       
<s id="id.2.1.95.10.4.1.0"> PROPOSITIO VIII. </s>
    </p>       
<p id="id.2.1.95.11.0.0.0" type="main">        
<s id="id.2.1.95.11.1.1.0"> Potentia pondus &longs;u&longs;tinens centrum grauitatis <lb/>
&longs;upra vectem horizonti &aelig;quidi&longs;tantem habens, <lb/>
<expan abbr="qu&ograve;">quo</expan> magis pondus ab hoc &longs;itu vecte eleuabitur; <lb/>
minori &longs;emper, vt &longs;u&longs;tineatur, egebit potentia: <lb/>
&longs;i <expan abbr="ver&ograve;">vero</expan> deprimetur, maiori. 
<pb/>
<arrow.to.target n="fig97"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.95.12.0.0.0" type="main">        
<s id="id.2.1.95.12.1.1.0"> Sit vectis AB horizonti &aelig;quidi&longs;tans, cuius fulcimentum C; <lb/>
pondus autem BD, eiu&longs;dem <expan abbr="ver&ograve;">vero</expan> grauitatis centrum &longs;it &longs;upra ve<lb/>
ctem vbi H: &longs;itq; potentia &longs;u&longs;tinens in A.  </s>
     
<s id="id.2.1.95.12.1.1.0.a"> moueatur deinde ve<lb/>
ctis AB in EF, &longs;itq; pondus motum in FG.  </s>
     
<s id="id.2.1.95.12.1.1.0.b"> Dico <expan abbr="prim&ugrave;m">primum</expan> mino <lb/>
rem <expan abbr="potenti&atilde;">potentiam</expan> in E &longs;u&longs;tinere pondus FG vecte EF, <expan abbr="qu&agrave;m">quam</expan> <expan abbr="pot&etilde;tia">potentia</expan> in <lb/>
A pondus BD vecte AB.  </s>
     
<s id="id.2.1.95.12.1.1.0.c"> &longs;it k centrum grauitatis ponderis FG; <lb/>
deinde <expan abbr="t&ugrave;m">tum</expan> ex H, <expan abbr="t&ugrave;m">tum</expan> ex K ducantur HL kM ip&longs;orum horizon<lb/>
tibus perpendiculares, qu&aelig; in <expan abbr="centr&utilde;">centrum</expan> mundi conuenient; &longs;itq; HL ip<lb/>
&longs;i quoq; AB perpendicularis.  </s>
             
<s id="id.2.1.95.12.1.2.0"> ducatur deinde kN ip&longs;i EF perpen&shy;<lb/>
dicularis, qu&aelig; ip&longs;i HL &aelig;qualis erit, &amp; CN ip&longs;i CL &aelig;qualis.  </s>
             
<s id="id.2.1.95.12.1.3.0"> Quo&shy;<lb/>
<arrow.to.target n="note153"></arrow.to.target> niam enim HL horizonti e&longs;t perpendicularis, potentia in A &longs;u<lb/>
&longs;tinens pondus BD ad ip&longs;um pondus eam habebit proportionem, <lb/>
quam CL ad CA.  </s>
     
<s id="id.2.1.95.12.1.3.0.a"> rur&longs;us quoniam kM horizonti e&longs;t perpendicu<lb/>
laris, potentia in E pondus FG &longs;u&longs;tinens ita erit ad pondus, vt <lb/>
CM ad CE.  </s>
     
<s id="id.2.1.95.12.1.3.0.b"> <expan abbr="C&ugrave;m">Cum</expan> autem CN NK ip&longs;is CL LH &longs;int &aelig;quales, <lb/>
<arrow.to.target n="note154"></arrow.to.target> angulosq; rectos contineant; erit CM minor ip&longs;a CL; ergo CM <lb/>
<arrow.to.target n="note155"></arrow.to.target> ad CA minorem habebit proportionem, quam CL ad CA; &amp; 
<pb n="45"/>
CA ip&longs;i CE e&longs;t &aelig;qualis, minorem igitur proportionem habebit <lb/>
CM ad CE. <expan abbr="qu&agrave;m">quam</expan> CL ad CA: &amp; <expan abbr="c&ugrave;m">cum</expan> pondera BD FG &longs;int <lb/>
&aelig;qualia, e&longs;t enim idem pondus; ergo minor erit proportio po<lb/>
tenti&aelig; in E pondus FG &longs;u&longs;tinentis ad ip&longs;um pondus, <expan abbr="qu&agrave;m">quam</expan> po<lb/>
tenti&aelig; in A pondus BD &longs;u&longs;tinentis ad ip&longs;um pondus.  </s>
             
<s id="id.2.1.95.12.1.4.0"> Quare <arrow.to.target n="note156"></arrow.to.target><lb/>
minor potentia in E &longs;u&longs;tinebit pondus FG, <expan abbr="qu&agrave;m">quam</expan> potentia in A <lb/>
pondus BD. &amp; <expan abbr="qu&ograve;">quo</expan> pondus magis eleuabitur; &longs;emper o&longs;tendetur <lb/>
minorem adhuc potentiam pondus &longs;u&longs;tinere; <expan abbr="c&ugrave;m">cum</expan> linea PC mi <arrow.to.target n="note157"></arrow.to.target><lb/>
nor &longs;it linea CM.  </s>
     
<s id="id.2.1.95.12.1.4.0.a"> &longs;it deinde vectis in QR, &amp; pondus in QS, <lb/>
cuius <expan abbr="centr&utilde;">centrum</expan> grauitatis &longs;it O.  </s>
     
<s id="id.2.1.95.12.1.4.0.b"> dico maiorem requiri potentiam in R <lb/>
ad <expan abbr="&longs;u&longs;tinend&utilde;">&longs;u&longs;tinendum</expan> pondus QS, <expan abbr="qu&agrave;m">quam</expan> in A ad pondus BD. ducatur <expan abbr="&agrave;">a</expan> cen<lb/>
tro grauitatis O linea OT horizonti perpendicularis.  </s>
             
<s id="id.2.1.95.12.1.5.0"> &amp; quo&shy;<lb/>
niam HL OT, &longs;i ex parte L, atq; T producantur, in centrum <lb/>
mundi conuenient; erit CT maior CL: e&longs;t autem CA ip&longs;i CR <arrow.to.target n="note158"></arrow.to.target><lb/>
&aelig;qualis, habebit ergo TC ad CR maiorem proportionem, <expan abbr="qu&agrave;m">quam</expan> <lb/>
LC ad CA.  </s>
     
<s id="id.2.1.95.12.1.5.0.a"> Maior igitur erit potentia in R &longs;u&longs;tinens pondus <arrow.to.target n="note159"></arrow.to.target><lb/>
QS, <expan abbr="qu&agrave;m">quam</expan> in A &longs;u&longs;tinens BD. &longs;imiliter o&longs;tendetur; <expan abbr="qu&ograve;">quo</expan> vectis <lb/>
RQ magis <expan abbr="&agrave;">a</expan> vecte AB di&longs;tabit deor&longs;um vergens, &longs;emper maio&shy;<lb/>
rem potentiam requiri ad &longs;u&longs;tinendum pondus: di&longs;tantia enim CV <arrow.to.target n="note160"></arrow.to.target><lb/>
longior e&longs;t CT.  </s>
     
<s id="id.2.1.95.12.1.5.0.b"> <expan abbr="Qu&ograve;">Quo</expan> igitur pondus <expan abbr="&agrave;">a</expan> &longs;itu horizonti &aelig;quidi&longs;tan<lb/>
te magis eleuabitur <expan abbr="&agrave;">a</expan> minori &longs;emper potentia pondus &longs;u&longs;tinebitur; <lb/>
<expan abbr="qu&ograve;">quo</expan> <expan abbr="ver&ograve;">vero</expan> magis deprimetur, maiori, vt &longs;u&longs;tineatur, egebit potentia. <lb/>
quod demon&longs;trare oportebat. </s>
             
<s id="id.2.1.95.12.1.6.0"> [<lb/>
quod demon&longs;trare oportebat.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig97" place="text">       </figure>       
<p id="id.2.1.95.12.2.1.0" type="caption">        
<s id="id.2.1.95.12.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.96.1.0.0.0" type="margin">        
<s id="id.2.1.96.1.1.1.0"> <margin.target id="note153"></margin.target>5 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.96.1.1.2.0"> <margin.target id="note154"></margin.target>6 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.96.1.1.3.0"> <margin.target id="note155"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.96.1.1.4.0"> <margin.target id="note156"></margin.target>10 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.96.1.1.5.0"> <margin.target id="note157"></margin.target>6 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.96.1.1.6.0"> <margin.target id="note158"></margin.target>6 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.96.1.1.7.0"> <margin.target id="note159"></margin.target>8 <emph type="italics"/>Quinti.  </s>
             
<s id="id.2.1.96.1.1.8.0"> Ex<emph.end type="italics"/> 10 <emph type="italics"/>quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.96.1.1.9.0"> <margin.target id="note160"></margin.target>6 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.97.1.0.0.0" type="main">        
<s id="id.2.1.97.1.1.1.0"> Hinc facile elicitur potentiam in A ad poten&shy;<lb/>
tiam in E ita e&longs;&longs;e, vt CL ad CM. </s>
    </p>       
<p id="id.2.1.97.2.0.0.0" type="main">        
<s id="id.2.1.97.2.1.1.0"> Nam ita e&longs;t LC ad CA, vt potentia in A ad pondus; vt au&shy;<lb/>
tem CA, hoc e&longs;t CE ad CM, ita e&longs;t pondus ad potentiam in E; <lb/>
quare ex &aelig;quali potentia in A ad potentiam in E ita erit, vt CL <arrow.to.target n="note161"></arrow.to.target><lb/>
ad CM. </s>
    </p>       
<p id="id.2.1.98.1.0.0.0" type="margin">        
<s id="id.2.1.98.1.1.1.0"> <margin.target id="note161"></margin.target>22 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.99.1.0.0.0" type="main">        
<s id="id.2.1.99.1.1.1.0"> Similiq; ratione non &longs;olum o&longs;tendetur, potentiam in A ad po&shy;<lb/>
tentiam in R ita e&longs;&longs;e, vt CL ad CT; &longs;ed &amp; potentiam quoq; in E <lb/>
ad potentiam in R ita e&longs;&longs;e, vt CM ad CT. &amp; ita in reliquis. 
<pb/>
<arrow.to.target n="fig98"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.99.2.0.0.0" type="main">        
<s id="id.2.1.99.2.1.1.0"> Sit deinde vectis AB horizonti &aelig;quidi&longs;tans, cuius fulcimen&shy;<lb/>
tum B; &amp; centrum grauitatis H ponderis CD &longs;it &longs;upra vectem; <lb/>
moueaturq; vectis in BE, pondu&longs;q; in FG.  </s>
     
<s id="id.2.1.99.2.1.1.0.a"> dico minorem po&shy;<lb/>
tentiam in E &longs;u&longs;tinere pondus FG vecte EB, <expan abbr="qu&agrave;m">quam</expan> potentia in <lb/>
A pondus CD vecte AB.  </s>
     
<s id="id.2.1.99.2.1.1.0.b"> &longs;it k centrum grauitatis ponderis FG, <lb/>
&amp; <expan abbr="&agrave;">a</expan> centris grauitatum Hk ip&longs;orum horizontibus perpendicu&shy;<lb/>
<arrow.to.target n="note162"></arrow.to.target> lares ducantur HL kM.  </s>
     
<s id="id.2.1.99.2.1.1.0.c"> Quoniam enim (ex &longs;upra demon&longs;tratis) <lb/>
<arrow.to.target n="note163"></arrow.to.target> BM minor e&longs;t BL, &amp; BE ip&longs;i BA &aelig;qualis; minorem habebit <lb/>
<arrow.to.target n="note164"></arrow.to.target> proportionem BM ad BE, <expan abbr="qu&agrave;m">quam</expan> BL ad BA. &longs;ed vt BM ad <lb/>
BE, ita potentia in E &longs;u&longs;tinens pondus FG ad ip&longs;um pondus; &amp; <lb/>
vt BL ad BA, ita potentia in A ad pondus CD; minorem <lb/>
habebit proportionem potentia in E ad pdndus FG, <expan abbr="qu&agrave;m">quam</expan> poten <lb/>
<arrow.to.target n="note165"></arrow.to.target> tia in A ad pondus CD.  </s>
     
<s id="id.2.1.99.2.1.1.0.d"> Ergo potentia in E minor erit poten&shy;<lb/>
tia in A. &longs;imiliter o&longs;tendetur, <expan abbr="qu&ograve;">quo</expan> magis pondus eleuabitur, &longs;em&shy;<lb/>
per minorem potentiam pondus &longs;u&longs;tinere.  </s>
             
<s id="id.2.1.99.2.1.2.0"> Sit autem vectis in <lb/>
BO, &amp; pondus in PQ, cuius cenrtum grauitatis &longs;it R.  </s>
     
<s id="id.2.1.99.2.1.2.0.a"> dico maio<lb/>
rem potentiam in O requiri ad &longs;u&longs;tinendum pondus PQ vecte BO, <lb/>
<expan abbr="qu&agrave;m">quam</expan> pondus CD vecte BA.  </s>
     
<s id="id.2.1.99.2.1.2.0.b"> ducatur <expan abbr="&agrave;">a</expan> puncto R horizonti per&shy;<lb/>
<arrow.to.target n="note166"></arrow.to.target> pendicularis RS.  </s>
     
<s id="id.2.1.99.2.1.2.0.c"> &amp; quoniam BS maior e&longs;t BL, habebit BS ad <lb/>
BO maiorem proportionem, <expan abbr="qu&agrave;m">quam</expan> BL ad BA; quare maior erit <lb/>
potentia in O &longs;u&longs;tinens pondus PQ, <expan abbr="qu&agrave;m">quam</expan> potentia in A &longs;u&longs;ti<lb/>
nens pondus CD. &amp; hoc modo o&longs;tendetur' <expan abbr="qu&ograve;">quo</expan> vectis BO ma<lb/>
gis <expan abbr="&agrave;">a</expan> vecte AB deor&longs;um tendens di&longs;tabit, &longs;emper maiorem ponderi 
<pb n="51"/>
&longs;u&longs;tinendo requiri potentiam. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig98" place="text">       </figure>       
<p id="id.2.1.99.2.2.1.0" type="caption">        
<s id="id.2.1.99.2.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.100.1.0.0.0" type="margin">        
<s id="id.2.1.100.1.1.1.0"> <margin.target id="note162"></margin.target>6 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.100.1.1.2.0"> <margin.target id="note163"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.100.1.1.3.0"> <margin.target id="note164"></margin.target>5 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.100.1.1.4.0"> <margin.target id="note165"></margin.target>10 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.100.1.1.5.0"> <margin.target id="note166"></margin.target>6 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.101.1.0.0.0" type="main">        
<s id="id.2.1.101.1.1.1.0"> Hinc quoq; vt &longs;upra patet pontentiam in A ad potentiam in E e&longs; <lb/>
&longs;e, vt BL ad BM; potentiamq; in A ad potentiam in O, vt BL <lb/>
ad BS. atque potentiam in E ad potentiam in O, vt BM <lb/>
ad BS. </s>
    </p>       
<p id="id.2.1.101.2.0.0.0" type="main">        
<s id="id.2.1.101.2.1.1.0"> Pr&aelig;terea &longs;i in B alia intelligatur potentia, ita vt du&aelig; &longs;int poten<lb/>
ti&aelig; pondus &longs;u&longs;tinentes; minor erit potentia in B &longs;u&longs;tinens pon&shy;<lb/>
dus PQ vecte BO, <expan abbr="qu&agrave;m">quam</expan> pondus CD vecte B32x aduer&longs;o au<lb/>
tem maior requiritur potentia in B ad &longs;u&longs;tinendum pondus FG ve <lb/>
cte BE, <expan abbr="qu&agrave;m">quam</expan> pondus CD vecte AB. ducta enim kN ip&longs;i EB <lb/>
perpendicularis, erit EN ip&longs;i AL &aelig;qualis: quare EM ip&longs;a LA <lb/>
maior erit.  </s>
             
<s id="id.2.1.101.2.1.2.0"> ergo maiorem habebit proportionem EM ad E<emph type="italics"/>B<emph.end type="italics"/>, <arrow.to.target n="note167"></arrow.to.target><expan abbr="qu&agrave;m"><lb/>
quam</expan> LA ad A<emph type="italics"/>B<emph.end type="italics"/>; &amp; LA ad A<emph type="italics"/>B<emph.end type="italics"/> maiorem, <expan abbr="qu&agrave;m">quam</expan> SO ad O<emph type="italics"/>B<emph.end type="italics"/>; <arrow.to.target n="note168"></arrow.to.target><lb/>
qu&aelig; &longs;unt proportiones potenti&aelig; ad pondus. </s>
    </p>       
<p id="id.2.1.102.1.0.0.0" type="margin">        
<s id="id.2.1.102.1.1.1.0"> <margin.target id="note167"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.102.1.1.2.0"> <margin.target id="note168"></margin.target>5 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.103.1.0.0.0" type="main">        
<s id="id.2.1.103.1.1.1.0"> Similiter o&longs;tendetur potentiam in <emph type="italics"/>B<emph.end type="italics"/> pondus vecte A<emph type="italics"/>B<emph.end type="italics"/> &longs;u&longs;ti&shy;<lb/>
nentem ad potentiam in eodem puncto <emph type="italics"/>B<emph.end type="italics"/> vecte E<emph type="italics"/>B<emph.end type="italics"/> &longs;u&longs;tinentem <lb/>
e&longs;&longs;e, vt LA ad EM; ad potentiam autem in B pondus vecte O<emph type="italics"/>B<emph.end type="italics"/><lb/>
&longs;u&longs;tinentem ita e&longs;&longs;e, vt AL ad OS. qu&aelig; <expan abbr="ver&ograve;">vero</expan> vectibus E<emph type="italics"/>B<emph.end type="italics"/> OB <lb/>
&longs;u&longs;tinent inter &longs;e &longs;e e&longs;&longs;e, vt EM ad OS. </s>
    </p>       
<p id="id.2.1.103.2.0.0.0" type="main">        
<s id="id.2.1.103.2.1.1.0"> Deinde vt in iis, qu&aelig; &longs;uperius dicta &longs;unt, demon&longs;trabimus po&shy;<lb/>
tentiam in <emph type="italics"/>B<emph.end type="italics"/> ad potentiam in E eam habere proportionem, quam <arrow.to.target n="note169"></arrow.to.target><lb/>
EM ad M<emph type="italics"/>B<emph.end type="italics"/>; &amp; potentiam in <emph type="italics"/>B<emph.end type="italics"/> ad potentiam in A ita e&longs;&longs;e, vt AL ad <arrow.to.target n="note170"></arrow.to.target><lb/>
L<emph type="italics"/>B<emph.end type="italics"/>, potentiamq; in <emph type="italics"/>B<emph.end type="italics"/> ad potentiam in O, vt OS ad S<emph type="italics"/>B.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.104.1.0.0.0" type="margin">        
<s id="id.2.1.104.1.1.1.0"> <margin.target id="note169"></margin.target>3 <emph type="italics"/>Cor.<emph.end type="italics"/> </s>
             
<s id="id.2.1.104.1.1.2.0"> <margin.target id="note170"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.105.1.0.0.0" type="main">        
<s id="id.2.1.105.1.1.1.0"> Sit autem vectis A<emph type="italics"/>B<emph.end type="italics"/><lb/>
horizonti &aelig;quidi&longs;tans, <lb/>
cuius fulcimentum <emph type="italics"/>B<emph.end type="italics"/>, <lb/>
grauitati&longs;q; centrum H <lb/>
ponderis AC &longs;it &longs;upra <lb/>
vectem: moueaturq; ve<lb/>
ctis in <emph type="italics"/>B<emph.end type="italics"/>E, ac pondus <lb/>
in EF, potentiaq; in G. <lb/>
&longs;imiliter vt &longs;upra o&longs;ten&shy;<lb/>
detur potentiam in G <lb/>
pondus EF &longs;ui&longs;tinen&shy;<lb/>
<arrow.to.target n="fig99"></arrow.to.target><lb/>
tem minorem e&longs;&longs;e potentia in D pondus AC &longs;u&longs;tinente.  </s>
             
<s id="id.2.1.105.1.1.2.0"> <expan abbr="c&ugrave;m">cum</expan> 
<pb/>
enim minor &longs;it BM ip&longs;a <lb/>
BL, minorem habebit <lb/>
proportionem MB ad <lb/>
BG, <expan abbr="qu&agrave;m">quam</expan> LB ad BD. <lb/>
atq; hoc modo o&longs;ten&shy;<lb/>
detur, <expan abbr="qu&ograve;">quo</expan> pondus ve&shy;<lb/>
cte magis eleuabitur, mi<lb/>
norem &longs;emper. ad pon&shy;<lb/>
dus &longs;u&longs;tinendum requi&shy;<lb/>
ri potentiam. </s>
             
<s id="id.2.1.105.1.1.3.0"> [ad pon&shy;<lb/>
dus &longs;u&longs;tinendum requi&shy;<lb/>
ri potentiam.]  </s>
             
<s id="id.2.1.105.1.1.4.0"> Simili&shy;<lb/>
ter &longs;i moucatur vectis <lb/>
in BO, potentiaq; &longs;u&shy;<lb/>
<arrow.to.target n="fig100"></arrow.to.target><lb/>
&longs;tinens in N, o&longs;tendetur potentiam in N maiorem e&longs;&longs;e potentia in <lb/>
D. maiorem enim habet proportionem SB ad BN, <expan abbr="qu&agrave;m">quam</expan> LB <lb/>
ad BD. o&longs;tendetur etiam, <expan abbr="qu&ograve;">quo</expan> magis pondus deprimetur; ma&shy;<lb/>
iorem &longs;emper (vt &longs;u&longs;tineatur) requiri potentiam. quod demon <lb/>
&longs;trare oportebat. </s>
             
<s id="id.2.1.105.1.1.5.0"> [quod demon<lb/>
&longs;trare oportebat.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig99" place="text">       </figure>       
<p id="id.2.1.105.1.2.1.0" type="caption">        
<s id="id.2.1.105.1.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig100" place="text">       </figure>       
<p id="id.2.1.105.1.2.3.0" type="caption">        
<s id="id.2.1.105.1.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.105.2.0.0.0" type="main">        
<s id="id.2.1.105.2.1.1.0"> Hinc quoq; liquet potentias in GDN inter &longs;e &longs;e ita e&longs;&longs;e, vt <lb/>
BM ad BL, atq; vt BL ad BS, deniq; vt BM ad BS. </s>
    </p>       
<p id="id.2.1.105.3.0.0.0" type="head">        
<s id="id.2.1.105.3.1.1.0"> COROLLARIVM. </s>
    </p>       
<p id="id.2.1.105.4.0.0.0" type="main">        
<s id="id.2.1.105.4.1.1.0"> Ex his manife&longs;tum e&longs;t; &longs;i potentia vecte &longs;ur&shy;<lb/>
&longs;um moueat pondus, cuius centrum grauitatis <lb/>
&longs;it &longs;upra vectem, <expan abbr="qu&ograve;">quo</expan> magis pondus eleuabitur; <lb/>
&longs;emper minorem potentiam requiri vt pondus <lb/>
moueatur. </s>
    </p>       
<p id="id.2.1.105.5.0.0.0" type="main">        
<s id="id.2.1.105.5.1.1.0"> Vbi enim potentia pondus &longs;u&longs;tinens e&longs;t &longs;emper minor, erit <lb/>
quoq; potentia ip&longs;um mouens &longs;emper minor. 
<pb n="52"/>
<arrow.to.target n="fig101"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.105.6.0.0.0" type="main">        
<s id="id.2.1.105.6.1.1.0"> Ex iis etiam demon&longs;trabitur, &longs;i centrum grauitatis eiu&longs;dem pon<lb/>
deris, &longs;iue propinquius, &longs;iue remotius fuerit <expan abbr="&agrave;">a</expan> vecte AB horizon&shy;<lb/>
ti &aelig;quidi&longs;tante, eandem potentiam in A pondus nihilominus <lb/>
&longs;u&longs;tinere: vt&longs;i centrum grauitatis H ponderis BD longius ab&longs;it <lb/>
<expan abbr="&agrave;">a</expan> vecte BA, <expan abbr="qu&agrave;m">quam</expan> centrum grauitatis N ponderis PV, dum&shy;<lb/>
modo ducta <expan abbr="&agrave;">a</expan> puncto H perpendicularis HL horizonti, vectiq; <lb/>
AB tran&longs;eat per N; &longs;itq; pondus PV ponderi BD &aelig;quale; <lb/>
erit <expan abbr="t&ugrave;m">tum</expan> pondus BD, <expan abbr="t&ugrave;m">tum</expan> pondus PV, ac &longs;i ambo in L e&longs;&shy;<lb/>
&longs;ent appen&longs;a; atque &longs;unt &aelig;qualia, <expan abbr="c&ugrave;m">cum</expan> loco vnius ponderis ac&shy;<lb/>
cipiantur, eadem igitur potentia in A &longs;u&longs;tinens pondus BD, <lb/>
pondus quoq; PV &longs;u&longs;tinebit.  </s>
             
<s id="id.2.1.105.6.1.2.0"> Vecte autem EF, <expan abbr="qu&ograve;">quo</expan> centrum <lb/>
grauitatis longius fuerit <expan abbr="&agrave;">a</expan> vecte, <expan abbr="e&ograve;">eo</expan> facilius potentia idem pon&shy;<lb/>
dus &longs;u&longs;tinebit: vt &longs;i centrum grauitatis k ponderis FG longius <lb/>
&longs;it <expan abbr="&agrave;">a</expan> vecte EF, <expan abbr="qu&agrave;m">quam</expan> centrum grauitatis X ponderis YZ; ita ta<lb/>
men vt ducta <expan abbr="&agrave;">a</expan> puncto k vecti FE perpendicularis tran&longs;eat per <lb/>
X; &longs;itq; pondus FG ponderi YZ &aelig;quale; &amp; <expan abbr="&agrave;">a</expan> punctis kX ip&shy;<lb/>
&longs;o&lt;*&gt;um horizontibus perpendiculares ducantur KM X9; erit C9 <lb/>
maior CM; ac propterea pondus FG in vecte erit, ac &longs;i in M e&longs; <lb/>
&longs;et appen&longs;um, &amp; pondus YZ, ac &longs;i in 9 e&longs;&longs;et appen&longs;um.  </s>
             
<s id="id.2.1.105.6.1.3.0"> quo 
<pb/>
<arrow.to.target n="fig102"></arrow.to.target><lb/>
<arrow.to.target n="note171"></arrow.to.target> niam autem maiorem habet proportionem C9 ad CE, <expan abbr="qu&agrave;m">quam</expan> <lb/>
CM ad CE, maior potentia in E &longs;u&longs;tinebit pondus YZ, <expan abbr="qu&agrave;m">quam</expan> <lb/>
FG.  </s>
     
<s id="id.2.1.105.6.1.3.0.a"> In vecte autem QR <expan abbr="&egrave;">e</expan> conuer&longs;o demon&longs;trabitur, &longs;cilicet <lb/>
<expan abbr="qu&ograve;">quo</expan> centrum grauitatis eiu&longs;dem ponderis &longs;it longius <expan abbr="&agrave;">a</expan> vecte, <expan abbr="e&ograve;">eo</expan> <lb/>
maiorem e&longs;&longs;e potentiam pondus &longs;u&longs;tinentem.  </s>
             
<s id="id.2.1.105.6.1.4.0"> maior enim e&longs;t <lb/>
CT, <expan abbr="qu&agrave;m">quam</expan> CI; &amp; ob id maiorem habebit proportionem CT <lb/>
ad CR, <expan abbr="qu&agrave;m">quam</expan> CI ad CR.  </s>
     
<s id="id.2.1.105.6.1.4.0.a"> Similiter demon&longs;trabitur, &longs;i pondus <lb/>
intra potentiam, &amp; fulcimentum fuerit collocatum; vel poten&shy;<lb/>
tia intra fulcimentum, &amp; pondus.  </s>
             
<s id="id.2.1.105.6.1.5.0"> Quod idem etiam potenti&aelig; <lb/>
eueniet mouenti.  </s>
             
<s id="id.2.1.105.6.1.6.0"> vbi enim minor potentia &longs;u&longs;tinet pondus, ibi <lb/>
minor potentia mouebit; &amp; vbi maior in &longs;u&longs;tinendo, ibi maior <lb/>
quoq; in mouendo requiretur.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig101" place="text">       </figure>       
<p id="id.2.1.105.6.2.1.0" type="caption">        
<s id="id.2.1.105.6.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig102" place="text">       </figure>       
<p id="id.2.1.105.6.2.3.0" type="caption">        
<s id="id.2.1.105.6.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.106.1.0.0.0" type="margin">        
<s id="id.2.1.106.1.1.1.0"> <margin.target id="note171"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.107.1.0.0.0" type="head">        
<s id="id.2.1.107.1.1.1.0"> RROPOSITIO VIIII. </s>
    </p>       
<p id="id.2.1.107.2.0.0.0" type="main">        
<s id="id.2.1.107.2.1.1.0"> Potentia pondus &longs;u&longs;tinens infra vectem ho&shy;<lb/>
rizonti &aelig;quidi&longs;tantem ip&longs;ius centrum grauitatis 
<pb n="53"/>
habens, <expan abbr="qu&ograve;">quo</expan> magis ab hoc &longs;itu vecte pondus ele<lb/>
uabitur maiori &longs;emper potentia, vt &longs;u&longs;tineatur, <lb/>
egebit.  </s>
             
<s id="id.2.1.107.2.1.2.0"> &longs;i <expan abbr="ver&ograve;">vero</expan> deprimetur, minori. <arrow.to.target n="fig103"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.107.3.0.0.0" type="main">        
<s id="id.2.1.107.3.1.1.0"> Sit vectis AB horizonti &aelig;quidi&longs;tans, cuius fulcimentum C; <lb/>
&longs;itq; pondus AD, cuius centrum grauitatis L &longs;it infra vectem; <lb/>
&longs;itq; potentia in B &longs;u&longs;tinens pondus AD: moueatur deinde ve&shy;<lb/>
ctis in FG, &amp; pondus in FH.  </s>
     
<s id="id.2.1.107.3.1.1.0.a"> Dico primum maiorem requiri <lb/>
potentiam in G ad &longs;u&longs;tinendum pondus FH vecte FG, <expan abbr="qu&agrave;m">quam</expan> <lb/>
&longs;it potentia in B pondere exi&longs;tente AD vecte autem AB.  </s>
     
<s id="id.2.1.107.3.1.1.0.b"> &longs;it M <lb/>
grauitatis centrum ponderis FH, &amp; <expan abbr="&agrave;">a</expan> punctis LM ip&longs;orum ho&shy;<lb/>
rizontibus perpendiculares ducantur Lk MN: ip&longs;i <expan abbr="ver&ograve;">vero</expan> FG per&shy;<lb/>
pendicularis ducatur MS, qu&aelig; &aelig;qualis erit LK, &amp; CK ip&longs;i CS <lb/>
erit etiam &aelig;qualis.  </s>
             
<s id="id.2.1.107.3.1.2.0"> Quoniam igitur CN maior e&longs;t Ck, habe&shy;<lb/>
bit  <arrow.to.target n="note172"></arrow.to.target> NC ad CG maiorem proportionem, <expan abbr="qu&agrave;m">quam</expan> Ck ad CB; po<arrow.to.target n="note173"></arrow.to.target><lb/>
tentia <expan abbr="uer&ograve;">uero</expan> in B ad pondus AD eandem habet, quam kC ad CB: <arrow.to.target n="note174"></arrow.to.target><lb/>
&amp; vt potentia in G ad pondus FH, ita e&longs;t NC ad CG; ergo <lb/>
maiorem habebit proportionem potentia in G ad pondus FH, <lb/>
<expan abbr="qu&agrave;m">quam</expan> potentia in B ad pondus AD.  </s>
     
<s id="id.2.1.107.3.1.2.0.a"> maior igitur e&longs;t potentia <arrow.to.target n="note175"></arrow.to.target><lb/>
in G ip&longs;a potentia in B. &longs;i <expan abbr="ver&ograve;">vero</expan> vectis &longs;it in OP, &amp; pondus in <lb/>
OQ; erit potentia in B maior, <expan abbr="qu&agrave;m">quam</expan> in P. eodem enim mo&shy;<lb/>
do o&longs;tendetur CR minorem e&longs;&longs;e Ck, &amp; CR ad CP minorem <arrow.to.target n="note176"></arrow.to.target>
<pb/>
<arrow.to.target n="fig104"></arrow.to.target><lb/>
habere proportionem, <expan abbr="qu&agrave;m">quam</expan> Ck ad CB; &amp; ob id potentiam in <lb/>
B maiorem e&longs;&longs;e potentia in P. &amp; hoc modo o&longs;tendetur, <expan abbr="qu&ograve;">quo</expan> ma&shy;<lb/>
gis <expan abbr="&agrave;">a</expan> &longs;itu AB pondus eleuabitur, &longs;emper maiorem potentiam ad <lb/>
pondus &longs;u&longs;tinendum requiri. <expan abbr="&egrave;">e</expan> contra <expan abbr="ver&ograve;">vero</expan> &longs;i deprimetur. quod <lb/>
demon&longs;trare oportebat. </s>
             
<s id="id.2.1.107.3.1.3.0"> [<expan abbr="&egrave;">e</expan> contra <expan abbr="ver&ograve;">vero</expan> &longs;i deprimetur.]  </s>
             
<s id="id.2.1.107.3.1.4.0"> [quod <lb/>
demon&longs;trare oportebat.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig103" place="text">       </figure>       
<p id="id.2.1.107.3.2.1.0" type="caption">        
<s id="id.2.1.107.3.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig104" place="text">       </figure>       
<p id="id.2.1.107.3.2.3.0" type="caption">        
<s id="id.2.1.107.3.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.108.1.0.0.0" type="margin">        
<s id="id.2.1.108.1.1.1.0"> <margin.target id="note172"></margin.target>7 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.108.1.1.2.0"> <margin.target id="note173"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.108.1.1.3.0"> <margin.target id="note174"></margin.target>5 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.108.1.1.4.0"> <margin.target id="note175"></margin.target>10 <emph type="italics"/>Quinti<emph.end type="italics"/> </s>
             
<s id="id.2.1.108.1.1.5.0"> <margin.target id="note176"></margin.target>7 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.109.1.0.0.0" type="main">        
<s id="id.2.1.109.1.1.1.0"> Hinc quoq; <expan abbr="facil&egrave;">facile</expan> elici pote&longs;t potentias in PBG inter &longs;e &longs;e ita <lb/>
e&longs;&longs;e, vt CR ad Ck; &amp; vt Ck ad CN; atq; vt CN ad CR. <lb/>
<arrow.to.target n="fig105"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.109.2.0.0.0" type="main">        
<s id="id.2.1.109.2.1.1.0"> Sit deinde vectis AB horizonti &aelig;quidi&longs;tans, cuius fulcimentum <lb/>
B; pondu&longs;q; CD habeat centrum grauitatis O infra vectem; &longs;itq; <lb/>
potentia in A &longs;u&longs;tinens pondus CD.  </s>
     
<s id="id.2.1.109.2.1.1.0.a"> Moueatur deinde vectis in 
<pb n="54"/>
BE BF, pondu&longs;q; transferatur in GH kL.  </s>
     
<s id="id.2.1.109.2.1.1.0.b"> Dico maiorem re&shy;<lb/>
quiri potentiam in E, vt pondus &longs;u&longs;tineatur, <expan abbr="qu&agrave;m">quam</expan> in A; &amp; ma<lb/>
iorem in A, <expan abbr="qu&agrave;m">quam</expan> in F. ducantur <expan abbr="&agrave;">a</expan> centris grauitatum horizon&shy;<lb/>
tibus perpendiculares NM OP QR, qu&aelig; ex parte NOQ <lb/>
protract&aelig; in centrum mundi conuenient.  </s>
             
<s id="id.2.1.109.2.1.2.0"> &longs;imiliter vt &longs;upra o&longs;ten <lb/>
detur BM <expan abbr="maior&etilde;">maiorem</expan> e&longs;&longs;e BP, &amp; <emph type="italics"/>B<emph.end type="italics"/>P maiorem BR; &amp; BM ad BE ma&shy;<lb/>
iorem  <arrow.to.target n="note177"></arrow.to.target> habere proportionem, <expan abbr="qa&agrave;m">qaam</expan> BP ad BA; &amp; BP ad BA ma&shy;<lb/>
iorem, <expan abbr="qu&agrave;m">quam</expan> BR ad BF: &amp; propter hoc potentiam in E maio&shy;<lb/>
rem e&longs;&longs;e potentia in A; &amp; potentiam in A maiorem potentia in <lb/>
F. &amp; <expan abbr="qu&ograve;">quo</expan> vectis magis <expan abbr="&agrave;">a</expan> &longs;itu AB eleuabitur, &longs;emper o&longs;tendetur, <lb/>
maiorem requiri potentiam ponderi &longs;u&longs;tinendo. &longs;i <expan abbr="ver&ograve;">vero</expan> depri&shy;<lb/>
metur, minorem. </s>
             
<s id="id.2.1.109.2.1.3.0"> [&longs;i <expan abbr="ver&ograve;">vero</expan> depri&shy;<lb/>
metur, minorem.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig105" place="text">       </figure>       
<p id="id.2.1.109.2.2.1.0" type="caption">        
<s id="id.2.1.109.2.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.110.1.0.0.0" type="margin">        
<s id="id.2.1.110.1.1.1.0"> <margin.target id="note177"></margin.target>7 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.111.1.0.0.0" type="main">        
<s id="id.2.1.111.1.1.1.0"> Hinc patet etiam potentias in EAF inter &longs;e &longs;e ita e&longs;&longs;e, vt BM ad <lb/>
BP; &amp; vt BP ad BR; ac vt BM ad BR. </s>
    </p>       
<p id="id.2.1.111.2.0.0.0" type="main">        
<s id="id.2.1.111.2.1.1.0"> In&longs;uper &longs;i in B altera &longs;it potentia, ita vt du&aelig; &longs;int potenti&aelig; pondus <lb/>
&longs;u&longs;tinentes, maiore opus e&longs;t potentia in B pondus kL &longs;u&longs;tinente <lb/>
vecte BF, <expan abbr="qu&agrave;m">quam</expan> pondus CD vecte AB. &amp; adhuc maiore vecte <lb/>
AB, <expan abbr="qu&agrave;m">quam</expan> vecte BE.  </s>
     
<s id="id.2.1.111.2.1.1.0.a"> maiorem enim habet proportionem RF <lb/>
ad FB, <expan abbr="qu&agrave;m">quam</expan> PA ad AB; &amp; PA ad AB maiorem habet, <expan abbr="qu&agrave;m">quam</expan> <lb/>
EM ad EB. </s>
    </p>       
<p id="id.2.1.111.3.0.0.0" type="main">        
<s id="id.2.1.111.3.1.1.0"> Similiterq; o&longs;tendetur potentias in B pondus vectibus &longs;u&longs;tinen&shy;<lb/>
tes inter &longs;e &longs;e ita e&longs;&longs;e, vt EM ad AP; &amp; ut <lb/>
AP ad FR; atque ut <lb/>
EM ad FR. </s>
    </p>       
<p id="id.2.1.111.4.0.0.0" type="main">        
<s id="id.2.1.111.4.1.1.0"> Pr&aelig;terea potentia in B ad potentiam in F ita erit, ut RF ad <arrow.to.target n="note178"></arrow.to.target><lb/>
RB; &amp; potentia in B ad potentiam in A, ut PA ad PB, &amp; po&shy;<lb/>
tentia  <arrow.to.target n="note179"></arrow.to.target> in <emph type="italics"/>B<emph.end type="italics"/> ad potentiam in E, ut EM ad M<emph type="italics"/>B.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.112.1.0.0.0" type="margin">        
<s id="id.2.1.112.1.1.1.0"> <margin.target id="note178"></margin.target>3 <emph type="italics"/>Cor.<emph.end type="italics"/> </s>
             
<s id="id.2.1.112.1.1.2.0"> <margin.target id="note179"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.113.1.0.0.0" type="main">        
<pb/>
       
<s id="id.2.1.113.1.2.1.0"> Sit autem vectis <lb/>
AB horizonti &aelig;qui&shy;<lb/>
di&longs;tans, cuius fulci&shy;<lb/>
mentum B; &amp; pon&shy;<lb/>
dus AC, cuius cen&shy;<lb/>
trum grauitatis &longs;it in&shy;<lb/>
fra vectem: &longs;itq; po&shy;<lb/>
tentia in D pondus <lb/>
<expan abbr="&longs;u&longs;tin&etilde;s">&longs;u&longs;tinens</expan>; moueaturq; <lb/>
vectis in BE BF, &amp; <lb/>
potentia in GH: &longs;i&shy;<lb/>
militer o&longs;tendetur po<lb/>
<arrow.to.target n="fig106"></arrow.to.target><lb/>
tentiam in G maiorem e&longs;&longs;e debere potentia in D; &amp; potentiam in <lb/>
D maiorem potentia in H.  </s>
     
<s id="id.2.1.113.1.2.1.0.a"> maiorem enim proportionem habet <lb/>
KB ad BG, <expan abbr="qu&agrave;m">quam</expan> BL ad BD; &amp; BL ad BD maiorem, <expan abbr="qu&agrave;m">quam</expan> <lb/>
MB ad BH.  </s>
     
<s id="id.2.1.113.1.2.1.0.b"> &amp; hoc modo o&longs;tendetur, <expan abbr="qu&ograve;">quo</expan> vectis magis <expan abbr="&agrave;">a</expan> &longs;itu <lb/>
AB eleuabitur, adhuc &longs;emper maiorem e&longs;&longs;e debere potentiam pon<lb/>
dus &longs;u&longs;tinentem. <expan abbr="qu&ograve;">quo</expan> autem magis deprimetur; minorem. quod <lb/>
demon&longs;trare oportebat. </s>
             
<s id="id.2.1.113.1.2.2.0"> [<expan abbr="qu&ograve;">quo</expan> autem magis deprimetur; minorem.]  </s>
             
<s id="id.2.1.113.1.2.3.0"> [quod <lb/>
demon&longs;trare oportebat.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig106" place="text">       </figure>       
<p id="id.2.1.113.1.3.1.0" type="caption">        
<s id="id.2.1.113.1.3.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.113.2.0.0.0" type="main">        
<s id="id.2.1.113.2.1.1.0"> Similiter in his potenti&aelig; in GDH inter &longs;e &longs;e ita. erunt, vt BK <lb/>
ad BL; &amp; vt BL ad BM; deniq; vt Bk ad BM. </s>
             
<s id="id.2.1.113.2.1.2.0"> [erunt, vt BK <lb/>
ad BL; &amp; vt BL ad BM; deniq; vt Bk ad BM.] </s>
    </p>       
<p id="id.2.1.113.3.0.0.0" type="head">        
<s id="id.2.1.113.3.1.1.0"> COROLLARIVM. </s>
    </p>       
<p id="id.2.1.113.4.0.0.0" type="main">        
<s id="id.2.1.113.4.1.1.0"> Ex his patet etiam, &longs;i potentia vecte &longs;ur&longs;um <lb/>
moueat pondus, cuius centrum grauitatis &longs;it in&shy;<lb/>
fra vectem; <expan abbr="qu&ograve;">quo</expan> magis pondus eleuabitur, &longs;em<lb/>
per maiorem requiri potentiam, vt pondus mo<lb/>
ueatur. </s>
    </p>       
<p id="id.2.1.113.5.0.0.0" type="main">        
<s id="id.2.1.113.5.1.1.0"> Nam &longs;i potentia pondus &longs;u&longs;tinens &longs;emper e&longs;t maior: erit quoq; <lb/>
potentia mouens &longs;emper maior. 
<pb n="55"/>
<arrow.to.target n="fig107"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.113.6.0.0.0" type="main">        
<s id="id.2.1.113.6.1.1.0"> Et his etiam <expan abbr="facil&egrave;">facile</expan> elicietur, &longs;i centrum grauitatis eiu&longs;dem pon&shy;<lb/>
deris, &longs;iue propius, &longs;iue remotius fuerit <expan abbr="&agrave;">a</expan> vecte AB horizonti &aelig;&shy;<lb/>
quidi&longs;tante; eandem potentiam in B pondus &longs;u&longs;tinere.  </s>
             
<s id="id.2.1.113.6.1.2.0"> vt &longs;i cen&shy;<lb/>
trum grauitatis L ponderis AD &longs;it remotius <expan abbr="&agrave;">a</expan> vecte BA, <expan abbr="qu&agrave;m">quam</expan> <lb/>
centrum grauitatis N ponderis PV; dummodo ducta <expan abbr="&agrave;">a</expan> puncto L <lb/>
perpendicularis LK horizonti, vectiq; AB tran&longs;eat per N: &longs;imili&shy;<lb/>
ter vt in pr&aelig;cedenti o&longs;tendetur, eandem potentiam in B, &amp; pondus <lb/>
AD, &amp; pondus PV &longs;u&longs;tinere.  </s>
             
<s id="id.2.1.113.6.1.3.0"> In vecte <expan abbr="aut&eacute;">aute</expan> EF, <expan abbr="qu&ograve;">quo</expan> <expan abbr="centr&utilde;">centrum</expan> grauitatis <lb/>
longius aberit <expan abbr="&agrave;">a</expan> vecte, <expan abbr="e&ograve;">eo</expan> maiori opus erit potentia ponderi &longs;u&longs;ti&shy;<lb/>
nendo.  </s>
             
<s id="id.2.1.113.6.1.4.0"> vt centrum grauitatis M ponderis FH remotius &longs;it <expan abbr="&agrave;">a</expan> ue<lb/>
cte EF, <expan abbr="qu&agrave;m">quam</expan> S centrum grauitatis ponderis XZ; ducantur <expan abbr="&agrave;">a</expan> pun<lb/>
ctis MS horizontibus perpendiculares MI SG; erit CI maior <lb/>
CG: ac propterea maior e&longs;&longs;e debet potentia in E pondus FH &longs;u<lb/>
&longs;tinens, <expan abbr="qu&agrave;m">quam</expan> pondus XZ.  </s>
     
<s id="id.2.1.113.6.1.4.0.a"> Contra <expan abbr="uer&ograve;">uero</expan> in uecte OR o&longs;tende<lb/>
tur, <expan abbr="qu&ograve;">quo</expan> &longs;cilicet centrum grauitatis eiu&longs;dem ponderis longius ab <lb/>
&longs;it <expan abbr="&agrave;">a</expan> uecte, <expan abbr="&agrave;">a</expan> minori potentia pondus &longs;u&longs;tineri.  </s>
             
<s id="id.2.1.113.6.1.5.0"> minor enim e&longs;t <lb/>
CY, <expan abbr="qu&agrave;m">quam</expan> CT.  </s>
     
<s id="id.2.1.113.6.1.5.0.a"> Simili quoq; modo demon&longs;trabitur, &longs;i pondus <lb/>
&longs;it intra potentiam, &amp; fulcimentum; uel potentia intra fulci&shy;<lb/>
mentum, &amp; pondus.  </s>
             
<s id="id.2.1.113.6.1.6.0"> Quod idem potenti&aelig; eueniet mouenti: 
<pb/>
vbi enim minor potentia &longs;u&longs;tinet pondus, ibi minor potentia mo&shy;<lb/>
uebit.  </s>
             
<s id="id.2.1.113.6.1.7.0"> &amp; vbi maior potentia in &longs;u&longs;tinendo; ibi quoq; maior in mo<lb/>
uendo aderit.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig107" place="text">       </figure>       
<p id="id.2.1.113.6.2.1.0" type="caption">        
<s id="id.2.1.113.6.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.113.7.0.0.0" type="head">        
<s id="id.2.1.113.7.1.1.0"> PROPOSITIO X. </s>
    </p>       
<p id="id.2.1.113.8.0.0.0" type="main">        
<s id="id.2.1.113.8.1.1.0"> Potentia pondus &longs;u&longs;tinens in ip&longs;o vecte cen&shy;<lb/>
trum grauitatis habens, quomodocunq; vecte <lb/>
transferatur pondus; eadem &longs;emper, vt &longs;u&longs;tinea&shy;<lb/>
tur, potentia opus erit. <arrow.to.target n="fig108"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.113.9.0.0.0" type="main">        
<s id="id.2.1.113.9.1.1.0"> Sit vectis AB horizonti <expan abbr="&aelig;quidi&longs;t&agrave;ns">&aelig;quidi&longs;tans</expan>, cuius fulcimentum C. <lb/>
E <expan abbr="ver&ograve;">vero</expan> centrum grauitatis ponderis in ip&longs;o &longs;it vecte.  </s>
             
<s id="id.2.1.113.9.1.2.0"> Moueatur <lb/>
deinde uectis in FG, Hk; &amp; centrum grauitatis in LM.  </s>
     
<s id="id.2.1.113.9.1.2.0.a"> dico ean<lb/>
dem potentiam in kBG idemmet &longs;emper &longs;u&longs;tinere pondus.  </s>
             
<s id="id.2.1.113.9.1.3.0"> <lb/>
Quoniam enim pondus in uecte AB perinde &longs;e habet, ac &longs;i e&longs;&longs;et <lb/>
<arrow.to.target n="note180"></arrow.to.target> appen&longs;um in E; &amp; in uecte GF, ac &longs;i e&longs;&longs;et appen&longs;um in L; &amp; in <lb/>
uecte Hk.  </s>
             
<s id="id.2.1.113.9.1.4.0"> ac &longs;i in M e&longs;&longs;et appen&longs;um; di&longs;tanti&aelig; <expan abbr="uer&ograve;">uero</expan> CL CE <lb/>
CM &longs;unt inter &longs;e &longs;e &aelig;quales; nec non CK CB CG inter &longs;e &aelig;&shy;<lb/>
quales; erit potentia in B ad pondus, ut CE ad CB; atque poten
<pb n="56"/>
tia in k ad pondus, ut CM ad Ck; &amp; potentia in G ad pondus, <lb/>
vt CL ad CG. eadem igitur potentia in k<emph type="italics"/>B<emph.end type="italics"/>G idem translatum <lb/>
pondus &longs;u&longs;tinebit. quod demon&longs;trare oportebat. </s>
             
<s id="id.2.1.113.9.1.5.0"> [quod demon&longs;trare oportebat.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig108" place="text">       </figure>       
<p id="id.2.1.113.9.2.1.0" type="caption">        
<s id="id.2.1.113.9.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.114.1.0.0.0" type="margin">        
<s id="id.2.1.114.1.1.1.0"> <margin.target id="note180"></margin.target>5 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.115.1.0.0.0" type="main">        
<s id="id.2.1.115.1.1.1.0"> Similiter o&longs;tendetur, &longs;i pondus e&longs;&longs;et intra potentiam, &amp; fulci&shy;<lb/>
mentum; vel potentia inter fulcimentum, &amp; pondus.  </s>
             
<s id="id.2.1.115.1.1.2.0"> quod idem <lb/>
potenti&aelig; mouenti eueniet.  </s>
    </p>       
<p id="id.2.1.115.2.0.0.0" type="head">        
<s id="id.2.1.115.2.1.1.0"> RROPOSITIO XI. </s>
    </p>       
<p id="id.2.1.115.3.0.0.0" type="main">        
<s id="id.2.1.115.3.1.1.0"> Si vectis di&longs;tantia inter fulcimentum, &amp; poten<lb/>
tiam ad di&longs;tantiam fulcimento, punctoq;, vbi <lb/>
<expan abbr="&agrave;">a</expan> centro grauitatis ponderis horizonti ducta <lb/>
perpendicularis vectem &longs;ecat, interiectam ma&shy;<lb/>
iorem habuerit proportionem, <expan abbr="qu&agrave;m">quam</expan> pondus <lb/>
ad potentiam; pondus vtiq; <expan abbr="&agrave;">a</expan> potentia moue&shy;<lb/>
bitur. </s>
    </p>       
<p id="id.2.1.115.4.0.0.0" type="main">        
<s id="id.2.1.115.4.1.1.0"> Sit <expan abbr="v&eacute;ctis">vectis</expan> AB, ex <lb/>
punctoq; A &longs;u&longs;penda<lb/>
tur pondus C; hoc e&longs;t <lb/>
punctum A &longs;emper &longs;it <lb/>
punctum, vbi perpen<lb/>
dicularis <expan abbr="&agrave;">a</expan> grauitatis <lb/>
centro ponderis du&shy;<lb/>
cta vectem &longs;ecat; &longs;itq; <lb/>
<arrow.to.target n="fig109"></arrow.to.target><lb/>
potentia in B, ac fulcimentum &longs;it D; &amp; DB ad DA maiorem <lb/>
habeat proportionem, <expan abbr="qu&agrave;m">quam</expan> pondus C ad potentiam in B.  </s>
     
<s id="id.2.1.115.4.1.1.0.a"> Di&shy;<lb/>
co pondus <expan abbr="C&agrave;">Ca</expan> potentia in B moueri.  </s>
             
<s id="id.2.1.115.4.1.2.0"> fiat vt BD ad DA, ita <lb/>
pondus E ad potentiam in B; atq; pondus E quoq; appendatur <lb/>
in A: patet potentiam in B &aelig;queponderare ip&longs;i E; hoc e&longs;t pon&shy;<lb/>
dus  <arrow.to.target n="note181"></arrow.to.target> E &longs;u&longs;tinere.  </s>
             
<s id="id.2.1.115.4.1.3.0"> &amp; quoniam BD ad DA maiorem habet pro&shy;<lb/>
portionem, <expan abbr="qu&agrave;m">quam</expan> Cad potentiam in B; &amp; vt BD ad DA, ita 
<pb/>
e&longs;t pondus E ad po&shy;<lb/>
tentiam: igitur E ad <lb/>
potentiam maiorem <lb/>
habebit proportio&shy;<lb/>
nem, <expan abbr="qu&agrave;m">quam</expan> pondus <lb/>
C ad eandem poten&shy;<lb/>
<arrow.to.target n="note182"></arrow.to.target> tiam.  </s>
             
<s id="id.2.1.115.4.1.4.0"> quare pondus <lb/>
E maius erit ponde&shy;<lb/>
<arrow.to.target n="fig110"></arrow.to.target><lb/>
re C. &amp; <expan abbr="c&ugrave;m">cum</expan> potentia ip&longs;&lt;*&gt; E &aelig;queponderet, potentia igitur ip&longs;i <lb/>
C non &aelig;queponderabit, &longs;ed &longs;ua ui deor&longs;um verget.  </s>
             
<s id="id.2.1.115.4.1.5.0"> pondus igitur <lb/>
C <expan abbr="&agrave;">a</expan> potentia in B mouebitur vecte AB, cuius fulcimentum <lb/>
e&longs;t D. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig109" place="text">       </figure>       
<p id="id.2.1.115.4.2.1.0" type="caption">        
<s id="id.2.1.115.4.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig110" place="text">       </figure>       
<p id="id.2.1.115.4.2.3.0" type="caption">        
<s id="id.2.1.115.4.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.116.1.0.0.0" type="margin">        
<s id="id.2.1.116.1.1.1.0"> <margin.target id="note181"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.116.1.1.2.0"> <margin.target id="note182"></margin.target>10 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.117.1.0.0.0" type="main">        
<s id="id.2.1.117.1.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> &longs;it vectis AB, &amp; <lb/>
fulcimentum A, pondu&longs;q; C <lb/>
in D appen&longs;um, &amp; potentia <lb/>
in B; &amp; BA ad AD maio&shy;<lb/>
rem habeat proportionem, <lb/>
<expan abbr="qu&agrave;m">quam</expan> pondus C ad poten&shy;<lb/>
tiam in B.  </s>
     
<s id="id.2.1.117.1.1.1.0.a"> dico pondus C <expan abbr="&agrave;">a</expan> <lb/>
<arrow.to.target n="fig111"></arrow.to.target><lb/>
potentia in B moueri.  </s>
             
<s id="id.2.1.117.1.1.2.0"> fiat vt BA ad AD; ita pondus E ad poten<lb/>
<arrow.to.target n="note183"></arrow.to.target>tiam in B: &amp; &longs;i E appendatur in D, potentia in B pondus E &longs;u&longs;ti<lb/>
nebit.  </s>
             
<s id="id.2.1.117.1.1.3.0"> &longs;ed <expan abbr="c&ugrave;m">cum</expan> BA ad AD maiorem habeat proportionem, <lb/>
<expan abbr="qu&agrave;m">quam</expan> pondus C ad potentiam in B; &amp; vt BA ad AD, ita e&longs;t <lb/>
pondus E ad potentiam in B: pondus igitur E ad potentiam, <lb/>
qu&aelig; e&longs;t in B, maiorem habebit proportionem, <expan abbr="qu&agrave;m">quam</expan> pondus C <lb/>
<arrow.to.target n="note184"></arrow.to.target> ad eandem potentiam.  </s>
             
<s id="id.2.1.117.1.1.4.0"> &amp; ideo pondus E maius erit pondere C. <lb/>
potentia <expan abbr="ver&ograve;">vero</expan> in B &longs;u&longs;tinet pondus E; ergo potentia in B pondus <lb/>
C minus pondere E in D appen&longs;um mouebit vecte AB, cuius fulci <lb/>
mentum e&longs;t A. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig111" place="text">       </figure>       
<p id="id.2.1.117.1.2.1.0" type="caption">        
<s id="id.2.1.117.1.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.118.1.0.0.0" type="margin">        
<s id="id.2.1.118.1.1.1.0"> <margin.target id="note183"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.118.1.1.2.0"> <margin.target id="note184"></margin.target>10 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.119.1.0.0.0" type="main">        
<pb n="57"/>
       
<s id="id.2.1.119.1.2.1.0"> Sit rur&longs;us vectis <lb/>
AB, cuius fulcimen <lb/>
<expan abbr="t&utilde;">tum</expan> A; &amp; pondus C in <lb/>
B &longs;it appen&longs;um; &longs;itq; <lb/>
potentia in D: &amp; <lb/>
DA ad AB maio&shy;<lb/>
rem habeat propor&shy;<lb/>
tionem, <expan abbr="qu&agrave;m">quam</expan> pon&shy;<lb/>
<arrow.to.target n="fig112"></arrow.to.target><lb/>
dus C ad potentiam, qu&aelig; e&longs;t in D.  </s>
     
<s id="id.2.1.119.1.2.1.0.a"> dico pondus C <expan abbr="&agrave;">a</expan> potentia <lb/>
in D moueri.  </s>
             
<s id="id.2.1.119.1.2.2.0"> fiat vt DA ad AB, ita pondus E ad potentiam in <lb/>
D; &amp; &longs;it pondus E ex puncto B &longs;u&longs;pen&longs;um: potentia in D pondus <lb/>
E &longs;u&longs;tinebit.  </s>
             
<s id="id.2.1.119.1.2.3.0"> &longs;ed DA ad AB maiorem habet proportionem, <lb/>
<expan abbr="qu&agrave;m">quam</expan> C ad potentiam in D; &amp; vt DA ad AB, ita e&longs;t pondus E <lb/>
ad potentiam in D; pondus igitur E ad potentiam, qu&aelig; e&longs;t in D, <lb/>
maiorem habebit proportionem, <expan abbr="qu&agrave;m">quam</expan> pondus C ad eandem po<lb/>
tentiam.  </s>
             
<s id="id.2.1.119.1.2.4.0"> quare pondus E maius e&longs;t pondere C. &amp; <expan abbr="c&ugrave;m">cum</expan> poten&shy;<lb/>
tia in D pondus E &longs;u&longs;tineat, potentia igitur in D pondus C in B <lb/>
appen&longs;um vecte AB, cuius fulcimentum e&longs;t A, mouebit. quod <lb/>
demon&longs;trare oportebat. </s>
             
<s id="id.2.1.119.1.2.5.0"> [quod <lb/>
demon&longs;trare oportebat.]  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig112" place="text">       </figure>       
<p id="id.2.1.119.1.3.1.0" type="caption">        
<s id="id.2.1.119.1.3.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.119.2.0.0.0" type="head">        
<s id="id.2.1.119.2.1.1.0"> ALITER. </s>
    </p>       
<p id="id.2.1.119.3.0.0.0" type="main">        
<s id="id.2.1.119.3.1.1.0"> Sit vectis AB, &amp; <lb/>
pondus C in A ap&shy;<lb/>
pen&longs;um &amp; poten&shy;<lb/>
tia in B; &longs;it <expan abbr="qu&eacute;">que</expan> fulci&shy;<lb/>
mentum D: &amp; DB <lb/>
<arrow.to.target n="fig113"></arrow.to.target><lb/>
ad DA maiorem habeat proportionem, <expan abbr="qu&agrave;m">quam</expan> pondus C ad po<lb/>
tentiam in B.  </s>
     
<s id="id.2.1.119.3.1.1.0.a"> dico pondus C <expan abbr="&agrave;">a</expan> potentia in B moueri.  </s>
             
<s id="id.2.1.119.3.1.2.0"> fiat BE ad <lb/>
EA, vt pondus C ad potentiam, erit punctum E inter BD.  </s>
     
<s id="id.2.1.119.3.1.2.0.a"> opor<lb/>
tet enim BE ad EA minorem habere proportionem, <expan abbr="qu&agrave;m">quam</expan> DB <lb/>
ad DA, &amp; ideo BE minor erit BD.  </s>
     
<s id="id.2.1.119.3.1.2.0.b"> &amp; quoniam potentia in B &longs;u<arrow.to.target n="note185"></arrow.to.target><lb/>
&longs;tinet pondus C in A appen&longs;um uecte AB, cuius <expan abbr="fulciment&utilde;">fulcimentum</expan> E; minor <lb/>
igitur potentia in B, <expan abbr="qu&agrave;m">quam</expan> data, idem pondus &longs;u&longs;tinebit fulcimen<lb/>
to D. data ergo potentia in B pondus C mouebit uecte AB, cuius <lb/>
fulcimentum e&longs;t D. 
<pb/>
 </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig113" place="text">       </figure>       
<p id="id.2.1.119.3.2.1.0" type="caption">        
<s id="id.2.1.119.3.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.120.1.0.0.0" type="margin">        
<s id="id.2.1.120.1.1.1.0"> <margin.target id="note185"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.121.1.0.0.0" type="main">        
<s id="id.2.1.121.1.1.1.0"> Sit deinde vectis AB, &amp; fulci <lb/>
mentum A, &amp; pondus C in D <lb/>
appen&longs;um, &longs;itq; potentia in B; &amp; <lb/>
AB ad AD maiorem habeat pro&shy;<lb/>
portionem, <expan abbr="qu&agrave;m">quam</expan> pondus C ad <lb/>
potentiam in B.  </s>
     
<s id="id.2.1.121.1.1.1.0.a"> dico pondus C <lb/>
<arrow.to.target n="fig114"></arrow.to.target><expan abbr="&agrave;"><lb/>
a</expan> potentia in B moueri.  </s>
             
<s id="id.2.1.121.1.1.2.0"> Fiat AB ad AE, vt pondus C ad poten <lb/>
<arrow.to.target n="note186"></arrow.to.target> tiam; erit &longs;imiliter punctum E inter BD. nece&longs;&longs;e e&longs;t enim AE <lb/>
maiorem e&longs;&longs;e AD. &amp; &longs;i pondus C e&longs;&longs;et in E appen&longs;um, potentia <lb/>
<arrow.to.target n="note187"></arrow.to.target> in B illud &longs;u&longs;tineret.  </s>
             
<s id="id.2.1.121.1.1.3.0"> minor autem potentia in B, <expan abbr="qu&agrave;m">quam</expan> data, &longs;u&longs;ti&shy;<lb/>
<arrow.to.target n="note188"></arrow.to.target> net pondus C in D appen&longs;um; data ergo potentia in B pondus C in <lb/>
<arrow.to.target n="note189"></arrow.to.target> D appen&longs;um vecte AB, cuius fulcimentum e&longs;t A, mouebit. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig114" place="text">       </figure>       
<p id="id.2.1.121.1.2.1.0" type="caption">        
<s id="id.2.1.121.1.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.122.1.0.0.0" type="margin">        
<s id="id.2.1.122.1.1.1.0"> <margin.target id="note186"></margin.target>8 <emph type="italics"/>quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.122.1.1.2.0"> <margin.target id="note187"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.122.1.1.3.0"> <margin.target id="note188"></margin.target>1 <emph type="italics"/>Cor.<emph.end type="italics"/> </s>
             
<s id="id.2.1.122.1.1.4.0"> <margin.target id="note189"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.123.1.0.0.0" type="main">        
<s id="id.2.1.123.1.1.1.0"> Sit rur&longs;us vectis AB, cu<lb/>
ius fulcimentum A, &amp; pon<lb/>
dus C in B &longs;it appen&longs;um; <lb/>
&longs;itq; potentia in D; &amp; DA <lb/>
ad AB maiorem habeat <lb/>
<arrow.to.target n="fig115"></arrow.to.target><lb/>
proportionem, <expan abbr="qu&agrave;m">quam</expan> pondus C ad potentiam in D.  </s>
     
<s id="id.2.1.123.1.1.1.0.a"> dico pon&shy;<lb/>
dus C <expan abbr="&agrave;">a</expan> potentia in D moueri.  </s>
             
<s id="id.2.1.123.1.1.2.0"> fiat vt pondus C ad potentiam, <lb/>
<arrow.to.target n="note190"></arrow.to.target> ita DA ad AE; erit AE maior AB; <expan abbr="c&ugrave;m">cum</expan> maior &longs;it proportio <lb/>
DA ad AB, <expan abbr="qu&agrave;m">quam</expan> DA ad AE. &amp; &longs;i pondus C appendatur in <lb/>
<arrow.to.target n="note191"></arrow.to.target> E, patet potentiam in D &longs;u&longs;tinere pondus C in E appen&longs;um.  </s>
             
<s id="id.2.1.123.1.1.3.0"> mi&shy;<lb/>
<arrow.to.target n="note192"></arrow.to.target> nor autem potentia, <expan abbr="qu&agrave;m">quam</expan> data, &longs;u&longs;tinet idem pondus C in B; <lb/>
<arrow.to.target n="note193"></arrow.to.target> data igitur potentia in D pondus C in B appen&longs;um mouebit ve&shy;<lb/>
cte AB, cuius fulcimentum e&longs;t A. quod oportebat demon&shy;<lb/>
&longs;trare.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig115" place="text">       </figure>       
<p id="id.2.1.123.1.2.1.0" type="caption">        
<s id="id.2.1.123.1.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.124.1.0.0.0" type="margin">        
<s id="id.2.1.124.1.1.1.0"> <margin.target id="note190"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.124.1.1.2.0"> <margin.target id="note191"></margin.target>3 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.124.1.1.3.0"> <margin.target id="note192"></margin.target>1 <emph type="italics"/>Cor.<emph.end type="italics"/> </s>
             
<s id="id.2.1.124.1.1.4.0"> <margin.target id="note193"></margin.target>3 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.125.1.0.0.0" type="head">        
<s id="id.2.1.125.1.1.1.0"> PROPOSITIO XII. </s>
     <lb/>
       
<s id="id.2.1.125.1.3.1.0"> PROBLEMA. </s>
    </p>       
<p id="id.2.1.125.2.0.0.0" type="main">        
<s id="id.2.1.125.2.1.1.0"> Datum pondus <expan abbr="&agrave;">a</expan> data potentia dato vecte <lb/>
moueri. 
<pb n="58"/>
<arrow.to.target n="fig116"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.125.3.0.0.0" type="main">        
<s id="id.2.1.125.3.1.1.0"> Sit pondus A vt centum, potentia <expan abbr="ver&ograve;">vero</expan> mouens &longs;it vt decem; <lb/>
&longs;itq; datus vectis BC.  </s>
     
<s id="id.2.1.125.3.1.1.0.a"> oportet potentiam, qu&aelig; e&longs;t decem pondus <lb/>
A centum vecte BC mouere.  </s>
             
<s id="id.2.1.125.3.1.2.0"> Diuidatur BC in D, ita vt CD <lb/>
ad DB eandem habeat proportionem, <expan abbr="qu&agrave;m">quam</expan> habet centum ad <lb/>
decem, hoc e&longs;t decem ad vnum; etenim &longs;i D ficret fulcimentum, <lb/>
con&longs;tat potentiam vt decem in C &aelig;queponderare ponderi A in B <arrow.to.target n="note194"></arrow.to.target><lb/>
appen&longs;o: hoc e&longs;t pondus A &longs;u&longs;tinere.  </s>
             
<s id="id.2.1.125.3.1.3.0"> accipiatur inter BD quod <lb/>
uis punctum E, &amp; fiat E fulcimentum.  </s>
             
<s id="id.2.1.125.3.1.4.0"> Quoniam enim maior <arrow.to.target n="note195"></arrow.to.target><lb/>
e&longs;t proportio CE ad EB, <expan abbr="qu&agrave;m">quam</expan> CD ad DB; maiorem habebit <lb/>
proportionem CE ad EB, <expan abbr="qu&agrave;m">quam</expan> pondus A ad potentiam decem <lb/>
in C: potentia igitur decem in C pondus A centum in B appen&shy;<lb/>
&longs;um vecte BC, cuius fulcimentum &longs;it E, mouebit. <arrow.to.target n="note196"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.125.4.0.0.0" type="main">        
<s id="id.2.1.125.4.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> &longs;it vectis <lb/>
BC, &amp; fulcimen&shy;<lb/>
tum B. diuidatur CB <lb/>
in D, ita vt CB ad <lb/>
BD eandem habeat <lb/>
proportionem, <expan abbr="qu&atilde;">quam</expan> <lb/>
<arrow.to.target n="fig117"></arrow.to.target><lb/>
habet centum ad decem: &amp; &longs;i pondus A in D &longs;u&longs;pendatur, &amp; po&shy;<lb/>
tentia in C, potentia vt decem in C pondus A in D appen&longs;um &longs;u<arrow.to.target n="note197"></arrow.to.target><lb/>
&longs;tinebit.  </s>
             
<s id="id.2.1.125.4.1.2.0"> accipiatur inter DB quoduis punctum E, ponaturq; pon<lb/>
dus A in E; &amp; <expan abbr="c&ugrave;m">cum</expan> &longs;it maior proportio CB ad BE, <expan abbr="qu&agrave;m">quam</expan> <arrow.to.target n="note198"></arrow.to.target><lb/>
BC ad BD; maiorem habebit proportionem CB ad BE, <expan abbr="qu&agrave;m">quam</expan> <lb/>
pondus A centum ad potentiam decem.  </s>
             
<s id="id.2.1.125.4.1.3.0"> potentia igitur decem <arrow.to.target n="note199"></arrow.to.target><lb/>
in C pondus A centum in E appen&longs;um mouebit vecte BC, cu<lb/>
ius fulcimentum e&longs;t B. quod facere oportebat.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig116" place="text">       </figure>       
<p id="id.2.1.125.4.2.1.0" type="caption">        
<s id="id.2.1.125.4.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig117" place="text">       </figure>       
<p id="id.2.1.125.4.2.3.0" type="caption">        
<s id="id.2.1.125.4.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.126.1.0.0.0" type="margin">        
<s id="id.2.1.126.1.1.1.0"> <margin.target id="note194"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.126.1.1.2.0"> <margin.target id="note195"></margin.target><emph type="italics"/>Lemma huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.126.1.1.3.0"> <margin.target id="note196"></margin.target>11 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.126.1.1.4.0"> <margin.target id="note197"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.126.1.1.5.0"> <margin.target id="note198"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.126.1.1.6.0"> <margin.target id="note199"></margin.target>11 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.127.1.0.0.0" type="main">        
<pb/>
       
<s id="id.2.1.127.1.2.1.0"> Hoc autem fieri non po&shy;<lb/>
te&longs;t exi&longs;tente vecte BC, cuius <lb/>
fulcimentum &longs;it B, &amp; pondus <lb/>
A centum in C appen&longs;um: po<lb/>
natur enim potentia &longs;u&longs;tinens <lb/>
pondus A vtcunq; inter BC, <lb/>
<arrow.to.target n="note200"></arrow.to.target> vt in D, &longs;emper potentia ma<lb/>
<arrow.to.target n="note201"></arrow.to.target> ior erit pondere A. quare opor<lb/>
<arrow.to.target n="fig118"></arrow.to.target><lb/>
tet datam potentiam maiorem e&longs;&longs;e pondere A. &longs;it igitur poten&shy;<lb/>
tia data vt centum quinquaginta.  </s>
             
<s id="id.2.1.127.1.2.2.0"> diuidatur BC in D, ita vt CB <lb/>
ad BD &longs;it, vt centum quinquaginta ad centum; hoc e&longs;t tria ad duo: <lb/>
<arrow.to.target n="note202"></arrow.to.target> &amp; &longs;i ponatur potentia in D, patet potentiam in D &longs;u&longs;tinere pon&shy;<lb/>
dus A in C appep&longs;um.  </s>
             
<s id="id.2.1.127.1.2.3.0"> accipiatur itaq; inter DC quoduis pun&shy;<lb/>
<arrow.to.target n="note203"></arrow.to.target> ctum E, ponaturq; potentia mouens in E; &amp; <expan abbr="c&ugrave;m">cum</expan> maior &longs;it pro&shy;<lb/>
portio EB ad BC, <expan abbr="qu&agrave;m">quam</expan> DB ad BC; habebit EB ad BC maio<lb/>
rem proportionem, <expan abbr="qu&agrave;m">quam</expan> pondus A ad potentiam in E.  </s>
     
<s id="id.2.1.127.1.2.3.0.a"> poten<lb/>
<arrow.to.target n="note204"></arrow.to.target>tia igitur vt centum quinquaginta in E pondus A centum in C <lb/>
appen&longs;um vecte BC, cuius fulcimentum e&longs;t B, mouebit. quod <lb/>
facere oportebat. </s>
             
<s id="id.2.1.127.1.2.4.0"> [quod <lb/>
facere oportebat.]  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig118" place="text">       </figure>       
<p id="id.2.1.127.1.3.1.0" type="caption">        
<s id="id.2.1.127.1.3.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.128.1.0.0.0" type="margin">        
<s id="id.2.1.128.1.1.1.0"> <margin.target id="note200"></margin.target>2 <emph type="italics"/>Cor.<emph.end type="italics"/> </s>
             
<s id="id.2.1.128.1.1.2.0"> <margin.target id="note201"></margin.target>3 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.128.1.1.3.0"> <margin.target id="note202"></margin.target>3 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.128.1.1.4.0"> <margin.target id="note203"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.128.1.1.5.0"> <margin.target id="note204"></margin.target>11 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.129.1.0.0.0" type="head">        
<s id="id.2.1.129.1.1.1.0"> COROLLARIVM. </s>
    </p>       
<p id="id.2.1.129.2.0.0.0" type="main">        
<s id="id.2.1.129.2.1.1.0"> Hinc manife&longs;tum e&longs;t &longs;i data potentia &longs;it dato <lb/>
pondere maior; hoc fieri po&longs;&longs;e, &longs;iue ita exi&longs;ten<lb/>
te vecte, vt eius fulcimentum &longs;it inter pondus, <lb/>
&amp; potentiam; &longs;iue pondus inter fulcimentum, <lb/>
&amp; potentiam habente; &longs;iue demum potentia in&shy;<lb/>
ter pondus, &amp; fulcimentum con&longs;tituta. </s>
    </p>       
<p id="id.2.1.129.3.0.0.0" type="main">        
<s id="id.2.1.129.3.1.1.0"> Sin autem data potentia minor, vel &aelig;qualis <lb/>
dato pondere fuerit; palam quoq; e&longs;t id ip&longs;um <lb/>
dumtaxat a&longs;&longs;e qui po&longs;&longs;e vecte ita exi&longs;tente, vt eius <lb/>
fulcimentum &longs;it inter pondus, &amp; pontentiam; 
<pb n="59"/>
vel pondus intra fulcimentum, &amp; potentiam <lb/>
habente.  </s>
    </p>       
<p id="id.2.1.129.4.0.0.0" type="head">        
<s id="id.2.1.129.4.1.1.0"> PROPOSITIO XIII. </s>
     <lb/>
       
<s id="id.2.1.129.4.3.1.0"> PROBLEMA. </s>
    </p>       
<p id="id.2.1.129.5.0.0.0" type="main">        
<s id="id.2.1.129.5.1.1.0"> Quotcunq; datis in vecte ponderibus <expan abbr="vbicun&shy;qu&egrave;">vbicun&shy;<lb/>
que</expan> appen&longs;is, cuius fulcimentum &longs;it quoq; da&shy;<lb/>
tum, potentiam inuenire, qu&aelig; in dato puncto <lb/>
data pondera &longs;u&longs;tineat. <arrow.to.target n="fig119"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.129.6.0.0.0" type="main">        
<s id="id.2.1.129.6.1.1.0"> Sint data pondera ABC in vecte DE, cuius fulcimentum F, <lb/>
vbicunq; in punctis DGH appen&longs;a: collocandaq; &longs;it potentia in <lb/>
puncto E. potentiam inuenire oportet, qu&aelig; in E data pondera <lb/>
ABC vecte DE &longs;u&longs;tineat.  </s>
             
<s id="id.2.1.129.6.1.2.0"> diuidatur DG in k, ita vt Dk ad KG <lb/>
&longs;it, vt pondus B ad pondus A; deinde diuidatur kH in L, ita vt kL <lb/>
ad LH, &longs;it vt pondus C ad pondera BA; atq; vt FE ad FL, ita <lb/>
fiant pondera ABC &longs;imul ad potentiam, qu&aelig; ponatur in E.  </s>
     
<s id="id.2.1.129.6.1.2.0.a"> di&shy;<lb/>
co potentiam in E data pondera ABC in DGH appen&longs;a vecte <lb/>
DE, cuius fulcimentum e&longs;t F, &longs;u&longs;tinere.  </s>
             
<s id="id.2.1.129.6.1.3.0"> Quoniam enim &longs;i ponde<lb/>
ra ABC &longs;imul e&longs;&longs;ent in L appen&longs;a, potentia in E data pondera <arrow.to.target n="note205"></arrow.to.target><lb/>
in L appen&longs;a &longs;u&longs;tineret; pondera <expan abbr="ver&ograve;">vero</expan> ABC <expan abbr="t&agrave;m">tam</expan> in L ponderant, <arrow.to.target n="note206"></arrow.to.target><expan abbr="qu&agrave;m"><lb/>
quam</expan> &longs;i C in H, &amp; BA &longs;imul in K e&longs;&longs;ent appen&longs;a; &amp; AB in k <expan abbr="t&agrave;m">tam</expan> 
<pb/>
<arrow.to.target n="fig120"></arrow.to.target><lb/>
ponderant, <expan abbr="qu&agrave;m">quam</expan> &longs;i A in D, &amp; B in G appen&longs;a e&longs;&longs;ent; ergo po&shy;<lb/>
tentia in E data pondera ABC in DGH appen&longs;a vecte DE, cu&shy;<lb/>
ius fulcimentum e&longs;t F, &longs;u&longs;tinebit.  </s>
             
<s id="id.2.1.129.6.1.4.0"> Si autem potentia in quouis <lb/>
alio puncto vectis DE (pr&aelig;terqu&agrave;m in F) con&longs;tituenda e&longs;&longs;et, <lb/>
vt in k; fiat vt Fk ad FL, ita pondera ABC ad potentiam: &longs;i&shy;<lb/>
<arrow.to.target n="note207"></arrow.to.target> militer demon&longs;trabimus potentiam in k pondera ABC in pun&shy;<lb/>
ctis DGH appen&longs;a &longs;u&longs;tinere. quod facere oportebat. <arrow.to.target n="fig121"></arrow.to.target> </s>
             
<s id="id.2.1.129.6.1.5.0"> [quod facere oportebat. <arrow.to.target n="fig121"></arrow.to.target>] </s>
    </p>       
<p id="id.2.1.129.7.0.0.0" type="main">        
<s id="id.2.1.129.7.1.1.0"> Ex hac, &amp; ex quinta huius, &longs;i pondera ABC &longs;int in vecte <lb/>
DE quomodocunq; po&longs;ita; oporteatq; potentiam inuenire, qu&aelig; <lb/>
in E data pondera &longs;u&longs;tinere debeat: ducantur <expan abbr="&agrave;">a</expan> centris grauita&shy;<lb/>
tum ponderum ABC horizontibus perpendiculares, qu&aelig; ve&shy;<lb/>
ctem DE in DGH punctis &longs;ecent; c&aelig;teraq; eodem modo fiant: <lb/>
Manife&longs;tum e&longs;t, potentiam in E, vel in K data pondera &longs;u&longs;tinere.  </s>
             
<s id="id.2.1.129.7.1.2.0"> <lb/>
idem enim e&longs;t, ac &longs;i pondera in DGH e&longs;&longs;ent appen&longs;a.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig119" place="text">       </figure>       
<p id="id.2.1.129.7.2.1.0" type="caption">        
<s id="id.2.1.129.7.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig120" place="text">       </figure>       
<p id="id.2.1.129.7.2.3.0" type="caption">        
<s id="id.2.1.129.7.2.3.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig121" place="text">       </figure>       
<p id="id.2.1.129.7.2.5.0" type="caption">        
<s id="id.2.1.129.7.2.5.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.130.1.0.0.0" type="margin">        
<s id="id.2.1.130.1.1.1.0"> <margin.target id="note205"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.130.1.1.2.0"> <margin.target id="note206"></margin.target>5 <emph type="italics"/>Huius. de libra.<emph.end type="italics"/> </s>
             
<s id="id.2.1.130.1.1.3.0"> [de libra.<emph.end type="italics"/>] </s>
             
<s id="id.2.1.130.1.1.4.0"> <margin.target id="note207"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.131.1.0.0.0" type="head">        
<pb n="60"/>
       
<s id="id.2.1.131.1.2.1.0"> PROPOSITIO XIIII. </s>
     <lb/>
       
<s id="id.2.1.131.1.4.1.0"> PROBLEMA. </s>
    </p>       
<p id="id.2.1.131.2.0.0.0" type="main">        
<s id="id.2.1.131.2.1.1.0"> Data quotcunq; pondera in dato vecte vbi&shy;<lb/>
cunq; &amp; quomodocunq; po&longs;ita <expan abbr="&agrave;">a</expan> data potentia <lb/>
moueri. <arrow.to.target n="fig122"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.131.3.0.0.0" type="main">        
<s id="id.2.1.131.3.1.1.0"> Sit datus vectis DE, &amp; &longs;int data pondera vt in pr&aelig;cedenti co<lb/>
rollario; &longs;itq; A vt centum, B vt quinquaginta, C vt triginta; <lb/>
dataq; potentia &longs;it vt triginta.  </s>
             
<s id="id.2.1.131.3.1.2.0"> exponantur eadem, inueniaturq; <lb/>
punctum L; deinde diuidatur LE in F, ita vt FE ad FL &longs;it, vt <lb/>
centum octoginta ad triginta, hoc e&longs;t &longs;ex ad vnum: &amp; &longs;i F fieret <lb/>
fulcimentum, potentia vt triginta in E &longs;u&longs;tineret pondera ABC.  </s>
     
<s id="id.2.1.131.3.1.2.0.a"> <arrow.to.target n="note208"></arrow.to.target><lb/>
accipiatur igitur inter LF quoduis punctum M, fiatq; M fulci&shy;<lb/>
mentum: manife&longs;tum e&longs;t potentiam in E vt triginta pondera <arrow.to.target n="note209"></arrow.to.target><lb/>
ABC vt centum octoginta vecte DE mouere. quod facere <lb/>
oportebat. </s>
             
<s id="id.2.1.131.3.1.3.0"> [quod facere <lb/>
oportebat.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig122" place="text">       </figure>       
<p id="id.2.1.131.3.2.1.0" type="caption">        
<s id="id.2.1.131.3.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.132.1.0.0.0" type="margin">        
<s id="id.2.1.132.1.1.1.0"> <margin.target id="note208"></margin.target>13 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.132.1.1.2.0"> <margin.target id="note209"></margin.target>11 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.133.1.0.0.0" type="main">        
<s id="id.2.1.133.1.1.1.0"> Hoc autem <expan abbr="vniuers&egrave;">vniuerse</expan> a&longs;&longs;equi <expan abbr="minim&egrave;">minime</expan> poterimus, &longs;i in extremita&shy;<lb/>
te vectis fulcimentum e&longs;&longs;et, vt in D; quia proportio DE, ad DL <lb/>
hoc e&longs;t proportio ponderum ABC ad potentiam, qu&aelig; pondera <lb/>
&longs;u&longs;tinere debeat, &longs;emper e&longs;t data.  </s>
             
<s id="id.2.1.133.1.1.2.0"> quod multo quoq; minus fieri <lb/>
po&longs;&longs;et, &longs;i ponenda e&longs;&longs;et potentia inter DL. </s>
    </p>       
<pb/>
       
<p id="id.2.1.133.3.0.0.0" type="head">        
<s id="id.2.1.133.3.1.1.0"> PROPOSITIO XV. </s>
     <lb/>
       
<s id="id.2.1.133.3.3.1.0"> PROBLEMA. </s>
    </p>       
<p id="id.2.1.133.4.0.0.0" type="main">        
<s id="id.2.1.133.4.1.1.0"> Quia <expan abbr="ver&ograve;">vero</expan> dum pondera vecte mouentur, <lb/>
vectis quoq; grauitatem habet, cuius nulla ha&shy;<lb/>
ctenus mentio facta e&longs;t: idcirco <expan abbr="prim&ugrave;m">primum</expan> quo&shy;<lb/>
modo inueniatur potentia, qu&aelig; in dato puncto <lb/>
datum vectem, cuius fulcimentum &longs;it quoq; da&shy;<lb/>
tum, &longs;u&longs;tineat, o&longs;tendamus. <arrow.to.target n="fig123"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.133.5.0.0.0" type="main">        
<s id="id.2.1.133.5.1.1.0"> Sit datus vectis AB, cuius fulcimentum &longs;it datum C; &longs;itq; <lb/>
punctum D, in quo collocanda &longs;it potentia, qu&aelig; vectem AB &longs;u<lb/>
&longs;tinere debeat, ita vt immobilis per&longs;i&longs;tat.  </s>
             
<s id="id.2.1.133.5.1.2.0"> ducatur <expan abbr="&agrave;">a</expan> puncto C <lb/>
linea CE horizonti perpendicularis, qu&aelig; vectem AB in duas di&shy;<lb/>
uidat partes AE EF, &longs;itq; partis AE centrum grauitatis G, &amp; <lb/>
partis EF centrum grauitatis H; <expan abbr="&agrave;">a</expan> <expan abbr="punctisqu&eacute;">punctisque</expan> GH horizon&shy;<lb/>
tibus perpendiculares ducantur Gk HL, qu&aelig; lineam AF <lb/>
in punctis KL &longs;ecent.  </s>
             
<s id="id.2.1.133.5.1.3.0"> quoniam enim vectis AB <expan abbr="&agrave;">a</expan> linea CE in duas <lb/>
diuiditur partes AE EF; ideo vectis AB nihil aliud erit, ni&longs;i <lb/>
duo pondera AE EF in vecte, &longs;iue libra AF po&longs;ita; cuius &longs;u&shy;<lb/>
&longs;pen&longs;io, &longs;iue fulcimentum e&longs;t C. quare pondera AE EF ita erunt <lb/>
po&longs;ita, ac &longs;i in kL e&longs;&longs;ent appen&longs;a.  </s>
             
<s id="id.2.1.133.5.1.4.0"> diuidatur ergo kL in M, <lb/>
ita vt kM ad ML, &longs;it vt grauitas partis EF ad grauitatem par&shy;<lb/>
tis AE; &amp; vt CA ad CM, ita fiat grauitas totius vectis AB ad <lb/>
potentiam, qu&aelig; &longs;i collocetur in D (dummodo DA horizonti 
<pb n="61"/>
perpendicularis exi&longs;tat) vecti &aelig;queponderabit; hoc e&longs;t vectem <arrow.to.target n="note210"></arrow.to.target><lb/>
AB deor&longs;um premendo &longs;u&longs;tinebit. quod inuenire oportebat.  </s>
             
<s id="id.2.1.133.5.1.5.0"> [quod inuenire oportebat.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig123" place="text">       </figure>       
<p id="id.2.1.133.5.2.1.0" type="caption">        
<s id="id.2.1.133.5.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.134.1.0.0.0" type="margin">        
<s id="id.2.1.134.1.1.1.0"> <margin.target id="note210"></margin.target>13 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.135.1.0.0.0" type="main">        
<s id="id.2.1.135.1.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> potentia in puncto B ponenda e&longs;&longs;et.  </s>
             
<s id="id.2.1.135.1.1.2.0"> fiat vt CF ad CM <lb/>
ita pondus AB ad potentiam.  </s>
             
<s id="id.2.1.135.1.1.3.0"> &longs;imili modo o&longs;tendetur poten&shy;<lb/>
tiam in B vectem AB &longs;u&longs;tinere.  </s>
             
<s id="id.2.1.135.1.1.4.0"> &longs;imiliterq; demon&longs;trabitur in quo&shy;<lb/>
cunq; alio &longs;itu (pr&aelig;terqu&agrave;m in e) ponenda fuerit potentia, vt in <lb/>
N. fiat enim vt CO ad CM, ita AB ad potentiam; qu&aelig; &longs;i pona&shy;<lb/>
tur in N, vectem AB &longs;u&longs;tinebit. </s>
    </p>       
<p id="id.2.1.135.2.0.0.0" type="main">        
<s id="id.2.1.135.2.1.1.0"> Adiiciatur autem pondus in vecte appen&longs;um, <lb/>
&longs;iue po&longs;itum; vt iisdem po&longs;itis &longs;it pondus P in <lb/>
A appen&longs;um; potentiaq; &longs;it ponenda in B, ita <lb/>
vt vectem AB <expan abbr="vn&agrave;">vna</expan> cum pondere P &longs;u&longs;tineat. <arrow.to.target n="fig124"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.135.3.0.0.0" type="main">        
<s id="id.2.1.135.3.1.1.0"> Diuidatur AM in Q, ita vt AQ ad QM &longs;it, ut grauitas ue&shy;<lb/>
ctis AB ad grauitatem ponderis P; deinde ut CF ad CQ, ita fat <lb/>
grauitas AB, &amp; P &longs;imul ad potentiam, qu&aelig; ponatur in B: patet <lb/>
potentiam in B uectem AB <expan abbr="un&agrave;">una</expan> cum pondere P &longs;u&longs;tinere.  </s>
             
<s id="id.2.1.135.3.1.2.0"> Si ue-<arrow.to.target n="note211"></arrow.to.target><expan abbr="r&ograve;"><lb/>
ro</expan> e&longs;&longs;et CA ad CM, vt AB ad P; e&longs;&longs;et punctum C eorum centrum <arrow.to.target n="note212"></arrow.to.target><lb/>
grauitatis, &amp; ideo vectis AB <expan abbr="vn&aacute;">vna</expan> cum pondere P ab&longs;q; potentia in <arrow.to.target n="note213"></arrow.to.target><lb/>
B manebit.  </s>
             
<s id="id.2.1.135.3.1.3.0"> &longs;ed &longs;i ponderum grauitatis centrum e&longs;&longs;et inter CF, vt <lb/>
in O; fiat vt CF ad CO, ita AB&amp;P &longs;imul ad potentiam, qu&aelig; <lb/>
in B, &amp; vectem AB, &amp; pondus P &longs;u&longs;tinebit. 
<pb/>
<arrow.to.target n="fig125"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.135.4.0.0.0" type="main">        
<s id="id.2.1.135.4.1.1.0"> Similiter o&longs;tendetur, &longs;i plura e&longs;&longs;ent pondera in vecte AB ubi&shy;<lb/>
cunq;, &amp; quomodocunq; po&longs;ita. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig124" place="text">       </figure>       
<p id="id.2.1.135.4.2.1.0" type="caption">        
<s id="id.2.1.135.4.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig125" place="text">       </figure>       
<p id="id.2.1.135.4.2.3.0" type="caption">        
<s id="id.2.1.135.4.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.136.1.0.0.0" type="margin">        
<s id="id.2.1.136.1.1.1.0"> <margin.target id="note211"></margin.target>13 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.136.1.1.2.0"> <margin.target id="note212"></margin.target><emph type="italics"/>Ex &longs;exta<emph.end type="italics"/> </s>
             
<s id="id.2.1.136.1.1.3.0"> <margin.target id="note213"></margin.target>1 <emph type="italics"/>Arch. de &aelig;quep.<emph.end type="italics"/> </s>
             
<s id="id.2.1.136.1.1.4.0"> [de &aelig;quep.<emph.end type="italics"/>] </s>
    </p>       
<p id="id.2.1.137.1.0.0.0" type="main">        
<s id="id.2.1.137.1.1.1.0"> In&longs;uper ex his non &longs;olum, ut in decimaquarta huius docuimus, <lb/>
quomodo &longs;cilicet data pondera ubicunq; in uecte po&longs;ita data poten<lb/>
tia dato uecte mouere po&longs;&longs;umus, eodem modo grauitate uectis <lb/>
con&longs;iderata idem facere poterimus; <expan abbr="uer&ugrave;m">uerum</expan> etiam accidentia reli&shy;<lb/>
qua, qu&aelig; &longs;upra ab&longs;q; uectis grauitatis con&longs;ideratione demon&longs;tra&shy;<lb/>
ta &longs;unt; &longs;imili modo uectis grauitate con&longs;iderata <expan abbr="vn&aacute;">vna</expan> cum ponde<lb/>
ribus, uel &longs;ine ponderibus o&longs;tendentur.  </s>
    </p>       
      </chap>
      <pb n="62"/>
      <chap>

        <p id="id.2.1.137.2.0.0.0" type="head">        
          <s id="id.2.1.137.3.1.1.0"> DE TROCHLEA. </s>
    </p>       
        <p id="id.2.1.137.4.0.0.0" type="main">        
          <s id="id.2.1.137.4.1.1.0"> Trochleae in&longs;trumento pon<lb/>
dus multipliciter moueri pote&longs;t; <lb/>
quia <expan abbr="ver&ograve;">vero</expan> in omnibus e&longs;t eadem <lb/>
ratio: ideo (vt res euidentior ap&shy;<lb/>
pareat) in iis, qu&aelig; dicenda &longs;unt, <lb/>
intelligatur pondus &longs;ur&longs;um ad re<lb/>
ctos horizontis plano angulos hoc modo &longs;em&shy;<lb/>
per moueri.  </s>
    </p>       
        <pb/>
        
        <p id="id.2.1.137.6.0.0.0" type="main">        
          <s id="id.2.1.137.6.1.1.0"> Sit pondus A, quod ip&longs;i ho<lb/>
rizontis plano &longs;ur&longs;um ad rectos <lb/>
angulos &longs;it attollendum; &amp; vt <lb/>
fieri &longs;olet, trochlea duos habens <lb/>
orbiculos, quorum axiculi &longs;int <lb/>
in BC, <expan abbr="&longs;upern&egrave;">&longs;uperne</expan> appendatur; <lb/>
trochlea <expan abbr="ver&ograve;">vero</expan> duos &longs;imiliter ha<lb/>
bens orbiculos, quorum axicu&shy;<lb/>
li &longs;int in DE, ponderi alligetur: <lb/>
ac per omnes vt riu&longs;q; trochle&aelig; <lb/>
orbiculos circunducatur ducta&shy;<lb/>
rius funis, quem in altero eius ex <lb/>
tremo, <expan abbr="put&aacute;">puta</expan> in F, oportet e&longs;&longs;e <lb/>
religatum.  </s>
             
<s id="id.2.1.137.6.1.2.0"> potentia autem mo<lb/>
uens ponatur in G, qu&aelig; dum <lb/>
de&longs;cendit, pondus A &longs;ur&longs;um ex <lb/>
aduer&longs;o attolletur; quemadmo<lb/>
dum Pappus in octauo libro Ma<lb/>
thematicarum collectionum a&longs;&shy;<lb/>
&longs;erit; nec non Vitruuius in deci <lb/>
mo de Architectura, &amp; alii. <arrow.to.target n="fig126"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.137.7.0.0.0" type="main">        
<s id="id.2.1.137.7.1.1.0"> Quomodo autem hoc trochle&aelig; in&longs;trumen&shy;<lb/>
tum reducatur ad vectem; cur magnum pondus <lb/>
ab exigua virtute, &amp; quomodo, quantoq; in tem<lb/>
pore moueatur; cur funis in vno capite debeat <lb/>
e&longs;&longs;e religatus; quodq; &longs;uperioris, <expan abbr="inferioris&qacute;ue">inferiorisque</expan> <lb/>
trochle&aelig; fuerit officium; &amp; quomodo omnis in 
<pb n="63"/>
numeris data proportio inter potentiam, &amp; pon<lb/>
dus inueniri po&longs;sit; dicamus.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig126" place="text">       </figure>       
<p id="id.2.1.137.7.2.1.0" type="caption">        
<s id="id.2.1.137.7.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.137.8.0.0.0" type="head">        
<s id="id.2.1.137.8.1.1.0"> LEMMA. </s>
    </p>       
<p id="id.2.1.137.9.0.0.0" type="main">        
<s id="id.2.1.137.9.1.1.0"> Sint rect&aelig; line&aelig; AB CD parallel&aelig;, qu&aelig; in <lb/>
punctis AC circulum ACE contingant, cuius <lb/>
centrum F: &amp; FA FC connectantur.  </s>
             
<s id="id.2.1.137.9.1.2.0"> Dico <lb/>
AFC rectam lineam e&longs;&longs;e. </s>
    </p>       
<p id="id.2.1.137.10.0.0.0" type="main">        
<s id="id.2.1.137.10.1.1.0"> Ducatur FE ip&longs;is AB CD &aelig;quidi&longs;tans.  </s>
             
<s id="id.2.1.137.10.1.2.0"> <lb/>
&amp; quoniam AB, &amp; FE &longs;unt parallel&aelig;, &amp; <lb/>
angulus BAF e&longs;t rectus; erit &amp; AFE re&shy;<lb/>
ctus.  </s>
             
<s id="id.2.1.137.10.1.3.0"> eodemq; modo CFE rectus erit.  </s>
             
<s id="id.2.1.137.10.1.4.0"> li&shy;<lb/>
neaigitur  <arrow.to.target n="note214"></arrow.to.target> AFC recta e&longs;t.  </s>
             
<s id="id.2.1.137.10.1.5.0"> quod erat de&shy;<lb/>
mon&longs;trandum.  <arrow.to.target n="note215"></arrow.to.target> <arrow.to.target n="note216"></arrow.to.target><lb/>
 </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure place="text">       </figure>       
<p id="id.2.1.137.10.2.1.0" type="caption">        
<s id="id.2.1.137.10.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.138.1.0.0.0" type="margin">        
<s id="id.2.1.138.1.1.1.0"> <margin.target id="note214"></margin.target>18 <emph type="italics"/>Tertii.<emph.end type="italics"/> </s>
             
<s id="id.2.1.138.1.1.2.0"> <margin.target id="note215"></margin.target>29 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.138.1.1.3.0"> <margin.target id="note216"></margin.target>14 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.139.1.0.0.0" type="head">        <lb/>
       
<s id="id.2.1.139.1.2.1.0"> PROPOSITIO I. </s>
    </p>       
<p id="id.2.1.139.2.0.0.0" type="main">        
<s id="id.2.1.139.2.1.1.0"> Si funis trochle&aelig; <expan abbr="&longs;upern&egrave;">&longs;uperne</expan> appen&longs;&aelig; orbiculo <lb/>
circunducatur, alterumq; eius extremum pon&shy;<lb/>
deri alligetur, altero interim <expan abbr="&agrave;">a</expan> potentia pondus <lb/>
&longs;u&longs;tinente apprehen&longs;o: erit potentia ponderi <lb/>
&aelig;qualis.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.139.4.0.0.0" type="main">        
<s id="id.2.1.139.4.1.1.0"> Sit pondus A, <lb/>
cui alligatus &longs;it fu&shy;<lb/>
nis in B; trochleaq; <lb/>
habens orbiculum C <lb/>
EF, cuius centrum <lb/>
D, &longs;ur&longs;um appenda&shy;<lb/>
tur; &longs;itq; D quoq; <lb/>
centrum axiculi; &amp; <lb/>
circa orbiculum uo&shy;<lb/>
luatur funis BC EF <lb/>
G; &longs;itq; potentia <lb/>
in G &longs;u&longs;tinens pon&shy;<lb/>
dus A.  </s>
     
<s id="id.2.1.139.4.1.1.0.a"> dico poten&shy;<lb/>
tiam in G ponderi A <lb/>
&aelig;qualem e&longs;&longs;e.  </s>
             
<s id="id.2.1.139.4.1.2.0"> Sit FG <lb/>
&aelig;quidi&longs;tans CB.  </s>
     
<s id="id.2.1.139.4.1.2.0.a"> <lb/>
Quoniam igitur pon<lb/>
<arrow.to.target n="note217"></arrow.to.target>dus A manet; erit <lb/>
<arrow.to.target n="fig127"></arrow.to.target><lb/>
CB horizonti plano perpendicularis &lt;*&gt; quare FG eidem plano per&shy;<lb/>
<arrow.to.target n="note218"></arrow.to.target> pendicularis erit.  </s>
             
<s id="id.2.1.139.4.1.3.0"> Sint CF <expan abbr="p&utilde;cta">puncta</expan> in orbiculo, <expan abbr="&agrave;">a</expan> quibus funes CB FG <lb/>
in horizontis <expan abbr="plan&utilde;">planum</expan> ad rectos angulos de&longs;cendunt; tangent BC FG <lb/>
<expan abbr="orbicul&utilde;">orbiculum</expan> CEF in punctis CF. <expan abbr="orbicul&utilde;">orbiculum</expan> enim <expan abbr="&longs;ecaren&otilde;">&longs;ecarenon</expan> po&longs;&longs;unt.  </s>
             
<s id="id.2.1.139.4.1.4.0"> con<lb/>
nectantur DC DF; erit CF recta linea, &amp; anguli DCB DFG recti.  </s>
             
<s id="id.2.1.139.4.1.5.0"> <lb/>
<arrow.to.target n="note219"></arrow.to.target> <expan abbr="Quoni&atilde;">Quoniam</expan> <expan abbr="aut&etilde;">autem</expan> BC <expan abbr="t&ugrave;m">tum</expan> horizonti, <expan abbr="t&ugrave;m">tum</expan> ip&longs;i CF e&longs;t perpendicularis; <lb/>
erit linea CF horizonti &aelig;quidi&longs;tans.  </s>
             
<s id="id.2.1.139.4.1.6.0"> <expan abbr="c&ugrave;m">cum</expan> <expan abbr="ver&ograve;">vero</expan> <expan abbr="p&otilde;dus">pondus</expan> appen&longs;um &longs;it <lb/>
<arrow.to.target n="note220"></arrow.to.target> in BC, &amp; potentia &longs;it in G; quod idem e&longs;t, ac &longs;i e&longs;&longs;et in F; erit <lb/>
CF tanquam libra, &longs;iue vectis, cuius centrum, &longs;iue fulcimentum e&longs;t <lb/>
D; nam in axiculo orbuculus &longs;u&longs;tinetur; atq; punctum D, <expan abbr="c&ugrave;m">cum</expan> &longs;it <lb/>
centrum axiculi, &amp; orbiculi, etiam vtri&longs;que circumuolutis <lb/>
immobile remanet.  </s>
             
<s id="id.2.1.139.4.1.7.0"> Itaq; <expan abbr="c&ugrave;m">cum</expan> di&longs;tantia DC &longs;it &aelig;qualis di&longs;tanti&aelig; <lb/>
DF, potentiaq; in F ponderi A in C appen&longs;o &aelig;queponderet, <expan abbr="c&ugrave;m">cum</expan> <lb/>
<arrow.to.target n="note221"></arrow.to.target> pondus &longs;u&longs;tineat, ne deor&longs;um vergat; erit potentia in F, &longs;iue in G <lb/>
(nam idem e&longs;t) con&longs;tituta ponderi A &aelig;qualis.  </s>
             
<s id="id.2.1.139.4.1.8.0"> Idem enim effi&shy;<lb/>
cit potentia in G, ac &longs;i in G aliud e&longs;&longs;et appen&longs;um pondus &aelig;quale <lb/>
ponderi A; qu&aelig; pondera in CF appen&longs;a &aelig;qu&aelig;ponderabunt.  </s>
             
<s id="id.2.1.139.4.1.9.0"> Pr&aelig;&shy;<lb/>
terea, <expan abbr="c&ugrave;m">cum</expan> in neutram fiat motus partem, idem erit vnico exi&shy;
<pb n="64"/>
&longs;tente fune BC EFG hoc modo orbiculo circumuoluto, ac &longs;i duo <lb/>
e&longs;&longs;ent funes BC FG alligati in vecte, &longs;iue libra CF. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig127" place="text">       </figure>       
<p id="id.2.1.139.4.2.1.0" type="caption">        
<s id="id.2.1.139.4.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.140.1.0.0.0" type="margin">        
<s id="id.2.1.140.1.1.1.0"> <margin.target id="note217"></margin.target>1 <emph type="italics"/>Huius. de libra.<emph.end type="italics"/> </s>
             
<s id="id.2.1.140.1.1.2.0"> [de libra.<emph.end type="italics"/>] </s>
             
<s id="id.2.1.140.1.1.3.0"> <margin.target id="note218"></margin.target>8 <emph type="italics"/>Vndecimi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.140.1.1.4.0"> <margin.target id="note219"></margin.target>18 <emph type="italics"/>Tertii.<emph.end type="italics"/> </s>
             
<s id="id.2.1.140.1.1.5.0"> <margin.target id="note220"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 28 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.140.1.1.6.0"> <margin.target id="note221"></margin.target>1 <emph type="italics"/>Primi. Archim. de &aelig;quepond.<emph.end type="italics"/> </s>
             
<s id="id.2.1.140.1.1.7.0"> [Archim.]  </s>
             
<s id="id.2.1.140.1.1.8.0"> [de &aelig;quepond.<emph.end type="italics"/>] </s>
    </p>       
<p id="id.2.1.141.1.0.0.0" type="head">        
<s id="id.2.1.141.1.1.1.0"> COROLLARIVM. </s>
    </p>       
<p id="id.2.1.141.2.0.0.0" type="main">        
<s id="id.2.1.141.2.1.1.0"> Ex hoc manife&longs;tum e&longs;&longs;e pote&longs;t, idem pon&shy;<lb/>
dus ab eadem potentia ab&longs;q; ullo huius tro&shy;<lb/>
chle&aelig; auxilio nihilominus &longs;u&longs;tineri po&longs;&longs;e. </s>
    </p>       
<p id="id.2.1.141.3.0.0.0" type="main">        
<s id="id.2.1.141.3.1.1.0"> Sit enim pondus H &aelig;quale <lb/>
ponderi A, cui alligatus &longs;it funis <lb/>
kL; &longs;itq; potentia in L &longs;u&longs;tinens <lb/>
pondus H. <expan abbr="c&ugrave;m">cum</expan> autem pondus <lb/>
ab&longs;q; vllo adminiculo &longs;u&longs;tinere <lb/>
volentes tanta vi opus &longs;it, quanta <lb/>
ponderi e&longs;t &aelig;qualis; erit potentia <lb/>
in L ponderi H &aelig;qualis; pondus <lb/>
<expan abbr="ver&ograve;">vero</expan> H ip&longs;i ponderi A e&longs;t &aelig;quale, <lb/>
cui potentia in G e&longs;t &aelig;qualis; erit <lb/>
igitur potentia in G potenti&aelig; in L <lb/>
&aelig;qualis.  </s>
             
<s id="id.2.1.141.3.1.2.0"> quod idem e&longs;t, ac &longs;i <expan abbr="ead&etilde;">eadem</expan> <lb/>
potentia idem pondus &longs;u&longs;tineret. <arrow.to.target n="fig128"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.141.4.0.0.0" type="main">        
<s id="id.2.1.141.4.1.1.0"> Pr&aelig;terea &longs;i potenti&aelig; in G, &amp; <lb/>
in L inuicem fuerint &aelig;quales, &longs;eor<lb/>
&longs;um autem ponderibus minores; <lb/>
patet potentias ponderibus &longs;u&longs;ti&shy;<lb/>
nendis non &longs;ufficere.  </s>
             
<s id="id.2.1.141.4.1.2.0"> &longs;i <expan abbr="ver&ograve;">vero</expan> maiores, manife&longs;tum e&longs;t pondera <expan abbr="&agrave;">a</expan> <lb/>
pontentiis moueri.  </s>
             
<s id="id.2.1.141.4.1.3.0"> &amp; &longs;ic in eadem e&longs;&longs;e proportione potentiam in <lb/>
L. ad pondus H, veluti potentia in G ad pondus A. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig128" place="text">       </figure>       
<p id="id.2.1.141.4.2.1.0" type="caption">        
<s id="id.2.1.141.4.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.141.5.0.0.0" type="main">        
<s id="id.2.1.141.5.1.1.0"> Sed quoniam in demon&longs;tratione a&longs;&longs;umptum fuit axiculum cir&shy;<lb/>
cumuerti, qui vt plurimum immobilis manet; idcirco immobili <lb/>
quoq; manente axiculo idem o&longs;tendatur.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.141.7.0.0.0" type="main">        
<s id="id.2.1.141.7.1.1.0"> Sit orbiculus trochle&aelig; CEF, cu<lb/>
ius centrum D; &longs;itq; axiculus GHk, <lb/>
cuius idem &longs;it centrum D.  </s>
     
<s id="id.2.1.141.7.1.1.0.a"> Ducatur <lb/>
CG DkF diameter horizonti &aelig;&shy;<lb/>
quidi&longs;tans.  </s>
             
<s id="id.2.1.141.7.1.2.0"> &amp; quoniam dum orbi&shy;<lb/>
culus circumuertitur, circumferen&shy;<lb/>
tia circuli CEF &longs;emper e&longs;t &aelig;quidi&shy;<lb/>
&longs;tans circumferenti&aelig; axiculi GHk; <lb/>
circa enim axiculum circumuerti&shy;<lb/>
tur; &amp; circulorum &aelig;quidi&longs;tantes cir<lb/>
cumferenti&aelig; idem habent centrum; <lb/>
erit punctum D &longs;emper &amp; orbiculi, <lb/>
<arrow.to.target n="fig129"></arrow.to.target><lb/>
&amp; axiculi centrum.  </s>
             
<s id="id.2.1.141.7.1.3.0"> Itaq; <expan abbr="c&ugrave;m">cum</expan> DC &longs;it &aelig;qualis DF, &amp; DG ip&longs;i <lb/>
Dk; erit GC ip&longs;i kF &aelig;qualis.  </s>
             
<s id="id.2.1.141.7.1.4.0"> &longs;i igitur in vecte, &longs;iue libra CF <lb/>
pondera appendantur &aelig;qualia, &aelig;queponderabunt.  </s>
             
<s id="id.2.1.141.7.1.5.0"> di&longs;tantia enim <lb/>
CG &aelig;qualis e&longs;t di&longs;tanti&aelig; kF; axiculu&longs;&lt;*&gt;; GHK immobilis gerit <lb/>
vicem centri, &longs;iue fulcimenti.  </s>
             
<s id="id.2.1.141.7.1.6.0"> immobili igitur manente axicu&shy;<lb/>
lo, &longs;i ponatur in F potentia &longs;u&longs;tinens pondus in C appen&longs;um; erit <lb/>
potentia in F ip&longs;i ponderi &aelig;qualis. quod erat o&longs;tendendum. </s>
             
<s id="id.2.1.141.7.1.7.0"> [quod erat o&longs;tendendum.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig129" place="text">       </figure>       
<p id="id.2.1.141.7.2.1.0" type="caption">        
<s id="id.2.1.141.7.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.141.8.0.0.0" type="main">        
<s id="id.2.1.141.8.1.1.0"> Et <expan abbr="c&ugrave;m">cum</expan> idem pror&longs;us &longs;it, &longs;iue axiculus circumuertatur, &longs;iue mi&shy;<lb/>
nus; liceat propterea in iis, qu&aelig; dicenda &longs;unt, loco axiculi cen&shy;<lb/>
trum <expan abbr="tant&ugrave;m">tantum</expan> accipere.  </s>
    </p>       
<p id="id.2.1.141.9.0.0.0" type="head">        
<s id="id.2.1.141.9.1.1.0"> PROPOSITIO II. </s>
    </p>       
<p id="id.2.1.141.10.0.0.0" type="main">        
<s id="id.2.1.141.10.1.1.0"> Si funis orbiculo trochle&aelig; ponderi alligat&aelig; <lb/>
circumducatur, altero eius extremo alicubi reli&shy;<lb/>
gato, altero <expan abbr="uer&ograve;">uero</expan> <expan abbr="&agrave;">a</expan> potentia pondus &longs;u&longs;tinente <lb/>
apprehen&longs;o; erit potentia ponderis &longs;ubdupla.  </s>
    </p>       
<pb n="65"/>
       
<p id="id.2.1.141.12.0.0.0" type="main">        
<s id="id.2.1.141.12.1.1.0"> Si pondus A; &longs;it BCD <lb/>
orbiculus trochle&aelig; pon&shy;<lb/>
deri A alligate, cuius cen<lb/>
trum E; funis deinde FB <lb/>
CDG circa orbiculum <lb/>
voluatur, qui religetur in <lb/>
F; &longs;itq; potentia in G &longs;u<lb/>
&longs;tinens pondus A.  </s>
     
<s id="id.2.1.141.12.1.1.0.a"> dico <lb/>
potentiam in G &longs;ubdu&shy;<lb/>
plam e&longs;&longs;e ponderis A.  </s>
     
<s id="id.2.1.141.12.1.1.0.b"> &longs;int <lb/>
funes FB GD puncti E <lb/>
horizonti perpendicula&shy;<lb/>
res, qui inter &longs;e &longs;e &aelig;qui&shy;<lb/>
di&longs;tantes  <arrow.to.target n="note222"></arrow.to.target> erunt; tangantq; <lb/>
funes FB GD circulum <lb/>
BCD in BD punctis.  </s>
             
<s id="id.2.1.141.12.1.2.0"> <lb/>
connectatur BD; erit BD <lb/>
per centrum E ducta, <arrow.to.target n="note223"></arrow.to.target><lb/>
<arrow.to.target n="fig130"></arrow.to.target><expan abbr="ip&longs;iu&longs;qu&eacute;"><lb/>
ip&longs;iu&longs;que</expan> centri horizonti &aelig;quidi&longs;tans.  </s>
             
<s id="id.2.1.141.12.1.3.0"> <expan abbr="C&ugrave;m">Cum</expan> autem <expan abbr="pot&eacute;n&shy;tia">poten&shy;<lb/>
tia</expan> in G trochlea pondus A &longs;u&longs;tinere debeat, funem ex altero ex&shy;<lb/>
tremo religatum e&longs;&longs;e oportet, puta in F; ita vt F &aelig;qualiter &longs;altem <lb/>
potenti&aelig; in G re&longs;i&longs;tat, alioquin potentia in G nullatenus pondus <lb/>
&longs;u&longs;tinere po&longs;&longs;et.  </s>
             
<s id="id.2.1.141.12.1.4.0"> Et quoniam potentia fune &longs;u&longs;tinet orbiculum, <lb/>
qui reliquam trochle&aelig; partem, cui appen&longs;um e&longs;t pondus, &longs;u&longs;tinet <lb/>
axiculo; grauitabit h&aelig;c trochle&aelig; pars in axiculo, hoc e&longs;t in centro <lb/>
E. quare pondus A in eodem quoq; centro E ponderabit, ac &longs;i <lb/>
in E e&longs;&longs;et appen&longs;um.  </s>
             
<s id="id.2.1.141.12.1.5.0"> po&longs;ita igitur potentia, qu&aelig; in G, vbi D <lb/>
(idem enim pror&longs;us e&longs;t) erit BD tanquam vectis, cuius fulci<lb/>
mentum erit B, pondus in E appen&longs;um, &amp; potentia in D. con<lb/>
uenienter enim fulcimenti rationem ip&longs;um B &longs;ubire pote&longs;t, exi<lb/>
&longs;tente fune FB immobili.  </s>
             
<s id="id.2.1.141.12.1.6.0"> c&aelig;terum hoc po&longs;terius magis eluce&longs;cet.  </s>
             
<s id="id.2.1.141.12.1.7.0"> <lb/>
Quoniam autem potentia ad pondus eandem habet proportio&shy;<lb/>
nem,  <arrow.to.target n="note224"></arrow.to.target> <expan abbr="qu&agrave;m">quam</expan> BE ad BD; &amp; BE in &longs;ubdupla e&longs;t proportione <lb/>
ad BD: potentia igitur in G ponderis A &longs;ubdupla erit. quod de&shy;<lb/>
mon&longs;trare oportebat. </s>
             
<s id="id.2.1.141.12.1.8.0"> [quod de&shy;<lb/>
mon&longs;trare oportebat.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig130" place="text">       </figure>       
<p id="id.2.1.141.12.2.1.0" type="caption">        
<s id="id.2.1.141.12.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.142.1.0.0.0" type="margin">        
<s id="id.2.1.142.1.1.1.0"> <margin.target id="note222"></margin.target>6 <emph type="italics"/>Vndecimi<emph.end type="italics"/> </s>
             
<s id="id.2.1.142.1.1.2.0"> <margin.target id="note223"></margin.target><emph type="italics"/>Ex pr&aelig;cedenti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.142.1.1.3.0"> <margin.target id="note224"></margin.target>2 <emph type="italics"/>Huius de vecte.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.143.1.0.0.0" type="main">        
<pb/>
       
<s id="id.2.1.143.1.2.1.0"> Hoc igitur ita &longs;e ha&shy;<lb/>
bet vnico exi&longs;tent e fune <lb/>
FBC DG ip&longs;i orbi culo <lb/>
circumducto, ac &longs;i duo e&longs;<lb/>
&longs;ent funes BF GD ve&shy;<lb/>
cti BD alligati, cuius ful<lb/>
cimentum erit B, pon&shy;<lb/>
dus in E appen&longs;um, &amp; <lb/>
potentia &longs;u&longs;tinens in D, <lb/>
vel quod idem e&longs;t in G.  </s>
     <lb/>
       

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure place="text">       </figure>       
<p id="id.2.1.143.1.3.1.0" type="caption">        
<s id="id.2.1.143.1.3.1.0.capt"> YYY </s>
     <lb/>
       
<s id="id.2.1.143.1.5.1.0"> COROLLARIVM I. </s>
    </p>       
<p id="id.2.1.143.2.0.0.0" type="main">        
<s id="id.2.1.143.2.1.1.0"> Ex hoc itaq; manife&longs;tum e&longs;t, pondus hoc mo <lb/>
do <expan abbr="&agrave;">a</expan> minori in &longs;ubdupla proportione potentia <lb/>
&longs;u&longs;tineri, quam &longs;ine vllo huiu&longs;modi trochle&aelig; <lb/>
auxilio.  </s>
    </p>       
<pb n="66"/>
       
<p id="id.2.1.143.4.0.0.0" type="main">        
<s id="id.2.1.143.4.1.1.0"> Veluti &longs;it pondus H ponderi A <lb/>
&aelig;quale, cui religatus &longs;it funis kL; <lb/>
potentiaq; in L &longs;u&longs;tineat pondus H; <lb/>
erit potentia in L &longs;eor&longs;um ponderi <lb/>
H, &amp; ponderi A &aelig;qualis; &longs;ed poten<lb/>
tia in G &longs;ubdupla e&longs;t ponderis A, <lb/>
quare potentia in G &longs;ubdupla erit po<lb/>
tenti&aelig;, qu&aelig; e&longs;t in L. &amp; hoc modo in <lb/>
huiu&longs;cemodi reliquis omnibus pro <lb/>
portio inueniri poterit.  </s>
     <lb/>
       

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure place="text">       </figure>       
<p id="id.2.1.143.4.2.1.0" type="caption">        
<s id="id.2.1.143.4.2.1.0.capt"> YYY </s>
     <lb/>
       
<s id="id.2.1.143.4.4.1.0"> COROLLARIVM. II. </s>
    </p>       
<p id="id.2.1.143.5.0.0.0" type="main">        
<s id="id.2.1.143.5.1.1.0"> Manife&longs;tum e&longs;t etiam; &longs;i du&aelig; fuerint poten&shy;<lb/>
ti&aelig; vna in G, altera in F, pondus A &longs;u&longs;tinentes; <lb/>
vtra&longs;q; &longs;imul ponderi A &aelig;quales e&longs;&longs;e: &amp; vnam <lb/>
quamque &longs;u&longs;tinere dimidium ponderis A. </s>
    </p>       
<p id="id.2.1.143.6.0.0.0" type="main">        
<s id="id.2.1.143.6.1.1.0"> Hoc autem ex tertio, &amp; quarto corollario &longs;ecund&aelig; huius in <lb/>
tractatu de vecte patet.  </s>
    </p>       
<p id="id.2.1.143.7.0.0.0" type="head">        
<s id="id.2.1.143.7.1.1.0"> COROLLARIVM III. </s>
    </p>       
<p id="id.2.1.143.8.0.0.0" type="main">        
<s id="id.2.1.143.8.1.1.0"> Illud quoq; pr&aelig;terea innote&longs;cit, cur &longs;cilicet fu<lb/>
nis ex altero religatus e&longs;&longs;e debeat extremo.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.143.10.0.0.0" type="head">        
<s id="id.2.1.143.10.1.1.0"> PROPOSITIO III. </s>
    </p>       
<p id="id.2.1.143.11.0.0.0" type="main">        
<s id="id.2.1.143.11.1.1.0"> Si vtri&longs;q; duarum trochlearum &longs;ingulis or&shy;<lb/>
biculis, quarum altera <expan abbr="&longs;upern&egrave;">&longs;uperne</expan>, altera <expan abbr="ver&ograve;">vero</expan> <expan abbr="in&shy;fern&egrave;">in&shy;<lb/>
ferne</expan> con&longs;tituta, ponderiq; alligata fuerit, cir<lb/>
cunducatur funis; altero eius extremo alicubi <lb/>
religato, altero <expan abbr="ver&ograve;">vero</expan> <expan abbr="&agrave;">a</expan> potentia pondus &longs;u&longs;ti&shy;<lb/>
nente detento; erit potentia ponderis &longs;ub du&shy;<lb/>
pla. </s>
    </p>       
<p id="id.2.1.143.12.0.0.0" type="main">        
<s id="id.2.1.143.12.1.1.0"> Sit pondus A; &longs;it BCD orbiculus trochle&aelig; pon<lb/>
deri A alligat&aelig;, cuius centrum K; EFG <expan abbr="ver&ograve;">vero</expan> <lb/>
&longs;it trochle&aelig; &longs;ur&longs;um appen&longs;&aelig;, cuius centrum H. <lb/>
deinde LBC DME FGN funis circa orbicu&shy;<lb/>
los ducatur, qui religetur in L; &longs;itq; potentia in <lb/>
N &longs;u&longs;tinens pondus A.  </s>
     
<s id="id.2.1.143.12.1.1.0.a"> dico potentiam in N <lb/>
&longs;ubduplam e&longs;&longs;e ponderis A. &longs;i enim potentia &longs;u<lb/>
&longs;tinens pondus A vbi M collocata foret, e&longs;&longs;et <lb/>
vtiq; potentia in M &longs;ubdupla ponderis A. po&shy;<lb/>
<arrow.to.target n="note225"></arrow.to.target> tenti&aelig; <expan abbr="ver&ograve;">vero</expan> in M &aelig;qualis e&longs;t vis in N. e&longs;t e&shy;<lb/>
<arrow.to.target n="note226"></arrow.to.target> nim ac &longs;i potentia in M dimidium ponderis <lb/>
A &longs;ine trochlea &longs;u&longs;tineret, cui &aelig;queponderat <lb/>
pondus in N ponderis A dimidio &aelig;quale.  </s>
             
<s id="id.2.1.143.12.1.2.0"> <lb/>
quare vis in N &aelig;qualis dimidio ponderis A <lb/>
ip&longs;um A &longs;u&longs;tinebit.  </s>
             
<s id="id.2.1.143.12.1.3.0"> Potentia igitur in N &longs;u&longs;ti<lb/>
nens pondus A &longs;ubdupla e&longs;t ip&longs;ius A. quod <lb/>
demon&longs;trare oportebat. <arrow.to.target n="fig131"></arrow.to.target> </s>
    </p>       
<pb n="67"/>
       
<p id="id.2.1.143.14.0.0.0" type="main">        
<s id="id.2.1.143.14.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> vt in &longs;ecunda figura &longs;it fu<lb/>
nis BC DEF GHkL orbiculis cir<lb/>
cum uolutus, &amp; religatus in B; poten<lb/>
tiaq; in L pondus A &longs;u&longs;tineat: erit <lb/>
potentia in L &longs;imiliter ponderis &longs;ubdu<lb/>
pla.  </s>
             
<s id="id.2.1.143.14.1.2.0"> orbiculus enim trochle&aelig; &longs;upe&shy;<lb/>
rioris, <expan abbr="ip&longs;aqu&eacute;">ip&longs;aque</expan> trochlea penitus &longs;unt <lb/>
inutiles: &amp; idem e&longs;t, ac &longs;i funis reli<lb/>
gatus e&longs;&longs;et in F, &amp; potentia in L &longs;u<lb/>
&longs;tineret pondus &longs;ola trochlea ponderi <lb/>
alligata, qu&aelig; potentia ponderis A o&longs;ten<lb/>
&longs;a e&longs;t &longs;ubdupla.  </s>
     <lb/>
       

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig131" place="text">       </figure>       
<p id="id.2.1.143.14.2.1.0" type="caption">        
<s id="id.2.1.143.14.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.144.1.0.0.0" type="margin">        
<s id="id.2.1.144.1.1.1.0"> <margin.target id="note225"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.144.1.1.2.0"> <margin.target id="note226"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.145.1.0.0.0" type="main">        

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure place="text">       </figure>       
<p id="id.2.1.145.1.1.1.0" type="caption">        
<s id="id.2.1.145.1.1.1.0.capt"> YYY </s>
     <lb/>
       
<s id="id.2.1.145.1.3.1.0"> COROLLARIVM. </s>
    </p>       
<p id="id.2.1.145.2.0.0.0" type="main">        
<s id="id.2.1.145.2.1.1.0"> Ex his &longs;equitur, &longs;i du&aelig; &longs;int potenti&aelig; in BL; <lb/>
vtra&longs;q; inter &longs;e &longs;e &aelig;quales e&longs;&longs;e. </s>
    </p>       
<p id="id.2.1.145.3.0.0.0" type="main">        
<s id="id.2.1.145.3.1.1.0"> Vtraq; enim &longs;eor&longs;um e&longs;t ip&longs;ius A &longs;ubdupla.  </s>
    </p>       
<p id="id.2.1.145.4.0.0.0" type="head">        
<pb/>
       
<s id="id.2.1.145.5.1.1.0"> PROPOSITIO IIII. </s>
    </p>       
<p id="id.2.1.145.6.0.0.0" type="main">        
<s id="id.2.1.145.6.1.1.0"> Sit vectis AB, cuius fulcimentum &longs;it A; qui <lb/>
bifariam diuidatur in D: &longs;itq; pondus C in D <lb/>
appen&longs;um; du&aelig;q; &longs;int potenti&aelig; &aelig;quales in BD <lb/>
pondus C &longs;u&longs;tinentes.  </s>
             
<s id="id.2.1.145.6.1.2.0"> Dico unamquamq; poten<lb/>
tiam in BD ponderis C &longs;ubtriplam e&longs;&longs;e. </s>
    </p>       
<p id="id.2.1.145.7.0.0.0" type="main">        
<s id="id.2.1.145.7.1.1.0"> Quoniam enim altera <lb/>
potentia e&longs;t in D colloca<lb/>
ta, &amp; pondus C in eodem <lb/>
puncto D e&longs;t appen&longs;um; <lb/>
potentia in D partem <lb/>
ponderis C &longs;u&longs;t^{i}nebit ip&shy;<lb/>
&longs;i potenti&aelig; D &aelig;qualem.  </s>
             
<s id="id.2.1.145.7.1.2.0"> <lb/>
<arrow.to.target n="fig132"></arrow.to.target><lb/>
quare potentia in B partem &longs;u&longs;tinebit reliquam, qu&aelig; pars dupla erit <lb/>
ip&longs;ius potenti&aelig; in B; <expan abbr="c&ugrave;m">cum</expan> pondus ad potentiam eandem habeat <lb/>
proportionem, quam AB ad AD: &amp; potenti&aelig; in BD &longs;unt &aelig;qua&shy;<lb/>
les; ergo potentia in B duplam &longs;u&longs;tinebit partem eius, quam &longs;u&longs;ti<lb/>
net potentia in D.  </s>
     
<s id="id.2.1.145.7.1.2.0.a"> diuidatur ergo pondus C in duas partes, qua <lb/>
rum vna &longs;it reliqu&aelig; dupla; quod fiet, &longs;i in tres partes &aelig;quales EFG <lb/>
diui&longs;erimus: tunc enim FG dupla erit ip&longs;ius E.  </s>
     
<s id="id.2.1.145.7.1.2.0.b"> Itaq; potentia <lb/>
in D partem E &longs;u&longs;tinebit, &amp; potentiam in B reliquas FG. vtreq; <lb/>
igitur inter &longs;e &longs;e &aelig;quales potenti&aelig; in BD &longs;imul totum &longs;u&longs;tinebunt <lb/>
pondus C.  </s>
     
<s id="id.2.1.145.7.1.2.0.c"> &amp; quoniam potentia in D partem E &longs;u&longs;tinet, qu&aelig; ter<lb/>
tia e&longs;t pars ponderis C, ip&longs;iq; e&longs;t &aelig;qualis; erit potentia in D &longs;ub <lb/>
tripla ponderis C. &amp; <expan abbr="c&ugrave;m">cum</expan> potentia in B &longs;u&longs;tineat partes FG, qua <lb/>
rum potentia in B e&longs;t &longs;ubdupla; erit in B potentia vni partium FG, <lb/>
<expan abbr="put&agrave;">puta</expan> G &aelig;qualis.  </s>
             
<s id="id.2.1.145.7.1.3.0"> G <expan abbr="ver&ograve;">vero</expan> tertia e&longs;t pars ponderis C; potentia <lb/>
igitur in B &longs;ubtripla erit ponderis C.  </s>
     
<s id="id.2.1.145.7.1.3.0.a"> Vnaqu&aelig;q; ergo potentia in <lb/>
BD &longs;ubtripla e&longs;t ponderis C. quod demon&longs;trare oportebat. 
<pb n="68"/>
<arrow.to.target n="fig133"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.145.8.0.0.0" type="main">        
<s id="id.2.1.145.8.1.1.0"> Et &longs;i duo e&longs;&longs;ent vectes AB EF bifariam in GD diui&longs;i, quorum <lb/>
fulcimenta e&longs;&longs;ent AF, &amp; pondus C in DG vtriq; vecti appen&shy;<lb/>
&longs;um, ita tamen vt in vtroq; &aelig;qualiter ponderet; du&aelig;q; e&longs;&longs;ent <lb/>
&aelig;quales potenti&aelig; in BG: eadem pror&longs;us ratione o&longs;tendetur, <lb/>
vnamquamq; potentiam in B, &amp; G ponderis C &longs;ubtriplam <lb/>
e&longs;&longs;e.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig132" place="text">       </figure>       
<p id="id.2.1.145.8.2.1.0" type="caption">        
<s id="id.2.1.145.8.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig133" place="text">       </figure>       
<p id="id.2.1.145.8.2.3.0" type="caption">        
<s id="id.2.1.145.8.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.145.9.0.0.0" type="head">        
<s id="id.2.1.145.9.1.1.0"> PROPOSITIO V. </s>
    </p>       
<p id="id.2.1.145.10.0.0.0" type="main">        
<s id="id.2.1.145.10.1.1.0"> Si vtri&longs;q; duarum <expan abbr="trochlear&utilde;">trochlearum</expan> &longs;ingulis orbiculis, <lb/>
quarum altera <expan abbr="&longs;upern&egrave;">&longs;uperne</expan>, altera <expan abbr="ver&ograve;">vero</expan> <expan abbr="infern&egrave;">inferne</expan> con&longs;ti<lb/>
tuta, ponderiq; alligata fuerit, circumducatur fu<lb/>
nis; altero eius extremo inferiori trochle&aelig; reli&shy;<lb/>
gato, altero <expan abbr="ver&ograve;">vero</expan> <expan abbr="&agrave;">a</expan> potentia pondus &longs;u&longs;tinente <lb/>
detento: erit potentia ponderis &longs;ubtripla.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.145.12.0.0.0" type="main">        
<s id="id.2.1.145.12.1.1.0"> Sit pondus A; &longs;it BCD orbiculus tro&shy;<lb/>
chle&aelig; ponderi A alligate, cuius centrum <lb/>
E; &amp; FGH trochle&aelig; &longs;ur&longs;um appen&longs;&aelig;, cu&shy;<lb/>
ius centrum k; &amp; LFGHBCDM funis <lb/>
orbiculis circumducatur, qui religetur in L <lb/>
trochle&aelig; inferiori; &longs;itq; potentia in M &longs;u&shy;<lb/>
&longs;tinens pondus A.  </s>
     
<s id="id.2.1.145.12.1.1.0.a"> dico potentiam in M <lb/>
&longs;ubtriplam e&longs;&longs;e ponderis A.  </s>
     
<s id="id.2.1.145.12.1.1.0.b"> ducantur FH <lb/>
BD per centra kE horizonti &aelig;quidi&longs;tan&shy;<lb/>
tes, &longs;icut in pr&aelig;cedentibus dictum e&longs;t Quo&shy;<lb/>
niam enim funis FL trochleam &longs;u&longs;tinet in&shy;<lb/>
feriorem, qu&aelig; &longs;u&longs;tinet orbiculum in eius <lb/>
centro E; erit funis in L vt potentia &longs;u&longs;ti&shy;<lb/>
nens orbiculum, ac &longs;i in ip&longs;o E centro e&longs;&longs;et; <lb/>
potentia <expan abbr="ver&ograve;">vero</expan> in M e&longs;t, ac &longs;i e&longs;&longs;et in D; <lb/>
efficietur igitur DB tanquam vectis, cuius <lb/>
<arrow.to.target n="note227"></arrow.to.target> fulcimentum erit B; pondus <expan abbr="ver&ograve;">vero</expan> A (vt &longs;u<lb/>
pra o&longs;ten&longs;um e&longs;t) ex E &longs;u&longs;pen&longs;um <expan abbr="&agrave;">a</expan> dua&shy;<lb/>
bus potentiis altera in D, altera in E &longs;u&longs;ten<lb/>
tatum.  </s>
             
<s id="id.2.1.145.12.1.2.0"> <expan abbr="C&ugrave;m">Cum</expan> autem in pondere &longs;u&longs;tinendo <lb/>
vectes FH BD immobiles maneant, &longs;i in <lb/>
funibus FL HB appendantur pondera, e&shy;<lb/>
<arrow.to.target n="note228"></arrow.to.target> runt h&aelig;c ip&longs;a &aelig;qualia; <expan abbr="c&ugrave;m">cum</expan> vectis FH ha&shy;<lb/>
beat fulcimentum in medio; alioquin ex al<lb/>
tera parte deor&longs;um fieret motus, quod <expan abbr="tam&etilde;">tamen</expan> <lb/>
non contingit.  </s>
             
<s id="id.2.1.145.12.1.3.0"> tam igitur &longs;u&longs;tinet funis FL, <lb/>
<expan abbr="qu&agrave;m">quam</expan> HB. deinde quoniam ex medio ve&shy;<lb/>
<arrow.to.target n="fig134"></arrow.to.target><lb/>
cte BD pondus &longs;u&longs;penditur, idcirco &longs;i du&aelig; fuerint potenti&aelig; in BD <lb/>
<arrow.to.target n="note229"></arrow.to.target> pondus &longs;u&longs;tinentes, erunt inuicem &aelig;quales.  </s>
             
<s id="id.2.1.145.12.1.4.0"> &amp; quamquam funis 
<pb n="69"/>
FL ip&longs;e quoq; pondus &longs;u&longs;tineat, <expan abbr="c&ugrave;m">cum</expan> potenti&aelig; in E <expan abbr="vic&etilde;">vicem</expan> gerat; quia <lb/>
tamen ex eodemmet puncto &longs;u&longs;tinet, vbi appen&longs;um e&longs;t pondus, non <lb/>
efficiet propterea, quin potenti&aelig; in BD &longs;int inter &longs;e &longs;e &aelig;quales; <lb/>
opitulatur enim <expan abbr="t&agrave;m">tam</expan> vni, <expan abbr="qu&agrave;m">quam</expan> alteri.  </s>
             
<s id="id.2.1.145.12.1.5.0"> potenti&aelig; <expan abbr="ver&ograve;">vero</expan> in BD e&aelig;&shy;<lb/>
dem &longs;unt, ac &longs;i e&longs;&longs;ent in HM; quare <expan abbr="t&agrave;m">tam</expan> &longs;u&longs;tinebit funis MD, <lb/>
<expan abbr="qu&agrave;m">quam</expan> HB.  </s>
     
<s id="id.2.1.145.12.1.5.0.a"> ita <expan abbr="ver&ograve;">vero</expan> &longs;u&longs;tinet HB, atq; FL; funis igitur MD ita <lb/>
&longs;u&longs;tinebit, &longs;icut FL, hoc e&longs;t, ac &longs;i in D, &amp; L appen&longs;a e&longs;&longs;ent pon&shy;<lb/>
dera &aelig;qualia.  </s>
             
<s id="id.2.1.145.12.1.6.0"> <expan abbr="C&ugrave;m">Cum</expan> itaq; &aelig;qualia pondera <expan abbr="&agrave;">a</expan> potentiis &longs;u&longs;tinean&shy;<lb/>
tur &aelig;qualibus, potenti&aelig; in ML &aelig;quales erunt; quarum eadem pror<lb/>
&longs;us e&longs;t ratio, ac &longs;i e&longs;&longs;ent amb&aelig; in DE.  </s>
     
<s id="id.2.1.145.12.1.6.0.a"> Itaq; <expan abbr="c&ugrave;m">cum</expan> pondus A in <lb/>
medio vectis BD &longs;it appen&longs;um, du&aelig;q; potenti&aelig; &longs;int &aelig;quales in <lb/>
DE pondus &longs;u&longs;tinentes; erit B fulcimentum, ac vn aqu&aelig;q; potentia, <arrow.to.target n="note230"></arrow.to.target><lb/>
&longs;iue in DE, &longs;iue in ML &longs;ubtripla ponderis A. ergo potentia in M <lb/>
&longs;u&longs;tinens pondus &longs;ubtripla erit ponderis A. quod o&longs;tendere o&shy;<lb/>
portebat.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig134" place="text">       </figure>       
<p id="id.2.1.145.12.2.1.0" type="caption">        
<s id="id.2.1.145.12.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.146.1.0.0.0" type="margin">        
<s id="id.2.1.146.1.1.1.0"> <margin.target id="note227"></margin.target><emph type="italics"/>In<emph.end type="italics"/> 2 <emph type="italics"/>Huius<emph.end type="italics"/> </s>
             
<s id="id.2.1.146.1.1.2.0"> <margin.target id="note228"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.146.1.1.3.0"> <margin.target id="note229"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 3 <emph type="italics"/>Cor.<emph.end type="italics"/> 2 <emph type="italics"/>Huius vecte.<emph.end type="italics"/> </s>
             
<s id="id.2.1.146.1.1.4.0"> <margin.target id="note230"></margin.target>4 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.147.1.0.0.0" type="head">        
<s id="id.2.1.147.1.1.1.0"> COROLLARIVM. </s>
    </p>       
<p id="id.2.1.147.2.0.0.0" type="main">        
<s id="id.2.1.147.2.1.1.0"> Ex hoc manife&longs;tum e&longs;t, vnumquemq; funem <lb/>
MD FL HB tertiam &longs;u&longs;tinere partem pon&shy;<lb/>
deris A. 
<pb/>
 </s>
    </p>       
<p id="id.2.1.147.3.0.0.0" type="main">        
<s id="id.2.1.147.3.1.1.0"> Pr&aelig;terea, &longs;i funis ex M per a&shy;<lb/>
lium adhuc deferatur orbiculum &longs;u<lb/>
periorem in trochlea &longs;ur&longs;um &longs;imi&shy;<lb/>
liter appen&longs;a con&longs;titutum, cuius <lb/>
centrum N; ita vt perueniat in O; <lb/>
ibiq; <expan abbr="&agrave;">a</expan> potentia detineatur; erit po<lb/>
tentia in O &longs;u&longs;tinens pondus A iti <lb/>
dem &longs;ubtripla ip&longs;ius ponderis.  </s>
             
<s id="id.2.1.147.3.1.2.0"> fu<lb/>
nis enim MD <expan abbr="tant&ugrave;m">tantum</expan> ponderis &longs;u<lb/>
&longs;tinet, ac &longs;i in D appen&longs;um e&longs;&longs;et <lb/>
pondus &aelig;quale terti&aelig; parti ponde<lb/>
<arrow.to.target n="note231"></arrow.to.target> ris A, cui &aelig;quiualet potentia in <lb/>
O ip&longs;i &aelig;qualis, hoc e&longs;t &longs;ubtripla <lb/>
ponderis A.  </s>
     
<s id="id.2.1.147.3.1.2.0.a"> Potentia igitur in O <lb/>
&longs;ubtripla e&longs;t ponderis A. <lb/>
<arrow.to.target n="fig135"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.147.4.0.0.0" type="main">        
<s id="id.2.1.147.4.1.1.0"> Et ne idem &longs;&aelig;pius repetatur, no<lb/>
ui&longs;&longs;e oportet potentiam in O &longs;em<lb/>
per &aelig;qualem e&longs;&longs;e ei, qu&aelig; e&longs;t in M; <lb/>
hoc e&longs;t &longs;i potentia in M e&longs;&longs;et &longs;ub <lb/>
quadrupla, &longs;ubquintupla, vel huiu&longs; <lb/>
modi aliter ip&longs;ius ponderis; poten<lb/>
tia quoq; in O erit itidem &longs;ubqua<lb/>
drupla, &longs;ubquintupla, atq; ita dein<lb/>
ceps eiu&longs;demmet ponderis, quem <lb/>
madmodum &longs;e habet potentia <lb/>
in M. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig135" place="text">       </figure>       
<p id="id.2.1.147.4.2.1.0" type="caption">        
<s id="id.2.1.147.4.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.148.1.0.0.0" type="margin">        
<s id="id.2.1.148.1.1.1.0"> <margin.target id="note231"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.149.1.0.0.0" type="head">        
<pb n="70"/>
       
<s id="id.2.1.149.1.2.1.0"> PROPOSITIO VI. </s>
    </p>       
<p id="id.2.1.149.2.0.0.0" type="main">        
<s id="id.2.1.149.2.1.1.0"> Sint duo vectes AB CD bifariam diui&longs;i in <lb/>
EF, quorum fulcimenta &longs;int.  </s>
             
<s id="id.2.1.149.2.1.2.0"> in BD; &longs;itq; pon<lb/>
dus G in EF vtriq; vecti appen&longs;um, ita ut ex <lb/>
vtroq; &aelig;qualiter ponderet; du&aelig;q; &longs;int potenti&aelig; <lb/>
in AC &aelig;quales pondus &longs;u&longs;tinentes.  </s>
             
<s id="id.2.1.149.2.1.3.0"> Dico unam <lb/>
quamq; potentiam in AC &longs;ubquadruplam e&longs;&shy;<lb/>
&longs;e ponderis G. </s>
    </p>       
<p id="id.2.1.149.3.0.0.0" type="main">        
<s id="id.2.1.149.3.1.1.0"> <expan abbr="C&ugrave;m">Cum</expan> enim potenti&aelig; in <lb/>
AC totum &longs;u&longs;tineant pon&shy;<lb/>
dus G, potentiaq; in A ad <lb/>
partem ponderis, quod &longs;u&longs;ti<lb/>
net, &longs;it vt BE ad BA; po&shy;<lb/>
tentia  <arrow.to.target n="note232"></arrow.to.target> <expan abbr="ver&ograve;">vero</expan> in C ad partem <lb/>
ip&longs;ius G, quod &longs;u&longs;tinet, ita <lb/>
&longs;it vt DF ad DC; &amp; vt BE <lb/>
ad BA, ita e&longs;t DF ad DC; <lb/>
<arrow.to.target n="fig136"></arrow.to.target><lb/>
erit potentia in A ad partem ponderis, quod &longs;u&longs;tinet, vt poten&shy;<lb/>
tia in C ad ip&longs;ius ponderis, quod &longs;u&longs;tinet, partem; &amp; potenti&aelig; <lb/>
in AC &longs;unt &aelig;quales; &aelig;quales igitur erunt partes ponderis G, <lb/>
qu&aelig; <expan abbr="&agrave;">a</expan> potentiis &longs;u&longs;tinentur.  </s>
             
<s id="id.2.1.149.3.1.2.0"> quare vnaqu&aelig;q; potentia in A C di&shy;<lb/>
midium &longs;u&longs;tinebit ponderis G.  </s>
     
<s id="id.2.1.149.3.1.2.0.a"> Potentia <expan abbr="ver&ograve;">vero</expan> in A &longs;ubdupla e&longs;t pon<lb/>
deris, quod &longs;u&longs;tinet: ergo potentia in A dimidio dimidii, hoc <lb/>
e&longs;t quart&aelig; portioni ponderis G &aelig;qualis erit; ideoq; &longs;ubquadrupla <lb/>
erit ponderis G.  </s>
     
<s id="id.2.1.149.3.1.2.0.b"> neq; aliter demon&longs;trabitur potentiam in C &longs;ub-quadruplam <lb/>
e&longs;&longs;e eiu&longs;dem ponderis G. quod demon&longs;trare opor&shy;<lb/>
tebat.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig136" place="text">       </figure>       
<p id="id.2.1.149.3.2.1.0" type="caption">        
<s id="id.2.1.149.3.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.150.1.0.0.0" type="margin">        
<s id="id.2.1.150.1.1.1.0"> <margin.target id="note232"></margin.target>2 <emph type="italics"/>Huius. de vecte.<emph.end type="italics"/> </s>
             
<s id="id.2.1.150.1.1.2.0"> [de vecte.<emph.end type="italics"/>] </s>
    </p>       
<p id="id.2.1.151.1.0.0.0" type="main">        
<pb/>
       
<s id="id.2.1.151.1.2.1.0"> Si <expan abbr="ver&ograve;">vero</expan> tres &longs;int vectes <lb/>
AB CD EF bifariam di&shy;<lb/>
ui&longs;i in GHk, quorum fulci <lb/>
menta &longs;int BDF; &amp; pondus <lb/>
L eodem modo in GHK <lb/>
appen&longs;um; &longs;intq; tres poten<lb/>
ti&aelig; in ACE &aelig;quales pondus <lb/>
&longs;u&longs;tinentes; &longs;imiliter o&longs;ten<lb/>
detur vnamquamque po&shy;<lb/>
tentiam &longs;ub&longs;excuplam e&longs;&longs;e <lb/>
ponderis L. atq; hoc ordi<lb/>
ne &longs;i quatuor e&longs;&longs;ent vectes, <lb/>
&amp; quatuor potenti&aelig;; erit vnaqu&aelig;q; potentia &longs;uboctupla ponderis. <lb/>
atq; ita deinceps in infinitum. </s>
             
<s id="id.2.1.151.1.2.2.0"> [<lb/>
atq; ita deinceps in infinitum.]  </s>
     <lb/>
       

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure place="text">       </figure>       
<p id="id.2.1.151.1.3.1.0" type="caption">        
<s id="id.2.1.151.1.3.1.0.capt"> YYY </s>
     <lb/>
       
<s id="id.2.1.151.1.5.1.0"> PROPOSITIO VII. </s>
    </p>       
<p id="id.2.1.151.2.0.0.0" type="main">        
<s id="id.2.1.151.2.1.1.0"> Si tribus duarum trochlearum orbiculis, <expan abbr="quar&utilde;">quarum</expan> <lb/>
altera <expan abbr="&longs;upern&egrave;">&longs;uperne</expan> vnico duntaxat, altera <expan abbr="ver&ograve;">vero</expan> <expan abbr="infer&shy;n&egrave;">infer&shy;<lb/>
ne</expan> duobus autem in&longs;ignita orbiculis, ponderiq; <lb/>
alligata con&longs;tituta fuerit, funis circumponatur; al<lb/>
tero eius extremo alicubi religato, altero <expan abbr="ver&ograve;">vero</expan> <expan abbr="&agrave;">a</expan> <lb/>
potentia pondus &longs;u&longs;tinente retento; erit potentia <lb/>
ponderis &longs;ubquadrupla.  </s>
    </p>       
<pb n="71"/>
       
<p id="id.2.1.151.4.0.0.0" type="main">        
<s id="id.2.1.151.4.1.1.0"> Sit pondus A; &longs;int tres orbiculi, quorum <lb/>
centra BCD; orbiculu&longs;q;, cuius centrum D, <lb/>
&longs;it trochle&aelig; &longs;ur&longs;um appen&longs;&aelig;; quorum <expan abbr="ver&ograve;">vero</expan> <lb/>
&longs;unt centra BC, &longs;int trochle&aelig; ponderi A alli<lb/>
gat&aelig;; funi&longs;q; EFGHkLNOP per omnes <lb/>
circumducatur orbiculos, qui religetur in E; <lb/>
&longs;itq; vis in P &longs;u&longs;tinens pondus A.  </s>
     
<s id="id.2.1.151.4.1.1.0.a"> dico po<lb/>
tentiam in P &longs;ubquadruplam e&longs;&longs;e ponderis <lb/>
A.  </s>
     
<s id="id.2.1.151.4.1.1.0.b"> ducantur kL GF ON per rotularum <lb/>
centra, &amp; horizonti &aelig;quidi&longs;tantes, qu&aelig; (ex <lb/>
iis, qu&aelig; dicta &longs;unt) tanquam vectes erunt.  </s>
             
<s id="id.2.1.151.4.1.2.0"> <lb/>
&amp; quoniam propter vectem, &longs;iue libram kL, <lb/>
cuius fulcimentum, &longs;iue centrum e&longs;t in me <lb/>
dio, <expan abbr="t&agrave;m">tam</expan> &longs;u&longs;tinet funis kG, <expan abbr="qu&agrave;m">quam</expan> LN, <expan abbr="c&ugrave;m">cum</expan> <arrow.to.target n="note233"></arrow.to.target><lb/>
in neutram partem fiat motus.  </s>
             
<s id="id.2.1.151.4.1.3.0"> nec non <lb/>
propter vectem GF, <expan abbr="&egrave;">e</expan> cuius medio veluti &longs;u<lb/>
&longs;pen&longs;um dependet onus; &longs;i du&aelig; e&longs;&longs;ent in GF <lb/>
potenti&aelig;, &longs;eu in HE (e&longs;t enim par vtriu&longs;q; <lb/>
&longs;itus ratio, vt iam &longs;epius dictum e&longs;t) e&longs;&longs;ent <arrow.to.target n="note234"></arrow.to.target><lb/>
vtiq; huiu&longs;modi potenti&aelig; inuicem &aelig;quales.  </s>
             
<s id="id.2.1.151.4.1.4.0"> <lb/>
quare ita &longs;u&longs;tinet funis HG, vt EF. &longs;imiliter <lb/>
o&longs;ten detur funem PO <expan abbr="t&agrave;m">tam</expan> &longs;u&longs;tinere, <expan abbr="qu&agrave;m">quam</expan> <lb/>
LN: quare funes PO kG EF LN &aelig;qua <lb/>
liter &longs;u&longs;tinent.  </s>
             
<s id="id.2.1.151.4.1.5.0"> &aelig;qualiter igitur funis PO &longs;u<lb/>
&longs;tinet, vt kG. &longs;i ergo du&aelig; intelligantur e&longs; <lb/>
<arrow.to.target n="fig137"></arrow.to.target><lb/>
&longs;e potenti&aelig; in OG, &longs;eu in PH, quod idem e&longs;t, pondus nihilomi<lb/>
nus &longs;u&longs;tinentes, quemadmodum funes &longs;u&longs;tinent, &aelig;quales vtiq; e&longs;<lb/>
&longs;ent; &amp; GF ON duorum vectium vires gerent; quorum fulci <lb/>
menta erunt FN, &amp; pondus A in BC medio vectium appen&longs;um.  </s>
             
<s id="id.2.1.151.4.1.6.0"> <lb/>
&amp; quoniam omnes funes &aelig;qualiter &longs;u&longs;tinent, <expan abbr="t&agrave;m">tam</expan> &longs;u&longs;tinebunt <lb/>
duo PO LN, <expan abbr="qu&agrave;m">quam</expan> duo KGEF; <expan abbr="t&agrave;m">tam</expan> igitur &longs;u&longs;tinebit vectis <lb/>
ON, <expan abbr="qu&agrave;m">quam</expan> vectis GF. quare in vtroq; vecte ON GF &aelig;quali <lb/>
ter pondus <expan abbr="p&otilde;derabit">ponderabit</expan>.  </s>
             
<s id="id.2.1.151.4.1.7.0"> erit ergo vnaqu&aelig;q; potentia in PH &longs;ubquadru<arrow.to.target n="note235"></arrow.to.target><lb/>
pla ponderis A. &amp; <expan abbr="c&ugrave;m">cum</expan> funis KG potenti&aelig; loco &longs;umatur, <expan abbr="quipp&egrave;">quippe</expan> <lb/>
qui haud &longs;ecus &longs;u&longs;tinet, <expan abbr="qu&agrave;m">quam</expan> PO; erit potentia in P &longs;u&longs;tinens pon&shy;<lb/>
dus A ip&longs;ius ponderis &longs;ubquadrupla. quod demon&longs;trare oportebat. </s>
             
<s id="id.2.1.151.4.1.8.0"> [quod demon&longs;trare oportebat.]  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig137" place="text">       </figure>       
<p id="id.2.1.151.4.2.1.0" type="caption">        
<s id="id.2.1.151.4.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.152.1.0.0.0" type="margin">        
<s id="id.2.1.152.1.1.1.0"> <margin.target id="note233"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.152.1.1.2.0"> <margin.target id="note234"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 2 <emph type="italics"/>Cor.<emph.end type="italics"/> 2 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.152.1.1.3.0"> <margin.target id="note235"></margin.target>6 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.153.1.0.0.0" type="head">        
<pb/>
       
<s id="id.2.1.153.1.2.1.0"> COROLLARIVM I. </s>
    </p>       
<p id="id.2.1.153.2.0.0.0" type="main">        
<s id="id.2.1.153.2.1.1.0"> Hinc manife&longs;tum e&longs;t vnumquemq; funem EF <lb/>
GK LN OP quartam &longs;u&longs;tinere partem pon&shy;<lb/>
deris A. </s>
    </p>       
<p id="id.2.1.153.3.0.0.0" type="head">        
<s id="id.2.1.153.3.1.1.0"> COROLLARIVM II. </s>
    </p>       
<p id="id.2.1.153.4.0.0.0" type="main">        
<s id="id.2.1.153.4.1.1.0"> Patet etiam orbiculum, cuius centrum C, <lb/>
non minus eo, cuius centrum e&longs;t B, &longs;u&longs;tinere.  </s>
    </p>       
<p id="id.2.1.153.5.0.0.0" type="head">        
<s id="id.2.1.153.5.1.1.0"> ALITER. </s>
    </p>       
<p id="id.2.1.153.6.0.0.0" type="main">        
<s id="id.2.1.153.6.1.1.0"> Adhuc ii&longs;dem po&longs;itis, &longs;i du&aelig; e&longs;&longs;ent poten<lb/>
ti&aelig; &aelig;quales pondus A &longs;u&longs;tinentes, vna in O <lb/>
<arrow.to.target n="note236"></arrow.to.target> altera in C; e&longs;&longs;et vnaqu&aelig;q; dictarum poten<lb/>
tiarum ponderis A &longs;ubtripla.  </s>
             
<s id="id.2.1.153.6.1.2.0"> &longs;ed quoniam <lb/>
vectis GF, cuius fulcimentum e&longs;t F bifariam <lb/>
diui&longs;us e&longs;t in C; &longs;i igitur ponatur in G poten<lb/>
tia idem pondus &longs;u&longs;tinens, vt potentia in C; <lb/>
erit potentia in G &longs;ubdupla potenti&aelig;, qu&aelig; e&longs; <lb/>
&longs;et in C; nam &longs;i potentia in C &longs;e ip&longs;a pon&shy;<lb/>
dus in C appen&longs;um &longs;u&longs;tineret, e&longs;&longs;et vtiq; ip<lb/>
&longs;i ponderi &aelig;qualis; &amp; idem pondus, &longs;i <expan abbr="&agrave;">a</expan> po<lb/>
<arrow.to.target n="note237"></arrow.to.target>tentia in G &longs;u&longs;tineretur, e&longs;&longs;et ip&longs;ius poten<lb/>
ti&aelig; in G duplum; potentia <expan abbr="ver&oacute;">vero</expan> in C &longs;ubtri<lb/>
pla e&longs;&longs;et ponderis A; ergo potentia in G <lb/>
&longs;ub&longs;excupla e&longs;&longs;et ponderis A.  </s>
     
<s id="id.2.1.153.6.1.2.0.a"> <expan abbr="C&ugrave;m">Cum</expan> itaq; <lb/>
potentia in O &longs;ubtripla &longs;it ponderis A, &amp; <lb/>
potentia in G &longs;ub&longs;excupla; erunt vtr&aelig;q; &longs;i&shy;<lb/>
mul potenti&aelig; in OG ip&longs;ius ponderis A &longs;ub <lb/>
dupl&aelig;.  </s>
             
<s id="id.2.1.153.6.1.3.0"> tertia enim pars cum &longs;exta dimi&shy;<lb/>
dium efficit.  </s>
             
<s id="id.2.1.153.6.1.4.0"> quoniam autem potenti&aelig; in <lb/>
OG, &longs;iue in PH (vt prius dictum e&longs;t) <lb/>
&longs;unt inter &longs;e &aelig;quales, ac vtr&aelig;q; &longs;imul &longs;ubdu<lb/>
pl&aelig; &longs;unt ponderis A. erit vnaqu&aelig;q; poten<lb/>
<arrow.to.target n="fig138"></arrow.to.target>
<pb n="72"/>
tia in P H ip&longs;ius A &longs;ubquadrupla.  </s>
             
<s id="id.2.1.153.6.1.5.0"> Potentia igitur in P &longs;u&longs;tinens pon<lb/>
dus A ip&longs;ius ponderis A &longs;ubquadrupla erit. quod erat o&longs;ten&shy;<lb/>
dendum. </s>
             
<s id="id.2.1.153.6.1.6.0"> [quod erat o&longs;ten&shy;<lb/>
dendum.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig138" place="text">       </figure>       
<p id="id.2.1.153.6.2.1.0" type="caption">        
<s id="id.2.1.153.6.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.154.1.0.0.0" type="margin">        
<s id="id.2.1.154.1.1.1.0"> <margin.target id="note236"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 4 <emph type="italics"/>Huius<emph.end type="italics"/> </s>
             
<s id="id.2.1.154.1.1.2.0"> <margin.target id="note237"></margin.target>2 <emph type="italics"/>Huius. de vecte.<emph.end type="italics"/> </s>
             
<s id="id.2.1.154.1.1.3.0"> [de vecte.<emph.end type="italics"/>] </s>
    </p>       
<p id="id.2.1.155.1.0.0.0" type="main">        
<s id="id.2.1.155.1.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> funis religetur in E, <lb/>
&amp; <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> quatuor adhuc <lb/>
circumuoluatur orbiculos, per <lb/>
ueniatq; ad P. &longs;imiliter o&longs;ten <lb/>
detur potentiam in P &longs;ubqua&shy;<lb/>
druplam e&longs;&longs;e ponderis A. <lb/>
idem enim e&longs;t, ac &longs;i funis re&shy;<lb/>
ligatus e&longs;&longs;et in L, potentiaq; <lb/>
&longs;u&longs;tineret pondus fune tribus <lb/>
<expan abbr="tant&ugrave;m">tantum</expan> orbiculis circumdu&shy;<lb/>
cto, quorum centra e&longs;&longs;ent B <lb/>
<expan abbr="Cq.">Cque</expan> orbiculus enim cuius <lb/>
centrum D e&longs;t p&oelig;nitus inu&shy;<lb/>
tilis. <arrow.to.target n="fig139"></arrow.to.target> </s>
     
<pb/>
       
<s id="id.2.1.155.1.3.1.0"> PROPOSITIO VIII. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig139" place="text">       </figure>       
<p id="id.2.1.155.1.4.1.0" type="caption">        
<s id="id.2.1.155.1.4.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.155.2.0.0.0" type="main">        
<s id="id.2.1.155.2.1.1.0"> Sint duo vetes AB CD bifariam diui&longs;i in EF, <lb/>
quorum fulcimenta &longs;int AC, &amp; pondus G in <lb/>
punctis EF vtriq; vecti &longs;it appen&longs;um, ita vt ex <lb/>
vtroq; &aelig;qualiter ponderet; tre&longs;q; &longs;int potenti&aelig; <lb/>
&aelig;quales in BDE pondus G &longs;u&longs;tinentes.  </s>
             
<s id="id.2.1.155.2.1.2.0"> Dico <lb/>
vnamquamq; &longs;eor&longs;um ex dictis potentiis &longs;ub&shy;<lb/>
quintuplam e&longs;&longs;e ponderis G. </s>
    </p>       
<p id="id.2.1.155.3.0.0.0" type="main">        
<s id="id.2.1.155.3.1.1.0"> Quoniam enim pondus G <lb/>
appen&longs;um e&longs;t in EF, &amp; tres <lb/>
&longs;unt potenti&aelig; in EBD &aelig;qua<lb/>
les; ideo potentia in E partem <lb/>
<expan abbr="tant&ugrave;m">tantum</expan> ponderis G &longs;u&longs;tinebit <lb/>
ip&longs;i potenti&aelig; in E &aelig;qualem; <lb/>
potenti&aelig; <expan abbr="ver&ograve;">vero</expan> in BD partem <lb/>
&longs;u&longs;tinebunt reliquam; &amp; pars, <lb/>
<arrow.to.target n="note238"></arrow.to.target> quam &longs;u&longs;tinet B, erit ip&longs;ius <lb/>
dupla; pars autem, quam &longs;u<lb/>
<arrow.to.target n="fig140"></arrow.to.target><lb/>
&longs;tinet D, erit &longs;imiliter ip&longs;ius D dupla; propter proportionem <lb/>
BA ad AE, &amp; DC ad CF.  </s>
     
<s id="id.2.1.155.3.1.1.0.a"> <expan abbr="C&ugrave;m">Cum</expan> itaq; potenti&aelig; in BD &longs;int &aelig;qua <lb/>
<arrow.to.target n="note239"></arrow.to.target> les, erunt (ex iis, qu&aelig; &longs;upra dictum e&longs;t) partes ponderis G, qu&aelig; <lb/>
<expan abbr="&agrave;">a</expan> potentiis BD &longs;u&longs;tinentur, inter &longs;e &longs;e &aelig;quales; &amp; vnaqu&aelig;q; du<lb/>
pla eius partis, qu&aelig; <expan abbr="&agrave;">a</expan> potentia in E &longs;u&longs;tinetur.  </s>
             
<s id="id.2.1.155.3.1.2.0"> diuidatur er&shy;<lb/>
go pondus G in tres partes, quarum du&aelig; &longs;int inter &longs;e &longs;e &aelig;quales, <lb/>
nec non vnaqu&aelig;q; &longs;eor&longs;um alterius terti&aelig; partis dupla.  </s>
             
<s id="id.2.1.155.3.1.3.0"> quod <lb/>
fiet, &longs;i in quinq; partes &aelig;quales HKLMN diuidatur; pars <lb/>
enim compo&longs;ita ex duabus partibus kL dupla e&longs;t partis H; pars <lb/>
quoq; MN eiu&longs;dem partis H e&longs;t &longs;imiliter dupla.  </s>
             
<s id="id.2.1.155.3.1.4.0"> quare &amp; pars <lb/>
kL parti MN erit &aelig;qualis.  </s>
             
<s id="id.2.1.155.3.1.5.0"> Su&longs;tineat autem potentia in E par<lb/>
tem H; &amp; potentia in B partes KL; potentia <expan abbr="ver&ograve;">vero</expan> in D partes 
<pb n="73"/>
MN: tres igitur potenti&aelig; &aelig;quales in BDE totum &longs;u&longs;tinebunt pon<lb/>
dus G; &amp; vnaqu&aelig;q; potentia in BD duplum &longs;u&longs;tinebit eius, quod <lb/>
&longs;u&longs;tinet potentia in E.  </s>
     
<s id="id.2.1.155.3.1.5.0.a"> <expan abbr="C&ugrave;m">Cum</expan> itaq; potentia in E partem H &longs;u&longs;ti&shy;<lb/>
neat, qu&aelig; quinta e&longs;t pars ponderis G, ip&longs;iq; &longs;it &aelig;qualis; erit po<lb/>
tentia in E &longs;ubquintupla ponderis G.  </s>
     
<s id="id.2.1.155.3.1.5.0.b"> &amp; quoniam potentia in B <lb/>
partes kL &longs;u&longs;tinet, qu&aelig; quidem dupl&aelig; &longs;unt potenti&aelig; B, &amp; partis H; <lb/>
erit quoq; potentia in B ip&longs;i H &aelig;qualis: quare &longs;ubquintupla erit <lb/>
ponderis G.  </s>
     
<s id="id.2.1.155.3.1.5.0.c"> Non aliter o&longs;tendetur potentiam in D &longs;ubquintu&shy;<lb/>
plam e&longs;&longs;e ponderis G. vnaqu&aelig;q; igitur potentia in BDE &longs;ubquin&shy;<lb/>
tupla e&longs;t ponderis G. quod demon&longs;trare oportebat. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig140" place="text">       </figure>       
<p id="id.2.1.155.3.2.1.0" type="caption">        
<s id="id.2.1.155.3.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.156.1.0.0.0" type="margin">        
<s id="id.2.1.156.1.1.1.0"> <margin.target id="note238"></margin.target>2 <emph type="italics"/>Huius. de vecte.<emph.end type="italics"/> </s>
             
<s id="id.2.1.156.1.1.2.0"> [de vecte.<emph.end type="italics"/>] </s>
             
<s id="id.2.1.156.1.1.3.0"> <margin.target id="note239"></margin.target><emph type="italics"/>In<emph.end type="italics"/> 6 <emph type="italics"/>Huius<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.157.1.0.0.0" type="main">        
<s id="id.2.1.157.1.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> &longs;int tres vectes AB <lb/>
CD EF bifariam diui&longs;i in <lb/>
GHk, quorum fulcimenta <lb/>
&longs;int ACE; &amp; pondus L eo <lb/>
dem modo in GHk &longs;it ap&shy;<lb/>
pen&longs;um; quatuorq; &longs;int po&shy;<lb/>
tenti&aelig; &aelig;quales in BDFG <lb/>
pondus L &longs;u&longs;tinentes; &longs;imili <lb/>
modo o&longs;tendetur vnam&shy;<lb/>
quamq; potentiam in BD <lb/>
FG &longs;ub&longs;eptuplam e&longs;&longs;e ponde<lb/>
ris L. &amp; &longs;i quatuor e&longs;&longs;ent vectes, &amp; quinq; potenti&aelig; &aelig;quales pon&shy;<lb/>
dus &longs;u&longs;tinentes; eodem quoq; modo o&longs;tendetur vnamquamq; <lb/>
potentiam &longs;ubnonuplam e&longs;&longs;e ponderis. atq; ita deinceps. </s>
             
<s id="id.2.1.157.1.1.2.0"> [atq; ita deinceps.]  </s>
     <lb/>
       

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure place="text">       </figure>       
<p id="id.2.1.157.1.2.1.0" type="caption">        
<s id="id.2.1.157.1.2.1.0.capt"> YYY </s>
     <lb/>
       
<s id="id.2.1.157.1.4.1.0"> PROPOSITIO VIIII. </s>
    </p>       
<p id="id.2.1.157.2.0.0.0" type="main">        
<s id="id.2.1.157.2.1.1.0"> Si quatuor duarum trochlearum binis orbi&shy;<lb/>
culis, quarum altera <expan abbr="&longs;upern&egrave;">&longs;uperne</expan>, altera vero <expan abbr="in&shy;fern&egrave;">in&shy;<lb/>
ferne</expan>, ponderiq; alligata, di&longs;po&longs;ita fuerit, cir<lb/>
cumducatur funis; altero eius extremo inferiori 
<pb/>
trochle&aelig; religato, altero <expan abbr="ver&ograve;">vero</expan> <expan abbr="&agrave;">a</expan> potentia pon&shy;<lb/>
dus &longs;u&longs;tinente retento: erit potentia ponderis <lb/>
&longs;ubquintupla. </s>
    </p>       
<p id="id.2.1.157.3.0.0.0" type="main">        
<s id="id.2.1.157.3.1.1.0"> Sit pondus A, cui alligata &longs;it trochlea duos <lb/>
habens orbiculos, quorum centra &longs;int BC; <lb/>
&longs;itq; trochlea &longs;ur&longs;um appen&longs;a duos alios ha&shy;<lb/>
bens orbiculos, quorum centra &longs;int DE; funi&longs;q; <lb/>
per omnes circumducatur orbiculos, qui tro&shy;<lb/>
chle&aelig; inferiori religetur in F; &longs;it <expan abbr="qu&eacute;">que</expan> poten<lb/>
tia in G &longs;u&longs;tinens pondus A.  </s>
     
<s id="id.2.1.157.3.1.1.0.a"> dico poten&shy;<lb/>
tiam in G &longs;ubquintuplam e&longs;&longs;e ponderis A. <lb/>
ducantur Hk LM per centra BC horizon&shy;<lb/>
ti &aelig;quidi&longs;tantes, quas eodem modo, quo &longs;u&shy;<lb/>
pra dictum e&longs;t, e&longs;&longs;e tanquam vectes o&longs;tende&shy;<lb/>
mus, quorum fulcimenta kM, &amp; pondus A <lb/>
ex medio vtriu&longs;q; vectis BC &longs;u&longs;pen&longs;um, &amp; tres <lb/>
potenti&aelig; in LHC pondus &longs;u&longs;tinentes, quas <lb/>
&longs;imili modo &aelig;quales e&longs;&longs;e demon&longs;trabimus; fu<lb/>
nes enim idem efficiunt, ac &longs;i e&longs;&longs;ent potenti&aelig;.  </s>
             
<s id="id.2.1.157.3.1.2.0"> <lb/>
&amp; quoniam pondus &aelig;qualiter ex vtroq; ve&shy;<lb/>
cte HK LM ponderat, quod quidem o&longs;ten&shy;<lb/>
detur quoque, vt in pr&aelig;cedentibus demon&shy;<lb/>
<arrow.to.target n="note240"></arrow.to.target> &longs;tratum e&longs;t: erit vnaqu&aelig;q; potentia, <expan abbr="t&ugrave;m">tum</expan> in <lb/>
L, &longs;eu in G, quod idem e&longs;t; <expan abbr="t&ugrave;m">tum</expan> in H, atq; <lb/>
in C, hoc e&longs;t in F, &longs;ubquintupla ponderis A.  </s>
     
<s id="id.2.1.157.3.1.2.0.a"> <lb/>
Potentia ergo in G &longs;u&longs;tinens pondus A ip&longs;ius <lb/>
A &longs;ubquintupla erit.  </s>
             
<s id="id.2.1.157.3.1.3.0"> quod o&longs;tendere opor&shy;<lb/>
tebat. <arrow.to.target n="fig141"></arrow.to.target> </s>
    </p>       
<pb n="74"/>
       
<p id="id.2.1.157.5.0.0.0" type="main">        
<s id="id.2.1.157.5.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> funis in F adhuc de&shy;<lb/>
feratur circa alium orbiculum, <lb/>
cuius centrum N, qui religetur <lb/>
in O; &longs;imiliter duplici medio <lb/>
(vt in &longs;eptima huius) demon<lb/>
&longs;trabitur potentiam in G pon&shy;<lb/>
dus A &longs;u&longs;tinentem &longs;ub&longs;excu<arrow.to.target n="note241"></arrow.to.target><lb/>
plam e&longs;&longs;e ponderis A.  </s>
     
<s id="id.2.1.157.5.1.1.0.a"> <expan abbr="Prim&ugrave;m">Primum</expan> <lb/>
quidem ex tribus vectibus LM <lb/>
Hk FP, quorum fulcimenta <lb/>
&longs;unt MkP, &amp; pondus in me <lb/>
dio vectium appen&longs;um; &amp; tres <lb/>
potenti&aelig; in LHF &aelig;quales pon<lb/>
dus &longs;u&longs;tin&eacute;res.  </s>
             
<s id="id.2.1.157.5.1.2.0"> deinde ex poten<arrow.to.target n="note242"></arrow.to.target><lb/>
tiis in LHN, quarum vnaqu&aelig;q; <lb/>
&longs;ubquintupla e&longs;&longs;et ponderis A. <lb/>
e&longs;&longs;ent enim amb&aelig; &longs;imul poten<lb/>
ti&aelig; in LH &longs;ubdupl&aelig; &longs;exquialte<lb/>
r&aelig; ip&longs;ius ponderis, <expan abbr="pot&etilde;tia">potentia</expan> <expan abbr="ver&ograve;">vero</expan> <lb/>
in F &longs;ubdecupla e&longs;&longs;et, <expan abbr="c&ugrave;m">cum</expan> &longs;it ip<lb/>
&longs;ius N &longs;ubdupla: &longs;ed du&aelig; quin <lb/>
t&aelig; <expan abbr="c&ugrave;m">cum</expan> decima dimidium ef<lb/>
ficiunt, <expan abbr="qu&ograve;d">quod</expan> &longs;i per terna diui <lb/>
datur, &longs;exta pars ponderis re<lb/>
&longs;pondebit vnicuiq; potenti&aelig; in <lb/>
LHF. ex quibus patet poten<lb/>
tiam in G &longs;ub&longs;excuplam e&longs;&longs;e <lb/>
ponderis A. &longs;imiliterq; demon<lb/>
&longs;trabitur vnumquemque orbi<lb/>
culum &aelig;qualem &longs;u&longs;tinere por&shy;<lb/>
tionem. <arrow.to.target n="fig142"></arrow.to.target> </s>
    </p>       
<pb/>
       
<p id="id.2.1.157.7.0.0.0" type="main">        
<s id="id.2.1.157.7.1.1.0"> <expan abbr="Qu&ograve;d">Quod</expan> &longs;i, vt in tertia figura <lb/>
funis in O protrahatur; per <lb/>
aliumq; circumducatur orbi&shy;<lb/>
culum, cuius centrum Q; qui <lb/>
deinde in R trochle&aelig; relige&shy;<lb/>
tur inferiori; erit potentia in <lb/>
<arrow.to.target n="note243"></arrow.to.target> G ponderis &longs;ub&longs;eptupla.  </s>
             
<s id="id.2.1.157.7.1.2.0"> atq; <lb/>
ita in infinitum procedendo <lb/>
proportio potenti&aelig; ad pon&shy;<lb/>
dus quotcunq; &longs;ubmulti&shy;<lb/>
plex inueniri poterit.  </s>
             
<s id="id.2.1.157.7.1.3.0"> dein&shy;<lb/>
de &longs;emper o&longs;tendetur vt in <lb/>
pr&aelig;cedentibus; &longs;i potentia <lb/>
pondus &longs;u&longs;tinens fuerit, vel <lb/>
&longs;ubquadrupla, vel &longs;ubquitu&shy;<lb/>
pla, vel quouis alio modo &longs;e <lb/>
habebit ad pondus; &longs;imiliter <lb/>
vnumquemque funem, vel <lb/>
quartam, vel quintam, vel <lb/>
quamuis aliam partem &longs;u&longs;ti&shy;<lb/>
nere ponderis, quemadmo&shy;<lb/>
dum potentia ip&longs;a; funes e&shy;<lb/>
nim idem efficiunt, ac &longs;i tot <lb/>
e&longs;&longs;ent potenti&aelig;: orbiculi ve <lb/>
<expan abbr="r&ograve;">ro</expan>, ac &longs;i tot e&longs;&longs;ent vectes.  </s>
     <lb/>
       

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig141" place="text">       </figure>       
<p id="id.2.1.157.7.2.1.0" type="caption">        
<s id="id.2.1.157.7.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig142" place="text">       </figure>       
<p id="id.2.1.157.7.2.3.0" type="caption">        
<s id="id.2.1.157.7.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.158.1.0.0.0" type="margin">        
<s id="id.2.1.158.1.1.1.0"> <margin.target id="note240"></margin.target>8 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.158.1.1.2.0"> <margin.target id="note241"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 6 <emph type="italics"/>huius<emph.end type="italics"/> </s>
             
<s id="id.2.1.158.1.1.3.0"> <margin.target id="note242"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 8 <emph type="italics"/>huius<emph.end type="italics"/> </s>
             
<s id="id.2.1.158.1.1.4.0"> <margin.target id="note243"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 8 <emph type="italics"/>Huius<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.159.1.0.0.0" type="main">        

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure place="text">       </figure>       
<p id="id.2.1.159.1.1.1.0" type="caption">        
<s id="id.2.1.159.1.1.1.0.capt"> YYY </s>
     <lb/>
       
<s id="id.2.1.159.1.3.1.0"> COROLLARIVM </s>
    </p>       
<p id="id.2.1.159.2.0.0.0" type="main">        
<s id="id.2.1.159.2.1.1.0"> Ex his manife&longs;tum e&longs;t orbiculos trochle&aelig;, cui <lb/>
e&longs;t alligatum pondus, efficere, vt pondus mino&shy;
<pb n="75"/>
re &longs;u&longs;tineatur potentia, <expan abbr="qu&agrave;m">quam</expan> &longs;it ip&longs;um pondus; <lb/>
quod quidem trochle&aelig; &longs;uperioris orbiculi non <lb/>
efficiunt. </s>
    </p>       
<p id="id.2.1.159.3.0.0.0" type="main">        
<s id="id.2.1.159.3.1.1.0"> Noui&longs;&longs;e tamen oportet, <expan abbr="qu&ograve;d">quod</expan> (vt fieri &longs;olet) inferioris tro<lb/>
chle&aelig; orbiculus, cuius centrum N, minor e&longs;&longs;e debet eo, cuius cen<lb/>
trum C; hic autem minor adhuc eo, cuius centrum B; ac deniq; <lb/>
&longs;i plures fuerint orbiculi in trochlea inferiori ponderi alligata, &longs;em<lb/>
per c&aelig;teris maior e&longs;&longs;e debet, qui annexo ponderi e&longs;t propinquior.  </s>
             
<s id="id.2.1.159.3.1.2.0"> <lb/>
oppo&longs;ito autem modo di&longs;ponendi &longs;unt in trochlea &longs;uperiori.  </s>
             
<s id="id.2.1.159.3.1.3.0"> quod <lb/>
fieri con&longs;ueuit, ne funes inuicem complicentur; nam <expan abbr="quant&ugrave;m">quantum</expan> <lb/>
ad orbiculos attinet, &longs;iue magni fuerint, &longs;iue parui, nihil refert; <lb/>
<expan abbr="c&ugrave;m">cum</expan> &longs;emper idem &longs;equatur. </s>
    </p>       
<p id="id.2.1.159.4.0.0.0" type="main">        
<s id="id.2.1.159.4.1.1.0"> Pr&aelig;terea notandum e&longs;t, quod etiam ex dictis <expan abbr="facil&egrave;">facile</expan> patet, &longs;i <lb/>
funis, &longs;iue religetur in R trochle&aelig; inferiori, &longs;iue in S, maximam <lb/>
<expan abbr="ind&egrave;">inde</expan> oriri differentiam inter potentiam, &amp; pondus: nam &longs;i relige<lb/>
tur in S, erit potentia in G ponderis &longs;ub&longs;excupla.  </s>
             
<s id="id.2.1.159.4.1.2.0"> &longs;i <expan abbr="ver&ograve;">vero</expan> in R, <lb/>
&longs;ub&longs;eptupla.  </s>
             
<s id="id.2.1.159.4.1.3.0"> quod trochle&aelig; &longs;uperiori non contingit, quia &longs;iue <lb/>
religetur funis (vt in pr&aelig;cedenti figura) in T, &longs;iue in O; &longs;em<lb/>
per potentia in G &longs;ub&longs;excupla erit ip&longs;ius ponderis. </s>
    </p>       
<p id="id.2.1.159.5.0.0.0" type="main">        
<s id="id.2.1.159.5.1.1.0"> Po&longs;t h&aelig;c con&longs;iderandum e&longs;t, quonam modo vis moueat pon<lb/>
dus; necnon potenti&aelig; mouentis, ponderi&longs;q; moti &longs;patium, atque <lb/>
tempus.  </s>
    </p>       
<p id="id.2.1.159.6.0.0.0" type="head">        
<s id="id.2.1.159.6.1.1.0"> PROPOSITIO X. </s>
    </p>       
<p id="id.2.1.159.7.0.0.0" type="main">        
<s id="id.2.1.159.7.1.1.0"> Si funis orbiculo trochle&aelig; &longs;ur&longs;um appen&longs;&aelig; <lb/>
fuerit circumuolutus, cuius altero extremo &longs;it al<lb/>
ligatum pondus; alteri autem mouens collocata <lb/>
&longs;it potentia: mouebit h&aelig;c vecte horizonti &longs;em&shy;<lb/>
per &aelig;quidi&longs;tante.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.159.9.0.0.0" type="main">        
<s id="id.2.1.159.9.1.1.0"> Sit pondus A, &longs;it orbiculus trochle&aelig; &longs;ur<lb/>
&longs;um appen&longs;&aelig;' cuius centrum K; &longs;it deinde <lb/>
funis HBCDEF aligatus ponderi A in H, <lb/>
orbiculoq; circumductus; &longs;itq; trochlea ita in <lb/>
L appen&longs;a, &amp; nullum alium habeat motum <lb/>
pr&aelig;ter liberam orbiculi circa axem ver&longs;ionem; <lb/>
&longs;itq; potentia in F mouens pondus A.  </s>
     
<s id="id.2.1.159.9.1.1.0.a"> Dico <lb/>
potentiam in F &longs;emper mouere pondus A <lb/>
vecte horizonti &aelig;quidi&longs;tante.  </s>
             
<s id="id.2.1.159.9.1.2.0"> ducatur BKE <lb/>
horizonti &aelig;quidi&longs;tans; &longs;intq; BE puncta, vbi <lb/>
funes BH, &amp; EF circulum tangunt; erit BkE <lb/>
<arrow.to.target n="note244"></arrow.to.target> vectis, cuius fulcimentum e&longs;t in eius medio <lb/>
k.  </s>
             
<s id="id.2.1.159.9.1.3.0"> &longs;icut &longs;upra o&longs;ten&longs;um e&longs;t.  </s>
             
<s id="id.2.1.159.9.1.4.0"> dum itaq; vis <lb/>
in F deor&longs;um tendit ver&longs;us M, vectis EB <lb/>
mouebitur, <expan abbr="c&ugrave;m">cum</expan> totus orbiculus moueatur, <lb/>
<arrow.to.target n="fig143"></arrow.to.target><lb/>
hoc e&longs;t circumuertatur.  </s>
             
<s id="id.2.1.159.9.1.5.0"> dum igitur F e&longs;t in M, &longs;it punctum E ve<lb/>
ctis v&longs;q; ad I motum; B autem v&longs;q; ad C, ita vt vectis &longs;it in <lb/>
CI.  </s>
     
<s id="id.2.1.159.9.1.5.0.a"> fiat deinde NM &aelig;qualis ip&longs;i FE: &amp; quando punctum E <lb/>
erit in I, tnnc funis punctum, quod erat in E, erit in N: quod au<lb/>
tem erat in B erit in C; ita vt ducta CI per centrum K tran&longs;eat.  </s>
             
<s id="id.2.1.159.9.1.6.0"> <lb/>
dum autem B e&longs;t in C, &longs;it punctum H in G; eritq; BH ip&longs;i <lb/>
CBG &aelig;qualis; <expan abbr="c&ugrave;m">cum</expan> &longs;it idem funis.  </s>
             
<s id="id.2.1.159.9.1.7.0"> &amp; quoniam dum EF tendit <lb/>
in NM, adhuc &longs;emper remanet EFM horizonti perpendicularis, <lb/>
circulumq; tangens in puncto E; ita vt ducta <expan abbr="&agrave;">a</expan> puncto E per cen<lb/>
trum k, &longs;it &longs;emper horizonti &aelig;quidi&longs;tans.  </s>
             
<s id="id.2.1.159.9.1.8.0"> quod idem euenit funi <lb/>
BG, &amp; puncto B. dum igitur circulus, &longs;iue orbiculus circumuer<lb/>
titur, &longs;emper mouetur vectis EB, &longs;emperq; adhuc remanet alius <lb/>
vectis in EB.  </s>
     
<s id="id.2.1.159.9.1.8.0.a"> &longs;iquidem ex ip&longs;ius rotul&aelig; natura, in qua &longs;emper <lb/>
dum mouetur, remanet diameter ex B in E (qu&aelig; vectis vicem ge<lb/>
rit) euenit, vt recedente vna, &longs;emper altera &longs;uccedat; eiu&longs;modi <lb/>
durante circumductione: atq; ita fit, vt potentia &longs;emper moueat <lb/>
pondus vecte EB horizonti &aelig;quidi&longs;tante. quod demon&longs;trare opor&shy;<lb/>
tebat. </s>
             
<s id="id.2.1.159.9.1.9.0"> [quod demon&longs;trare opor&shy;<lb/>
tebat.]  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig143" place="text">       </figure>       
<p id="id.2.1.159.9.2.1.0" type="caption">        
<s id="id.2.1.159.9.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.160.1.0.0.0" type="margin">        
<s id="id.2.1.160.1.1.1.0"> <margin.target id="note244"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.161.1.0.0.0" type="main">        
<pb n="76"/>
       
<s id="id.2.1.161.1.2.1.0"> Ii&longs;dem po&longs;itis, &longs;patium potenti&aelig; pondus <lb/>
mouentis e&longs;t &aelig;quale &longs;patio eiu&longs;dem ponderis <lb/>
moti. </s>
    </p>       
<p id="id.2.1.161.2.0.0.0" type="main">        
<s id="id.2.1.161.2.1.1.0"> Quoniam enim o&longs;ten&longs;um e&longs;t, dum F e&longs;t in M, pondus A, hoc <lb/>
e&longs;t punctum H e&longs;&longs;e in G; &amp; <expan abbr="c&ugrave;m">cum</expan> funis HBCDEF &longs;it &aelig;qualis <lb/>
GBCDENFM, e&longs;t enim idem funis; dempto igitur communi <lb/>
GBCDENF, erit HG ip&longs;i FM &aelig;qualis.  </s>
             
<s id="id.2.1.161.2.1.2.0"> &longs;imiliterq; o&longs;tende&shy;<lb/>
tur, de&longs;cen&longs;um F &longs;emper &aelig;qualem e&longs;&longs;e a&longs;cen&longs;ui H. ergo &longs;patium <lb/>
potenti&aelig; &aelig;quale e&longs;t &longs;patio ponderis. quod erat demon&longs;tran&shy;<lb/>
dum. </s>
             
<s id="id.2.1.161.2.1.3.0"> [quod erat demon&longs;tran&shy;<lb/>
dum.] </s>
    </p>       
<p id="id.2.1.161.3.0.0.0" type="main">        
<s id="id.2.1.161.3.1.1.0"> Pr&aelig;terea potentia idem pondus per &aelig;quale <lb/>
&longs;patium in &aelig;quali tempore mouet, <expan abbr="t&agrave;m">tam</expan> fune <lb/>
hoc modo orbiculo trochle&aelig; &longs;ur&longs;um appen&longs;&aelig; <lb/>
circumuoluto, <expan abbr="qu&agrave;m">quam</expan> &longs;ine trochlea: dummo&shy;<lb/>
do ip&longs;ius potenti&aelig; lationes in velocitate &longs;int &aelig;&shy;<lb/>
quales.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.161.5.0.0.0" type="main">        
<s id="id.2.1.161.5.1.1.0"> Ii&longs;dem po&longs;itis &longs;it aliud pondus P <lb/>
&aelig;quale ponderi A, cui alligatus &longs;it <lb/>
funis TQ <expan abbr="horiz&otilde;ti">horizonti</expan> <expan abbr="perp&etilde;dicularis">perpendicularis</expan>; <lb/>
et &longs;it TQ ip&longs;i HB &aelig;qualis; moueat <lb/>
<expan abbr="qu&eacute;">que</expan> <expan abbr="pot&etilde;tia">potentia</expan> in Q <expan abbr="p&otilde;dus">pondus</expan> P &longs;ur&longs;um <lb/>
ad rectos angulos horizonti, quem <lb/>
admodum mouetur pondus A.  </s>
     
<s id="id.2.1.161.5.1.1.0.a"> di<lb/>
co per &aelig;quale &longs;patium in eodem <lb/>
tempore potentiam in Q pondus <lb/>
P, &amp; potentiam in F pondus A <lb/>
mouere.  </s>
             
<s id="id.2.1.161.5.1.2.0"> quod idem e&longs;t, ac &longs;i e&longs;&longs;et <lb/>
idem pondus in &aelig;quali tempore <lb/>
motum; &longs;icut propo&longs;uimus.  </s>
             
<s id="id.2.1.161.5.1.3.0"> Pro&shy;<lb/>
ducatur EF in S, &amp; TQ in R; <lb/>
fiantq; QR FS non &longs;olum inter <lb/>
&longs;e &longs;e, <expan abbr="ver&ugrave;m">verum</expan> etiam ip&longs;i BH &aelig;qua<lb/>
les.  </s>
             
<s id="id.2.1.161.5.1.4.0"> <expan abbr="C&ugrave;m">Cum</expan> autem TQ QR &longs;int <lb/>
ip&longs;is HB FS &aelig;quales, &amp; vis in Q <lb/>
moueat pondus P per rectam T <lb/>
QR; vis autem in F moueat A <lb/>
per rectam HB, &amp; velocitates <lb/>
<arrow.to.target n="fig144"></arrow.to.target><lb/>
motuum vtriu&longs;q; potenti&aelig; &longs;int &aelig;quales; tunc in eodem tempore <lb/>
potentia in Q erit in R, &amp; potentia in F erit in S; <expan abbr="c&ugrave;m">cum</expan> &longs;patia &longs;int <lb/>
&aelig;qualia.  </s>
             
<s id="id.2.1.161.5.1.5.0"> &longs;ed dum potentia in Q e&longs;t in R, pondus P, hoc e&longs;t <lb/>
punctum T erit in Q; <expan abbr="c&ugrave;m">cum</expan> TQ &longs;it ip&longs;i QR &aelig;qualis.  </s>
             
<s id="id.2.1.161.5.1.6.0"> &amp; dum po<lb/>
tentia in F e&longs;t in S, pondus A, hoc e&longs;t punctum H erit in B; &longs;ed <lb/>
&longs;patium TQ &aelig;quale e&longs;t &longs;patio HB, potenti&aelig; ergo in FQ &aelig;quali <lb/>
ter mot&aelig; pondera PA &aelig;qualia per &aelig;qualia &longs;patia in eodem tempo<lb/>
re mouebunt. quod erat demon&longs;trandum  </s>
             
<s id="id.2.1.161.5.1.7.0"> [quod erat demon&longs;trandum] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig144" place="text">       </figure>       
<p id="id.2.1.161.5.2.1.0" type="caption">        
<s id="id.2.1.161.5.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.161.6.0.0.0" type="head">        
<s id="id.2.1.161.6.1.1.0"> PROPOSITIO XI. </s>
    </p>       
<p id="id.2.1.161.7.0.0.0" type="main">        
<s id="id.2.1.161.7.1.1.0"> Si funis orbiculo trochle&aelig; ponderi alligat&aelig; <lb/>
fuerit circumuolutus, qui in altero eius extre&shy;
<pb n="77"/>
mo alicubi religetur, altero autem <expan abbr="&agrave;">a</expan> potentia <lb/>
mouente pondus appr&aelig;hen&longs;o; vecte &longs;emper ho<lb/>
rizonti &aelig;qui&longs;tante potentia mouebit. </s>
    </p>       
<p id="id.2.1.161.8.0.0.0" type="main">        
<s id="id.2.1.161.8.1.1.0"> Sit pondus A; Sit orbiculus.  </s>
             
<s id="id.2.1.161.8.1.2.0"> <lb/>
CED trochle&aelig; ponderi A alli&shy;<lb/>
gat&aelig; ex kH; &longs;itq; KH ad rectos <lb/>
angulos horizonti, ita vt pon&shy;<lb/>
dus &longs;emper trochle&aelig; motum, &longs;i&shy;<lb/>
ue &longs;ur&longs;um, &longs;iue deor&longs;um factum <lb/>
&longs;equatur; &longs;itq; orbiculi centrum <lb/>
K; &amp; funis orbiculo circumuo&shy;<lb/>
lutus &longs;it BCDEF, qui relige&shy;<lb/>
tur in B, ita vt in B immobilis <lb/>
maneat; &amp; &longs;it potentia in F mo&shy;<lb/>
uens pondus A.  </s>
     
<s id="id.2.1.161.8.1.2.0.a"> dico potentia m <lb/>
in F &longs;emper mouere <expan abbr="p&otilde;dus">pondus</expan> A ve<lb/>
cte horizonti &aelig;quidi&longs;tante.  </s>
             
<s id="id.2.1.161.8.1.3.0"> &longs;int <lb/>
BC EF inter &longs;e &longs;e, ip&longs;iq; kH &aelig;&shy;<lb/>
quidi&longs;tantes, &amp; eiu&longs;dem kH ho<lb/>
rizonti perpendiculares, tangen<lb/>
te&longs;q; <expan abbr="circul&utilde;">circulum</expan> CED in EC <expan abbr="p&utilde;ctis">punctis</expan>; <lb/>
et connectatur EC, qu&aelig; per cen<arrow.to.target n="note245"></arrow.to.target><lb/>
trum k tran&longs;ibit, horizontiq; <lb/>
&aelig;quidi&longs;tans erit; &longs;icuti prius di<lb/>
ctum e&longs;t.  </s>
             
<s id="id.2.1.161.8.1.4.0"> Quoniam enim or<lb/>
biculus CED circa eius cen<lb/>
trum K vertitur; ideo dum vis <lb/>
in F trahit &longs;ur&longs;um punctum E, <lb/>
deberet punctum C de&longs;cende <lb/>
re, ac trahere deor&longs;um B; &longs;ed fu<lb/>
<arrow.to.target n="fig145"></arrow.to.target><lb/>
nis in B e&longs;t immobilis, &amp; BC de&longs;cedere non pote&longs;t; quare dum <lb/>
potentia in F trahit &longs;ur&longs;um E, totus orbiculus &longs;ur&longs;um mouebitur; <lb/>
ac per con&longs;equens tota trochlea, &amp; pondus; &amp; EkC erit tanquam <arrow.to.target n="note246"></arrow.to.target><lb/>
vectis, cuius fulcimentum erit C; e&longs;t enim punctum C propter BC <lb/>
<expan abbr="fer&egrave;">fere</expan> immobile, potentia <expan abbr="ver&ograve;">vero</expan> mouens vectem e&longs;t in F fune EF, 
<pb/>
&amp; pondus in k appen&longs;um.  </s>
             
<s id="id.2.1.161.8.1.5.0"> <lb/>
<expan abbr="qu&ograve;d">quod</expan> &longs;i punctum C omnino fue<lb/>
rit immobile, moueaturq; ve<lb/>
ctis EC in NC; &amp; diuidatur <lb/>
NC bifariam in L: erunt CL <lb/>
LN ip&longs;is Ck KE &aelig;quales.  </s>
             
<s id="id.2.1.161.8.1.6.0"> <lb/>
quare &longs;i vectis EC e&longs;&longs;et in CN, <lb/>
punctum k e&longs;&longs;et in L; &amp; &longs;i du<lb/>
catur LM horizonti perpendi<lb/>
cularis, qu&aelig; &longs;it etiam &aelig;qualis <lb/>
kH; e&longs;&longs;et pondus A, hoc e&longs;t <lb/>
punctum H in M.  </s>
     
<s id="id.2.1.161.8.1.6.0.a"> &longs;ed quoniam <lb/>
potentia in F dum tendit &longs;ur&shy;<lb/>
&longs;um mouendo orbiculum, &longs;em<lb/>
per mouetur &longs;uper rectam EFG, <lb/>
qu&aelig; &longs;emper e&longs;t quoq; &aelig;quidi<lb/>
&longs;tans BC; nece&longs;&longs;e erit orbicu<lb/>
lum trochle&aelig; &longs;emper inter li&shy;<lb/>
neas EG BC e&longs;&longs;e: &amp; centrum <lb/>
k, cum &longs;it in medio, &longs;uper <lb/>
rectam lineam HkT &longs;emper <lb/>
moueri.  </s>
             
<s id="id.2.1.161.8.1.7.0"> Itaq; ducatur per L li<lb/>
nea PTLQ horizonti, &amp; EC <lb/>
&aelig;quidi&longs;tans, qu&aelig; &longs;ecet Hk pro&shy;<lb/>
ductam in T; &amp; centro T, &longs;pa<lb/>
tio <expan abbr="ver&ograve;">vero</expan> TQ, circulus de&longs;criba<lb/>
<arrow.to.target n="fig146"></arrow.to.target><lb/>
tur QRPS, qui &aelig;qualis erit circulo CED; &amp; puncta PQ tangent fu<lb/>
<arrow.to.target n="note247"></arrow.to.target> nes FE BC in PQ punctis.  </s>
             
<s id="id.2.1.161.8.1.8.0"> rectangulum enim e&longs;t PECQ, &amp; <lb/>
PT TQ ip&longs;is EK kC &longs;unt &aelig;quales.  </s>
             
<s id="id.2.1.161.8.1.9.0"> deinde per T ducatur R <lb/>
TS diameter circuli PQS &aelig;quidi&longs;tans ip&longs;i NC; <expan abbr="fiatqu&eacute;">fiatque</expan> TO &aelig;qua <lb/>
lis kH.  </s>
     
<s id="id.2.1.161.8.1.9.0.a"> dum autem centrum k motum erit v&longs;q; ad lineam PQ, <lb/>
tunc centrum k erit in T. o&longs;ten&longs;um e&longs;t enim centrum orbiculi &longs;u<lb/>
per rectam HT &longs;emper moueri.  </s>
             
<s id="id.2.1.161.8.1.10.0"> idcirco vt centrum k &longs;it in li<lb/>
nea PQ ip&longs;i EC &aelig;quidi&longs;tante, nece&longs;&longs;e e&longs;t vt &longs;it in T. &amp; vt vectis <lb/>
EC eleuetur in angulo ECN, nece&longs;&longs;e e&longs;t, vt &longs;it in RS, non au&shy;<lb/>
<arrow.to.target n="note248"></arrow.to.target> tem in CN: angulus enim RSE angulo NCE e&longs;t &aelig;qualis, &amp; &longs;ic 
<pb n="78"/>
fulcimentum C non e&longs;t penitus immobile.  </s>
             
<s id="id.2.1.161.8.1.11.0"> <expan abbr="c&ugrave;m">cum</expan> totus orbiculus &longs;ur<lb/>
&longs;um moueatur, toru&longs;q; mutet totum locum; habet tamen C ratio <lb/>
nem fulcimenti, quia minus mouetur C, <expan abbr="qu&agrave;m">quam</expan> k, &amp; E: punctum <lb/>
enim E mouetur v&longs;q; ad R, &amp; K v&longs;q; ad T, punctum <expan abbr="ver&ograve;">vero</expan> C v&longs;q; <lb/>
ad S <expan abbr="tant&ugrave;m">tantum</expan>.  </s>
             
<s id="id.2.1.161.8.1.12.0"> quare dum centrum K e&longs;t in T, po&longs;itio orbiculi erit <lb/>
QR PS: &amp; pondus A. hoc e&longs;t punctum H erit in O; <expan abbr="c&ugrave;m">cum</expan> TO <lb/>
&longs;it &aelig;qualis kH; po&longs;itio <expan abbr="ver&ograve;">vero</expan> EC, &longs;cilicet vectis moti, erit RS, po<lb/>
tentiaq; in F mota erit &longs;ur&longs;um per rectam EFG.  </s>
     
<s id="id.2.1.161.8.1.12.0.a"> eodem autem <lb/>
tempore, quo k erit in T, &longs;it potentia in G: dum autem vectis EC <lb/>
hoc modo mouetur, adhuc &longs;emper remanent GP BQ inter &longs;e &longs;e &aelig;&shy;<lb/>
quidi&longs;tantes, atq; horizonti perpendiculares, ita vt vbi orbiculum <lb/>
tangunt, vt in punctis PQ; &longs;emper linea PQ erit diameter orbi <lb/>
culi, &amp; tanquam vectis horizonti &aelig;quidi&longs;tans.  </s>
             
<s id="id.2.1.161.8.1.13.0"> dum igitur orbi&shy;<lb/>
culus mouetur, &amp; circumuertitur, &longs;emper etiam mouetur vectis <lb/>
EC, &amp; &longs;emper remanet alius vectis in orbiculo horizonti &aelig;qui&longs;tans, <lb/>
vt PQ; ita vt potentia in F &longs;emper moueat pondus vecte hori<lb/>
zonti &aelig;quidi&longs;tante, cuius fulcimentum erit &longs;emper in linea CB; &amp; <lb/>
pondus in medio vectis appen&longs;um; potentiaq; in linea EG. quod <lb/>
erat o&longs;tendendum. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig145" place="text">       </figure>       
<p id="id.2.1.161.8.2.1.0" type="caption">        
<s id="id.2.1.161.8.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig146" place="text">       </figure>       
<p id="id.2.1.161.8.2.3.0" type="caption">        
<s id="id.2.1.161.8.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.162.1.0.0.0" type="margin">        
<s id="id.2.1.162.1.1.1.0"> <margin.target id="note245"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 1 <emph type="italics"/>huius<emph.end type="italics"/> </s>
             
<s id="id.2.1.162.1.1.2.0"> <margin.target id="note246"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 2 <emph type="italics"/>huius<emph.end type="italics"/> </s>
             
<s id="id.2.1.162.1.1.3.0"> <margin.target id="note247"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 34 <emph type="italics"/>primi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.162.1.1.4.0"> <margin.target id="note248"></margin.target>29 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.163.1.0.0.0" type="main">        
<s id="id.2.1.163.1.1.1.0"> Ii&longs;dem po&longs;itis, &longs;patium potenti&aelig; pondus <lb/>
mouentis duplum e&longs;t &longs;patii eiu&longs;dem ponderis <lb/>
moti. </s>
    </p>       
<p id="id.2.1.163.2.0.0.0" type="main">        
<s id="id.2.1.163.2.1.1.0"> <expan abbr="C&ugrave;m">Cum</expan> enim o&longs;ten&longs;um &longs;it, dum k e&longs;t in T, pondus A, hoc e&longs;t <lb/>
punctum H e&longs;&longs;e in O, &amp; in eodem etiam tempore potentiam in <lb/>
F e&longs;&longs;e in G: &amp; quoniam funis BCDEF e&longs;t &aelig;qualis funi BQS <lb/>
PG; funis enim e&longs;t idem; &amp; funis circa &longs;emicirculum CDE e&longs;t <lb/>
&aelig;qualis funi circa &longs;emicirculum QSP; demptis igitur communi<lb/>
bus BQ, &amp; FP; erit reliquus FG ip&longs;is CQ, &amp; EP &longs;imul &longs;umptis <lb/>
&aelig;qualis.  </s>
             
<s id="id.2.1.163.2.1.2.0"> &longs;ed EP ip&longs;i TK e&longs;t &aelig;qualis, &amp; CQ ip&longs;i quoq; Tk &aelig;qualis, <lb/>
&longs;unt enim Pk TC parallelogramma rectangula; quare line&aelig; EP <lb/>
CQ &longs;imul ip&longs;ius Tk dupl&aelig; erunt.  </s>
             
<s id="id.2.1.163.2.1.3.0"> funis igitur FC ip&longs;ius TK du<lb/>
plus erit.  </s>
             
<s id="id.2.1.163.2.1.4.0"> &amp; quoniam kH e&longs;t &aelig;qualis TO, dempto communi kO, <lb/>
erit kT ip&longs;i HO &aelig;qualis; quare funis FG ip&longs;ius HO duplus erit; 
<pb/>
hoc e&longs;t &longs;patium potenti&aelig; &longs;patii ponderis duplum. quod erat <lb/>
demon&longs;trandum. </s>
             
<s id="id.2.1.163.2.1.5.0"> [quod erat <lb/>
demon&longs;trandum.] </s>
    </p>       
<p id="id.2.1.163.3.0.0.0" type="main">        
<s id="id.2.1.163.3.1.1.0"> Potentia deinde idem pondus in &aelig;quali tem&shy;<lb/>
pore per dimidium &longs;patium mouebit fune circa <lb/>
orbiculum trochle&aelig; ponderi alligat&aelig; reuoluto, <lb/>
<expan abbr="qu&agrave;m">quam</expan> &longs;ine trochlea; dummodo ip&longs;ius potenti&aelig; <lb/>
velocitates motuum &longs;int &aelig;quales. </s>
    </p>       
<p id="id.2.1.163.4.0.0.0" type="main">        
<s id="id.2.1.163.4.1.1.0"> Sit enim (ii&longs;dem po&longs;i<lb/>
tis) aliud pondus V &aelig;qua <lb/>
le ponderi A, cui alligatus <lb/>
&longs;it funis 9X; &longs;itq; poten<lb/>
tia in X mouens pondus <lb/>
V.  </s>
     
<s id="id.2.1.163.4.1.1.0.a"> dico &longs;i vtriu&longs;q; poten<lb/>
ti&aelig; motuum velocitates <lb/>
&longs;int &aelig;quales, in eodem <lb/>
tempore potentiam in F <lb/>
mouere pondus A per di<lb/>
midium &longs;patium eius, per <lb/>
quod <expan abbr="&agrave;">a</expan> potentia in X mo<lb/>
uetur pondus V; quod <lb/>
idem e&longs;t, ac &longs;i e&longs;&longs;et idem <lb/>
pondus in &aelig;quali tempo <lb/>
re motum.  </s>
             
<s id="id.2.1.163.4.1.2.0"> Moueat po<lb/>
tentia in X pondus V, po<lb/>
tentiaq; perueniat in Y; <lb/>
&longs;itq; XY &aelig;qualis ip&longs;i FG; <lb/>
&amp; fiat YZ &aelig;qualis X9, ita <lb/>
vt quando potentia in X <lb/>
erit in Y, &longs;it pondus V, <lb/>
hoc e&longs;t punctum 9 in Z.  </s>
     
<s id="id.2.1.163.4.1.2.0.a"> <lb/>
&longs;ed 9 Z e&longs;t &aelig;qualis FG, <lb/>
<arrow.to.target n="fig147"></arrow.to.target>
<pb n="79"/>
<expan abbr="c&ugrave;m">cum</expan> &longs;it &aelig;qualis XY; ergo 9 Zip&longs;ius HO dupla erit.  </s>
             
<s id="id.2.1.163.4.1.3.0"> Itaq; dum poten<lb/>
ti&aelig; erunt in GY, pondera AV erunt in OZ. in eodem autem <lb/>
tempore erunt potenti&aelig; in GY, ip&longs;arum enim velocitates mo <lb/>
tuum &longs;unt &aelig;quales; quare vis in F pondus A in eodem tempore <lb/>
mouebit per dimidium &longs;patium eius, per quod mouetur <expan abbr="&agrave;">a</expan> poten<lb/>
tia in X pondus V: &amp; pondera &longs;unt &aelig;qualia; Potentia ergo idem <lb/>
pondus in &aelig;quali tempore per dimidium &longs;patium mouebit fune, <lb/>
trochleaq; hoc modo ponderi alligata, <expan abbr="qu&agrave;m">quam</expan> &longs;ine trochlea; dum <lb/>
modo potenti&aelig; motuum velocitates &longs;int &aelig;quales. quod erat de&shy;<lb/>
mon&longs;trandum. </s>
             
<s id="id.2.1.163.4.1.4.0"> [quod erat de&shy;<lb/>
mon&longs;trandum.]  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig147" place="text">       </figure>       
<p id="id.2.1.163.4.2.1.0" type="caption">        
<s id="id.2.1.163.4.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.163.5.0.0.0" type="head">        
<s id="id.2.1.163.5.1.1.0"> PROPOSITIO XII. </s>
    </p>       
<p id="id.2.1.163.6.0.0.0" type="main">        
<s id="id.2.1.163.6.1.1.0"> Si funis circa plures reuoluatur orbiculos, al&shy;<lb/>
tero eius extremo alicubi religato, altero au&shy;<lb/>
tem <expan abbr="&agrave;">a</expan> potentia pondus mouente detento; poten<lb/>
tia vectibus horizonti &longs;emper &aelig;quidi&longs;tantibus <lb/>
mouebit.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.163.8.0.0.0" type="main">        
<s id="id.2.1.163.8.1.1.0"> Sit pondus A, &longs;it orbiculus CED tro&shy;<lb/>
chle&aelig; ponderi alligat&aelig; ex kS ad rectos an<lb/>
gulos horizonti; ita vt pondus &longs;emper eius <lb/>
motum &longs;ur&longs;um, ac deor&longs;um factum &longs;equa&shy;<lb/>
tur.  </s>
             
<s id="id.2.1.163.8.1.2.0"> &longs;it deinde orbiculus circa centrum L <lb/>
trochle&aelig; &longs;ur&longs;um appen&longs;&aelig; &longs;itq; funis circa <lb/>
orbiculos reuolutus BCDEHMNO, <lb/>
qui religatus &longs;it in B; &longs;itq; vis in O mouens <lb/>
pondus A mouendo &longs;e deor&longs;um per OP.  </s>
     
<s id="id.2.1.163.8.1.2.0.a"> <lb/>
dico potentiam in O &longs;emper mouere pon&shy;<lb/>
dus A vectibus horizonti &longs;emper &aelig;quidi&shy;<lb/>
&longs;tantibus.   </s>
             
<s id="id.2.1.163.8.1.3.0"> [<lb/>
&longs;tantibus.]  </s>
             
<s id="id.2.1.163.8.1.4.0"> ducatur NH per centrum L ho<lb/>
<arrow.to.target n="note249"></arrow.to.target> rizonti &aelig;quidi&longs;tans, qu&aelig; erit vectis orbi&shy;<lb/>
culi, cuius centrum e&longs;t L. ducatur deinde <lb/>
EC per centrum k &longs;imiliter horizonti &aelig;qui <lb/>
<arrow.to.target n="note250"></arrow.to.target> di&longs;tans, qu&aelig; etiam erit vectis orbiculi, cu&shy;<lb/>
ius centrum e&longs;t k.  </s>
             
<s id="id.2.1.163.8.1.5.0"> Moueatur potentia in <lb/>
O deor&longs;um, qu&aelig; dum deor&longs;um mouetur, ve<lb/>
ctem NH mouebit; &amp; dum vectis moue&shy;<lb/>
<arrow.to.target n="note251"></arrow.to.target> tur, N deor&longs;um mouebitur, H <expan abbr="ver&ograve;">vero</expan> &longs;ur&shy;<lb/>
&longs;um, vti&longs;upra dictum e&longs;t.  </s>
             
<s id="id.2.1.163.8.1.6.0"> dum autem H <lb/>
mouetur &longs;ur&longs;um, mouet etiam &longs;ur&longs;um E; &amp; <lb/>
vectem EC, cuius fulcimentum e&longs;t C, &longs;ed <lb/>
fulcimentum C non pote&longs;t mouere deor&shy;<lb/>
&longs;um B; ideo orbiculus, cuius centrum K, &longs;ur<lb/>
<arrow.to.target n="fig148"></arrow.to.target><lb/>
&longs;um mouebitur, &amp; per con&longs;equens trochlea, &amp; pondus A; vt in <lb/>
pr&aelig;cedenti dictum e&longs;t.  </s>
             
<s id="id.2.1.163.8.1.7.0"> &amp; quoniam ob eandem cau&longs;am in pr&aelig;ce&shy;<lb/>
dentibus a&longs;signatam in HN, &amp; EC &longs;emper remanent vectes hori<lb/>
zonti &aelig;quidi&longs;tantes; potentia ergo mouens pondus A &longs;emper <lb/>
eum mouebit vectibus horizonti &aelig;quidi&longs;tantibus. quod erat o&shy;<lb/>
&longs;tendendum.  </s>
             
<s id="id.2.1.163.8.1.8.0"> [quod erat o&shy;<lb/>
&longs;tendendum.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig148" place="text">       </figure>       
<p id="id.2.1.163.8.2.1.0" type="caption">        
<s id="id.2.1.163.8.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.164.1.0.0.0" type="margin">        
<s id="id.2.1.164.1.1.1.0"> <margin.target id="note249"></margin.target>1, <emph type="italics"/>Et<emph.end type="italics"/> 10 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.164.1.1.2.0"> <margin.target id="note250"></margin.target>11 <emph type="italics"/>huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.164.1.1.3.0"> <margin.target id="note251"></margin.target>10 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.165.1.0.0.0" type="main">        
<s id="id.2.1.165.1.1.1.0"> Et &longs;i funis circa plures &longs;it reuolutus orbiculos; &longs;imiliter o&longs;tende&shy;<lb/>
tur, potentiam mouere pondus vectibus horizonti &longs;emper &aelig;qui&shy;<lb/>
di&longs;tantibus: &amp; vectes orbiculorum trochle&aelig; &longs;uperioris &longs;emper <lb/>
e&longs;&longs;e, vt HN, quorum fulcimenta erunt &longs;emper in medio: vectes au&shy;<lb/>
tem orbiculorum trochle&aelig; inferioris &longs;emper exi&longs;tere, vt EC; quo&shy;
<pb n="80"/>
rum fulcimenta erunt in extremitatibus vectium. </s>
    </p>       
<p id="id.2.1.165.2.0.0.0" type="main">        
<s id="id.2.1.165.2.1.1.0"> Ii&longs;dem po&longs;itis, &longs;patium potenti&aelig; duplum e&longs;t <lb/>
&longs;patii ponderis. </s>
    </p>       
<p id="id.2.1.165.3.0.0.0" type="main">        
<s id="id.2.1.165.3.1.1.0"> Sit motum centrum K v&longs;q; ad centrum R; &amp; orbiculus &longs;it FTG. <lb/>
deinde per centrum R ducatur GF ip&longs;i EC &aelig;quidi&longs;tans: tangent <lb/>
funes EH CB orbiculum in GF punctis.  </s>
             
<s id="id.2.1.165.3.1.2.0"> fiat deniq; RQ &aelig;qua <lb/>
lis KS. dum igitur k erit in R; pondus A, &longs;cilicet punctum S erit <lb/>
in <expan abbr="q.">que</expan> &amp; dum centrum orbiculi e&longs;t in R, &longs;it potentia in O mota <lb/>
in P.  </s>
     
<s id="id.2.1.165.3.1.2.0.a"> &amp; quoniam funis BCDEHMNO e&longs;t &aelig;qualis funi BFT <lb/>
GHMNP; e&longs;t enim idem funis; &amp; FTG &aelig;qualis e&longs;t CDE; dem<lb/>
ptis igitur communibus BF, &amp; GHMNO, erit reliquus OP ip<lb/>
&longs;is FCEG &longs;imul &longs;umptis &aelig;qualis: &amp; per con&longs;equens duplus kR, <lb/>
&amp; QS &amp; <expan abbr="c&ugrave;m">cum</expan> OP &longs;it &longs;patium potenti&aelig; mot&aelig;, &amp; SQ &longs;patium pon<lb/>
deris moti; erit &longs;patium potenti&aelig; duplum &longs;patii ponderis. quod <lb/>
erat o&longs;tendendum. </s>
             
<s id="id.2.1.165.3.1.3.0"> [quod <lb/>
erat o&longs;tendendum.] </s>
    </p>       
<p id="id.2.1.165.4.0.0.0" type="main">        
<s id="id.2.1.165.4.1.1.0"> Pr&aelig;terea potentia idem pondus in &aelig;quali <lb/>
tempore per dimidium &longs;patium mouebit fune <lb/>
circa duos orbiculos reuoluto, quorum vnus <lb/>
&longs;it trochle&aelig; &longs;uperioris, alter <expan abbr="ver&ograve;">vero</expan> &longs;it trochle&aelig; <lb/>
ponderi alligat&aelig;; <expan abbr="qu&agrave;m">quam</expan> &longs;ine trochleis: dummo&shy;<lb/>
do ip&longs;ius potenti&aelig; lationes &longs;int &aelig;qualiter ve&shy;<lb/>
loces </s>
    </p>       
<pb/>
       
<p id="id.2.1.165.6.0.0.0" type="main">        
<s id="id.2.1.165.6.1.1.0"> Ii&longs;dem namq; po&longs;itis, &longs;it pon<lb/>
dus V &aelig;quale ip&longs;i A, cui alliga&shy;<lb/>
tus &longs;it funis X9; &longs;itq; <expan abbr="pot&etilde;tia">potentia</expan> in X <lb/>
mouens <expan abbr="p&otilde;dus">pondus</expan> V; qu&aelig; dum pon <lb/>
dus mouet, perueniat in Y: fiant <lb/>
<expan abbr="qu&eacute;">que</expan> XY Z9 ip&longs;i OP &aelig;quales; <lb/>
erit Z9 dupla QS. &amp; &longs;i vtriu&longs;&shy;<lb/>
que potenti&aelig; velocitates mo&shy;<lb/>
tuum &longs;int &aelig;quales; patet pon&shy;<lb/>
dus V duplum pertran&longs;ire &longs;pa&shy;<lb/>
tium in eodem tempore <expan abbr="e&igrave;us">eius</expan>, <lb/>
quod pertran&longs;it pondus A.  </s>
     
<s id="id.2.1.165.6.1.1.0.a"> in eo <lb/>
dem enim tempore potentia in <lb/>
X peruenit ad Y, &amp; potentia in <lb/>
O ad P; ponderaq; &longs;imiliter in <lb/>
Z Q. quod erat demon&longs;tran&shy;<lb/>
dum.  </s>
     <lb/>
       

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure place="text">       </figure>       
<p id="id.2.1.165.6.2.1.0" type="caption">        
<s id="id.2.1.165.6.2.1.0.capt"> YYY </s>
     <lb/>
       
<s id="id.2.1.165.6.4.1.0"> PROPOSITIO XIII. </s>
    </p>       
<p id="id.2.1.165.7.0.0.0" type="main">        
<s id="id.2.1.165.7.1.1.0"> Fune circa &longs;ingulos duarum trochlearum <lb/>
orbiculos, quarum altera <expan abbr="&longs;upern&egrave;">&longs;uperne</expan>, altera <expan abbr="ver&ograve;">vero</expan> <lb/>
<expan abbr="infern&egrave;">inferne</expan>, ponderiq; alligata fuerit, reuoluto; <lb/>
altero etiam eius extremo inferiori trochle&aelig; re&shy;
<pb n="81"/>
ligata, altero autem <expan abbr="&agrave;">a</expan> mouente potentia deten&shy;<lb/>
to: erit decur&longs;um trahentis potenti&aelig; &longs;patium, mo<lb/>
ti ponderis &longs;patii triplum. </s>
    </p>       
<p id="id.2.1.165.8.0.0.0" type="main">        
<s id="id.2.1.165.8.1.1.0"> Sit pondus A; &longs;it BCD orbiculus tro<lb/>
chle&aelig; ponderi A ex EQ &longs;u&longs;pen&longs;o alligat&aelig;; <lb/>
&longs;itq; orbiculi centrum E; &longs;it deinde FGH <lb/>
orbiculus trochle&aelig; &longs;ur&longs;um appen&longs;&aelig;, cuius <lb/>
centrum k; &longs;itq; funis LFGHDCBM <lb/>
circa omnes reuolutus orbiculos, tro&shy;<lb/>
chle&aelig;q; inferiori in L religatus: &longs;itq; in <lb/>
M potentia mouens.  </s>
             
<s id="id.2.1.165.8.1.2.0"> dico &longs;patium de&shy;<lb/>
cur&longs;um <expan abbr="&agrave;">a</expan> potentia in M, dum mouet pon<lb/>
dus, triplum e&longs;&longs;e &longs;patii moti ponderis A.  </s>
     
<s id="id.2.1.165.8.1.2.0.a"> <lb/>
Moueatur potentia in M v&longs;q; ad N; &amp; <lb/>
centrum E &longs;it motum v&longs;q; ad O; &amp; L v&longs; <lb/>
que ad P; atq; pondus A, hoc e&longs;t pun&shy;<lb/>
ctum Q v&longs;q; ad R; orbiculu&longs;q; motus, &longs;it <lb/>
TSV. ducantur per EO line&aelig; ST BD <lb/>
horizonti &aelig;quidi&longs;tantes, qu&aelig; inter &longs;e &longs;e <lb/>
quoq; &aelig;quidi&longs;tantes erunt.  </s>
             
<s id="id.2.1.165.8.1.3.0"> quoniam au<lb/>
tem dum E e&longs;t in O, punctum Q e&longs;t in <lb/>
R; erit EQ &aelig;qualis OR, &amp; EO ip&longs;i QR <lb/>
&aelig;qualis; &longs;imiliter LQ &aelig;qualis erit PR, <lb/>
&amp; L P ip&longs;i QR &aelig;qualis.  </s>
             
<s id="id.2.1.165.8.1.4.0"> tres igitur QR <lb/>
EO LP inter &longs;e &longs;e &aelig;quales erunt; quibus <lb/>
etiam &longs;unt &aelig;quales BS DT.  </s>
     
<s id="id.2.1.165.8.1.4.0.a"> &amp; quoniam fu<lb/>
nis LFGHDCBM &aelig;qualis e&longs;t funi PF <lb/>
GHTVSN, <expan abbr="c&ugrave;m">cum</expan> &longs;it idem funis, &amp; qui <lb/>
circa &longs;emicirculum TVS e&longs;t &aelig;qualis funi <lb/>
circa &longs;emicirculum BCD; demptis igi<lb/>
tur communibus PFGHT' &amp; SM; erit <lb/>
reliquus MN tribus BS LP DT &longs;imul <lb/>
&longs;umptis &aelig;qualis.  </s>
             
<s id="id.2.1.165.8.1.5.0"> BS <expan abbr="ver&ograve;">vero</expan> LP DT &longs;imul <lb/>
tripli &longs;unt EO, &amp; ex con&longs;equenti QR. <lb/>
<arrow.to.target n="fig149"></arrow.to.target>
<pb/>
&longs;patium igitur MN translat&aelig; potenti&aelig; &longs;patii QR ponderis mo<lb/>
ti triplum erit. quod erat demon&longs;trandum.  </s>
             
<s id="id.2.1.165.8.1.6.0"> [quod erat demon&longs;trandum.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig149" place="text">       </figure>       
<p id="id.2.1.165.8.2.1.0" type="caption">        
<s id="id.2.1.165.8.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.165.9.0.0.0" type="main">        
<s id="id.2.1.165.9.1.1.0"> Tempus quoq; huius motus manife&longs;tum e&longs;t, eadem enim po<lb/>
tentia in &aelig;quali tempore &longs;patio <expan abbr="&longs;ecund&ugrave;m">&longs;ecundum</expan> triplum ampliori &longs;ine <lb/>
huiu&longs;modi trochleis idem pondus mouebit, <expan abbr="qu&agrave;m">quam</expan> cum ei&longs;dem <lb/>
hoc modo accomodatis.  </s>
             
<s id="id.2.1.165.9.1.2.0"> &longs;patium ponderis &longs;ine trochleis moti <lb/>
&aelig;quale e&longs;t &longs;patio potenti&aelig;. &amp; hoc modo in omnibus inueniemus <lb/>
tempus. </s>
             
<s id="id.2.1.165.9.1.3.0"> [&amp; hoc modo in omnibus inueniemus <lb/>
tempus.]  </s>
    </p>       
<p id="id.2.1.165.10.0.0.0" type="head">        
<s id="id.2.1.165.10.1.1.0"> PROPOSITIO XIIII. </s>
    </p>       
<p id="id.2.1.165.11.0.0.0" type="main">        
<s id="id.2.1.165.11.1.1.0"> Fune circa tres duarum trochlearum orbicu<lb/>
los, quarum altera <expan abbr="&longs;upern&egrave;">&longs;uperne</expan> vnico dumtaxat, al <lb/>
tera <expan abbr="ver&ograve;">vero</expan> <expan abbr="in&longs;ern&egrave;">in&longs;erne</expan>, duobus autem in&longs;ignita or&shy;<lb/>
biculis, <expan abbr="ponderi&qacute;ue">ponderique</expan> alligata fuerit, reuoluto; <lb/>
altero eius e&longs;tremo alicubi religato, altero autem <lb/>
<expan abbr="&agrave;">a</expan> potentia pondus mouente detento: erit decur&shy;<lb/>
&longs;um trahentis potenti&aelig; &longs;patium moti ponderis <lb/>
&longs;patii quadruplum.  </s>
    </p>       
<pb n="82"/>
       
<p id="id.2.1.165.13.0.0.0" type="main">        
<s id="id.2.1.165.13.1.1.0"> Sit pondus A, &longs;int duo orbiculi, <expan abbr="quor&umacr;">quorum</expan> <expan abbr="c&etilde;">cem</expan> <lb/>
tra k I trochle&aelig; ponderi alligat&aelig; k <foreign lang="greek">a</foreign>; ita vt <lb/>
pondus motum trochle&aelig; &longs;ur&longs;um, &amp; deor&longs;um <lb/>
&longs;emper &longs;equatur: &longs;it deinde orbiculus, cuius cen<lb/>
trum L, trochle&aelig; &longs;ur&longs;um appen&longs;&aelig; in &lt;35&gt;; &longs;itq; <lb/>
funis circa omnes orbiculos circumuolutus BC<lb/>
DEFGHZMNO, religatu&longs;q; in B; &longs;itq; po<lb/>
tentia in O mouens pondus A.  </s>
     
<s id="id.2.1.165.13.1.1.0.a"> dico &longs;patium, <lb/>
quod mouendo pertran&longs;it potentia in O, qua&shy;<lb/>
druplum e&longs;&longs;e &longs;patii moti ponderis A.  </s>
     
<s id="id.2.1.165.13.1.1.0.b"> mouean<lb/>
tur orbiculi trochle&aelig; ponderi alligat&aelig;; &amp; dum <lb/>
centrum k e&longs;t in R, centrum I &longs;it in S, &amp; pon<lb/>
dus A, hoc e&longs;t punctum <foreign lang="greek">a</foreign> in <foreign lang="greek">b</foreign>: erunt IS kR <lb/>
<foreign lang="greek">ab</foreign> inter &longs;e &longs;e &aelig;quales, itemq; k I ip&longs;i RS e&shy;<lb/>
rit &aelig;qualis.  </s>
             
<s id="id.2.1.165.13.1.2.0"> orbiculi enim inter &longs;e &longs;e eandem <lb/>
&longs;emper &longs;eruant di&longs;tantiam; &amp; k <foreign lang="greek">a</foreign> ip&longs;i R <foreign lang="greek">b</foreign> &aelig;&shy;<lb/>
qualis erit.  </s>
             
<s id="id.2.1.165.13.1.3.0"> ducantur per orbiculorum centra <lb/>
line&aelig; FH QT EC VX NZ horizonti &aelig;qui<lb/>
di&longs;tantes, qu&aelig; tangent funes in FHQTEC <lb/>
VX NZ punctis, &amp; inter &longs;e &longs;e quoq; &aelig;quidi<lb/>
&longs;tantes erunt: &amp; EQ CT VN XZ non &longs;o<lb/>
lum inter &longs;e &longs;e, &longs;ed etiam ip&longs;is IS KR <foreign lang="greek">ab</foreign> &aelig;qua<lb/>
les erunt.  </s>
             
<s id="id.2.1.165.13.1.4.0"> &amp; dum centra kI &longs;unt in RS, po<lb/>
tentia in O &longs;it mota in P.  </s>
     
<s id="id.2.1.165.13.1.4.0.a"> &amp; quoniam funis <lb/>
BCDEFGHZMNO e&longs;t &aelig;qualis funi BT9 <lb/>
QFGHXYVP, e&longs;t enim <expan abbr="id&etilde;">idem</expan> funis, &amp; funes cir<lb/>
<arrow.to.target n="fig150"></arrow.to.target><lb/>
ca T9Q XYV &longs;emicirculos &longs;unt &aelig;quales funibus, qui &longs;unt circa <lb/>
CDE ZMN; Demptis igitur communibus BT, QF GHX, <lb/>
&amp; VO; erit OP &aelig;qualis ip&longs;is VN XZ CT QE &longs;imul &longs;umptis.  </s>
             
<s id="id.2.1.165.13.1.5.0"> <lb/>
quatuor <expan abbr="ver&ograve;">vero</expan> VN ZX CT QE &longs;unt inter&longs;e &longs;e &aelig;quales, &amp; &longs;imul <lb/>
quadrupl&aelig; kR, &amp; <foreign lang="greek">ab</foreign>; quare OP quadrupla erit ip&longs;ius <foreign lang="greek">ab</foreign>.  </s>
             
<s id="id.2.1.165.13.1.6.0"> &longs;pa<lb/>
tium igitur potenti&aelig; quadruplum e&longs;t &longs;patii ponderis. quod erat <lb/>
o&longs;tendendum.  </s>
             
<s id="id.2.1.165.13.1.7.0"> [quod erat <lb/>
o&longs;tendendum.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig150" place="text">       </figure>       
<p id="id.2.1.165.13.2.1.0" type="caption">        
<s id="id.2.1.165.13.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.165.14.0.0.0" type="main">        
<s id="id.2.1.165.14.1.1.0"> Et &longs;i funis in P circa alium adhuc reuoluatur orbiculum ver&longs;us <lb/>
&lt;35&gt;, <expan abbr="potentiaqu&eacute;">potentiaque</expan> mouendo &longs;e deor&longs;um moueat &longs;ur&longs;um pondus; &longs;imi <lb/>
liter o&longs;tendetur &longs;patium potenti&aelig; quadruplum e&longs;&longs;e &longs;patii ponderis.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.165.16.0.0.0" type="main">        
<s id="id.2.1.165.16.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> funis in B circumuoluatur al<lb/>
teri orbiculo, qui deinde trochle&aelig; in&shy;<lb/>
<arrow.to.target n="note252"></arrow.to.target> feriori religetur; erit potentia in O <lb/>
&longs;u&longs;tinens pondus A &longs;ubquintupla pon<lb/>
deris.  </s>
             
<s id="id.2.1.165.16.1.2.0"> &amp; &longs;i in O &longs;it potentia mouens <lb/>
pondus A; &longs;imiliter demon&longs;trabitur <lb/>
&longs;patium potenti&aelig; in O quintuplum e&longs; <lb/>
&longs;e &longs;patii ponderis A. <lb/>
<arrow.to.target n="fig151"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.165.17.0.0.0" type="main">        
<s id="id.2.1.165.17.1.1.0"> Et &longs;i funis ita circa orbiculos apte&shy;<lb/>
tur, vt potentia in O &longs;u&longs;tinens pon&shy;<lb/>
dus &longs;it ponderis &longs;ub&longs;extupla; &amp; loco <lb/>
potenti&aelig; &longs;u&longs;tinentis ponatur in O po&shy;<lb/>
tentia mouens pondus: eodem modo <lb/>
o&longs;tendetur &longs;patium potenti&aelig; &longs;extu&shy;<lb/>
plum e&longs;&longs;e &longs;patii ponderis moti.  </s>
             
<s id="id.2.1.165.17.1.2.0"> &amp; &longs;ic <lb/>
procedendo in infinitum proportiones <lb/>
&longs;patii potenti&aelig; ad &longs;patium ponderis <lb/>
moti quotcunq; multiplices inuenien&shy;<lb/>
tur.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig151" place="text">       </figure>       
<p id="id.2.1.165.17.2.1.0" type="caption">        
<s id="id.2.1.165.17.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.166.1.0.0.0" type="margin">        
<s id="id.2.1.166.1.1.1.0"> <margin.target id="note252"></margin.target>9 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.167.1.0.0.0" type="head">        
<s id="id.2.1.167.1.1.1.0"> COROLLARIVM I. </s>
    </p>       
<p id="id.2.1.167.2.0.0.0" type="main">        
<s id="id.2.1.167.2.1.1.0"> Ex his manife&longs;tum e&longs;t ita &longs;e habere pondus <lb/>
ad potentiam ip&longs;um &longs;u&longs;tinentem, &longs;icuti &longs;patium <lb/>
potenti&aelig; mouentis ad &longs;patium ponderis moti. </s>
    </p>       
<p id="id.2.1.167.3.0.0.0" type="main">        
<s id="id.2.1.167.3.1.1.0"> Vt &longs;i pondus A quintuplum &longs;it potenti&aelig; in O pondus A &longs;u&longs;ti&shy;<lb/>
nentis; erit &amp; &longs;patium OP potenti&aelig; pondus mouentis quintuplum <lb/>
&longs;patii <foreign lang="greek">ab</foreign> ponderis moti.  </s>
    </p>       
<p id="id.2.1.167.4.0.0.0" type="head">        
<pb n="83"/>
       
<s id="id.2.1.167.5.1.1.0"> COROLLARIVM II. </s>
    </p>       
<p id="id.2.1.167.6.0.0.0" type="main">        
<s id="id.2.1.167.6.1.1.0"> Patet etiam per ea, qu&aelig; dicta &longs;unt, orbiculos <lb/>
trochle&aelig;, qu&aelig; ponderi e&longs;t alligata, efficere; vt <expan abbr="&agrave;">a</expan> <lb/>
moto pondere minus, <expan abbr="qu&agrave;m">quam</expan> <expan abbr="&agrave;">a</expan> trahente poten&shy;<lb/>
tia de&longs;cribatur &longs;patium; maioriq; tempore datum <lb/>
&aelig;quale &longs;patium de&longs;cribi, <expan abbr="qu&agrave;m">quam</expan> &longs;ine illis.  </s>
             
<s id="id.2.1.167.6.1.2.0"> quod <lb/>
quidem orbiculi trochle&aelig; &longs;uperioris non effi&shy;<lb/>
ciunt. </s>
    </p>       
<p id="id.2.1.167.7.0.0.0" type="main">        
<s id="id.2.1.167.7.1.1.0"> Multiplici o&longs;ten&longs;a ponderis ad potentiam proportione, iam ex <lb/>
aduer&longs;o potenti&aelig; ad pondus proportio multiplex o&longs;tendatur.  </s>
    </p>       
<p id="id.2.1.167.8.0.0.0" type="head">        
<s id="id.2.1.167.8.1.1.0"> PROPOSITIO XV. </s>
    </p>       
<p id="id.2.1.167.9.0.0.0" type="main">        
<s id="id.2.1.167.9.1.1.0"> Si funis orbiculo trochle&aelig; <expan abbr="&agrave;">a</expan> potentia &longs;ur&longs;um <lb/>
detent&aelig; fuerit circumuolutus; altero eius extre&shy;<lb/>
mo alicubi religato, alteri <expan abbr="ver&ograve;">vero</expan> pondere appen<lb/>
&longs;o; dupla erit ponderis potentia.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.167.11.0.0.0" type="main">        
<s id="id.2.1.167.11.1.1.0"> Sit trochlea habens orbiculum, cuius <lb/>
centrum A; &amp; &longs;it pondus B alligatum fu<lb/>
ni CDEFG, qui circa orbiculum &longs;it re&shy;<lb/>
uolutus, ac tandem religatus in G: &longs;itq; <lb/>
potentia in H &longs;u&longs;tinens pondus.  </s>
             
<s id="id.2.1.167.11.1.2.0"> dico po<lb/>
tentiam in H duplam e&longs;&longs;e ponderis B. du<lb/>
catur DF per <expan abbr="centr&utilde;">centrum</expan> A horizonti &aelig;quidi<lb/>
&longs;tans.  </s>
             
<s id="id.2.1.167.11.1.3.0"> <expan abbr="quoni&atilde;">quoniam</expan> igitur potentia in H &longs;u&longs;tinet <lb/>
<expan abbr="trochle&atilde;">trochleam</expan>, qu&aelig; &longs;u&longs;tinet <expan abbr="orbicul&utilde;in">orbiculunin</expan> eius <expan abbr="c&etilde;tro">centro</expan> <lb/>
A, qui pondus &longs;u&longs;tinet; erit potentia &longs;u&longs;ti<lb/>
nens <expan abbr="orbicul&utilde;">orbiculum</expan>, ac &longs;i in A <expan abbr="c&otilde;&longs;tituta">con&longs;tituta</expan> e&longs;&longs;et; ip&longs;a <lb/>
ergo in A exi&longs;tente, pondere <expan abbr="ver&ograve;">vero</expan> in D <lb/>
appen&longs;o, funiq; CD religato; erit DF <lb/>
tanquam vectis, cuius fulcimentum erit <lb/>
F, pondus in D, &amp; potentia in A.  </s>
     
<s id="id.2.1.167.11.1.3.0.a"> po&shy;<lb/>
<arrow.to.target n="note253"></arrow.to.target> tentia <expan abbr="ver&ograve;">vero</expan> ad pondus e&longs;t, vt DF ad <lb/>
ad FA, &amp; DF dupla e&longs;t ip&longs;ius FA; Po&shy;<lb/>
<arrow.to.target n="fig152"></arrow.to.target><lb/>
tentia igitur in A, &longs;iue in H, quod idem e&longs;t, ponderis B dupla erit. <lb/>
quod demon&longs;trare oportebat. </s>
             
<s id="id.2.1.167.11.1.4.0"> [<lb/>
quod demon&longs;trare oportebat.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig152" place="text">       </figure>       
<p id="id.2.1.167.11.2.1.0" type="caption">        
<s id="id.2.1.167.11.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.168.1.0.0.0" type="margin">        
<s id="id.2.1.168.1.1.1.0"> <margin.target id="note253"></margin.target>3 <emph type="italics"/>Huius. de vecte.<emph.end type="italics"/> </s>
             
<s id="id.2.1.168.1.1.2.0"> [de vecte.<emph.end type="italics"/>] </s>
    </p>       
<p id="id.2.1.169.1.0.0.0" type="main">        
<s id="id.2.1.169.1.1.1.0"> Pr&aelig;terea con&longs;iderandum occurrit, <expan abbr="c&ugrave;m">cum</expan> h&aelig;c omnia maneant, <lb/>
idem e&longs;&longs;e vnico exi&longs;tente fune CD EFG hoc modo orbiculo cicum <lb/>
uoluto, ac &longs;i duo e&longs;&longs;ent funes CD FG in vecte &longs;iue libra DF al&shy;<lb/>
ligati.  </s>
    </p>       
<p id="id.2.1.169.2.0.0.0" type="head">        
<s id="id.2.1.169.2.1.1.0"> ALITER. </s>
    </p>       
<p id="id.2.1.169.3.0.0.0" type="main">        
<s id="id.2.1.169.3.1.1.0"> Ii&longs;dem po&longs;itis, &longs;i in G appen&longs;um e&longs;&longs;et pondus k &aelig;quale pon&shy;<lb/>
deri B, pondera B k &aelig;queponderabunt in libra DF, cuius centrum <lb/>
A.  </s>
     
<s id="id.2.1.169.3.1.1.0.a"> potentia <expan abbr="ver&ograve;">vero</expan> in H &longs;u&longs;tinens pondera Bk e&longs;t ip&longs;is &longs;imul &longs;um<lb/>
ptis &aelig;qualis, &amp; pondera BK ip&longs;ius B &longs;unt dupla; potentia ergo in <lb/>
H ponderis B dupla erit.  </s>
             
<s id="id.2.1.169.3.1.2.0"> &amp; quoniam funis religatus in G nihil a&shy;<lb/>
liud efficit, ni&longs;i <expan abbr="qu&ograve;d">quod</expan> pondus B &longs;u&longs;tinet, ne de&longs;cendat; quod idem <lb/>
efficit pondus k in G appen&longs;um: potentia igitur in H &longs;u&longs;tinens <lb/>
pondus B, fune religato in G, dupla e&longs;t ponderis B. quod de&shy;<lb/>
mon&longs;trare oportebat.  </s>
    </p>       
<p id="id.2.1.169.4.0.0.0" type="head">        
<pb n="84"/>
       
<s id="id.2.1.169.5.1.1.0"> PROPOSITIO XVI. </s>
    </p>       
<p id="id.2.1.169.6.0.0.0" type="main">        
<s id="id.2.1.169.6.1.1.0"> Ii&longs;dem po&longs;itis &longs;i in H &longs;it potentia mouens pon<lb/>
dus, mouebit h&aelig;c eadem vecte horizonti &longs;em&shy;<lb/>
per &aelig;quidi&longs;tante </s>
    </p>       
<p id="id.2.1.169.7.0.0.0" type="main">        
<s id="id.2.1.169.7.1.1.0"> Hoc etiam (&longs;icut in &longs;uperioribus dictum <lb/>
e&longs;t) o&longs;tendetur.  </s>
             
<s id="id.2.1.169.7.1.2.0"> moueatur enim orbiculus <lb/>
&longs;ur&longs;um, po&longs;itionemq; habeat MNO, cuius <lb/>
centrum L: &amp; per L ducatur MLO ip&longs;i DF, <lb/>
&amp; horizonti &aelig;quidi&longs;tans.  </s>
             
<s id="id.2.1.169.7.1.3.0"> &amp; quoniam funes <lb/>
tangunt circulum MON in punctis MO; <lb/>
ideo <expan abbr="c&ugrave;m">cum</expan> potentia in A, &longs;eu in H, quod <lb/>
idem e&longs;t, moueat pondus B in D appen&longs;um <lb/>
vecte DF, cuius fulcimentum e&longs;t F; &longs;emper <lb/>
adhuc remanebit alius vectis, .vt MO hori<lb/>
zonti &aelig;quidi&longs;tans, ita vt &longs;emper potentia <lb/>
moueat pondus vecte horizonti &aelig;quidi&longs;tan<lb/>
te, cuius fulcimentum e&longs;t &longs;emper in linea <lb/>
OG, &amp; pondus in MC, potentiaq; in cen<lb/>
tro orbiculi. <arrow.to.target n="fig153"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.169.8.0.0.0" type="main">        
<s id="id.2.1.169.8.1.1.0"> Ii&longs;dem po&longs;itis, &longs;patium ponderis moti duplum <lb/>
e&longs;t &longs;patii potenti&aelig; mouentis.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig153" place="text">       </figure>       
<p id="id.2.1.169.8.2.1.0" type="caption">        
<s id="id.2.1.169.8.2.1.0.capt"> YYY </s>
    </p>       
<pb/>
       
<p id="id.2.1.169.10.0.0.0" type="main">        
<s id="id.2.1.169.10.1.1.0"> Sit motus orbiculus <expan abbr="&agrave;">a</expan> centro A <lb/>
v&longs;q; ad centrum L; &amp; pondus B, <lb/>
hoc e&longs;t punctum C, in eodem tem&shy;<lb/>
pore&longs;it motum in P; &amp; potentia in <lb/>
H v&longs;q; ad K; erit AH ip&longs;i LK &aelig;qua <lb/>
lis, &amp; AL ip&longs;i Hk.  </s>
             
<s id="id.2.1.169.10.1.2.0"> &amp; quoniam fu<lb/>
nis CDEFG e&longs;t &aelig;qualis funi PM <lb/>
NOG, idem enim e&longs;t funis, &amp; fu <lb/>
nis circa &longs;emicirculum MNO &aelig;&shy;<lb/>
qualis e&longs;t funi circa &longs;emicirculum <lb/>
DEF; demptis igitur communi&shy;<lb/>
bus DP FG, erit PC &aelig;qualis <lb/>
DM FO &longs;imul &longs;umptis, qui funes <lb/>
&longs;unt dupli ip&longs;ius AL, &amp; con&longs;equen&shy;<lb/>
ter ip&longs;ius Hk.  </s>
             
<s id="id.2.1.169.10.1.3.0"> &longs;patium ergo pon<lb/>
deris moti CP duplum e&longs;t &longs;patii <lb/>
Hk potenti&aelig;. quod oportebat de&shy;<lb/>
mon&longs;trare. </s>
             
<s id="id.2.1.169.10.1.4.0"> [quod oportebat de&shy;<lb/>
mon&longs;trare.]  </s>
     <lb/>
       

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure place="text">       </figure>       
<p id="id.2.1.169.10.2.1.0" type="caption">        
<s id="id.2.1.169.10.2.1.0.capt"> YYY </s>
     <lb/>
       
<s id="id.2.1.169.10.4.1.0"> COROLLARIVM </s>
    </p>       
<p id="id.2.1.169.11.0.0.0" type="main">        
<s id="id.2.1.169.11.1.1.0"> Ex hoc manife&longs;tum e&longs;t, idem pondus trahi <lb/>
ab eadem potentia in &aelig;quali tempore per du&shy;<lb/>
plum &longs;patium trochlea hoc modo accommoda<lb/>
ta, <expan abbr="qu&agrave;m">quam</expan> &longs;ine trochlea; dummodo ip&longs;ius poten<lb/>
ti&aelig; lationes in velocitate &longs;int &aelig;quales. </s>
    </p>       
<p id="id.2.1.169.12.0.0.0" type="main">        
<s id="id.2.1.169.12.1.1.0"> Spatium enim ponderis moti &longs;ine trochlea &aelig;quale e&longs;t &longs;patio <lb/>
potenti&aelig;.  </s>
    </p>       
<pb n="85"/>
       
<p id="id.2.1.169.14.0.0.0" type="main">        
<s id="id.2.1.169.14.1.1.0"> Si autem funis in G circa alium reuoluatur <lb/>
orbiculum, cuius centrum k; &longs;itq; huiu&longs;mo<lb/>
di orbiculi trochlea deor&longs;um affixa, qu&aelig; nul<lb/>
lum alium habeat motum, ni&longs;i liberam orbi <lb/>
culi circa axem reuolutionem; funi&longs;q; relige<lb/>
tur in M; erit potentia in H &longs;u&longs;tinens pondus <lb/>
B &longs;imiliter ip&longs;ius ponderis dupla.  </s>
             
<s id="id.2.1.169.14.1.2.0"> quod qui <lb/>
dem manife&longs;tum e&longs;t, <expan abbr="c&ugrave;m">cum</expan> idem pror&longs;us &longs;it, <lb/>
&longs;iue funis &longs;it religatus in M, &longs;iue in G. orbicu<lb/>
lus enim, cuius centrum k, nihil efficit; penitu&longs; <lb/>
<expan abbr="qu&eacute;">que</expan> inutilis e&longs;t. <arrow.to.target n="fig154"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.169.15.0.0.0" type="main">        
<s id="id.2.1.169.15.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> &longs;it potentia in M &longs;u&longs;tinens pon<lb/>
dus B, &amp; trochlea &longs;uperior &longs;it &longs;ur&longs;um appen<lb/>
&longs;a; erit potentia in M &aelig;qualis ponderi B. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig154" place="text">       </figure>       
<p id="id.2.1.169.15.2.1.0" type="caption">        
<s id="id.2.1.169.15.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.169.16.0.0.0" type="main">        
<s id="id.2.1.169.16.1.1.0"> Quoniam enim potentia in G &longs;u&longs;tinens <arrow.to.target n="note254"></arrow.to.target><lb/>
pondus B &aelig;qualis e&longs;t ponderi B, &amp; ip&longs;i po<lb/>
tenti&aelig; in G &aelig;qualis e&longs;t potentia in L; e&longs;t <lb/>
enim GL vectis, cuius fulcimentum e&longs;t k; <lb/>
&amp; di&longs;tantia Gk di&longs;tanti&aelig; kL e&longs;t &aelig;qualis; <lb/>
erit igitur potentia in L, &longs;iue (quod idem e&longs;t) <lb/>
in M, ponderi B &aelig;qualis. </s>
    </p>       
<p id="id.2.1.170.1.0.0.0" type="margin">        
<s id="id.2.1.170.1.1.1.0"> <margin.target id="note254"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.171.1.0.0.0" type="main">        
<s id="id.2.1.171.1.1.1.0"> Huiu&longs;modi autem motus fit vectibus DF LG, quorum fulci <lb/>
menta &longs;unt kA, &amp; pondus in D, &amp; potentia in F. &longs;ed in vecte <lb/>
LG potentia e&longs;t in L, pondus <expan abbr="ver&ograve;">vero</expan>, ac &longs;i e&longs;&longs;et in G. </s>
    </p>       
<p id="id.2.1.171.2.0.0.0" type="main">        
<s id="id.2.1.171.2.1.1.0"> Si deinde in M &longs;it potentia mouens pondus, transferaturq; po<lb/>
tentia in N, pondus autem motum fuerit v&longs;q; ad O; erit MN <lb/>
&longs;patium potenti&aelig; &aelig;quale &longs;patio CO ponderis.  </s>
             
<s id="id.2.1.171.2.1.2.0"> <expan abbr="C&ugrave;m">Cum</expan> enim funis <lb/>
MLGFDC &aelig;qualis &longs;it funi NLGFDO. e&longs;t enim idem funis; <lb/>
dempto communi MLGFDO; erit &longs;patium MN potenti&aelig; &aelig;&shy;<lb/>
quale &longs;patio CO ponderis. </s>
    </p>       
<p id="id.2.1.171.3.0.0.0" type="main">        
<s id="id.2.1.171.3.1.1.0"> Et &longs;i funis in M circa plures reuoluatur orbiculos, &longs;emper erit <lb/>
potentia altero eius extremo pondus &longs;u&longs;tinens &aelig;qualis ip&longs;i ponderi.  </s>
             
<s id="id.2.1.171.3.1.2.0"> <lb/>
&longs;patiaq; ponderis, atq; potenti&aelig; mouentis &longs;emper o&longs;tendentur <lb/>
&aelig;qualia.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.171.5.0.0.0" type="head">        
<s id="id.2.1.171.5.1.1.0"> PROPOSITIO XVII. </s>
    </p>       
<p id="id.2.1.171.6.0.0.0" type="main">        
<s id="id.2.1.171.6.1.1.0"> Si vtri&longs;q; duarum trochlearum &longs;ingulis orbicu<lb/>
lis, quarum vna <expan abbr="&longs;upern&egrave;">&longs;uperne</expan> <expan abbr="&agrave;">a</expan> potentia &longs;u&longs;tineatur, <lb/>
altera <expan abbr="ver&ograve;">vero</expan> <expan abbr="infern&egrave;">inferne</expan>, ibiq; affixa, con&longs;tituta fue&shy;<lb/>
rit, funis circumducatur; altero eius extremo &longs;u<lb/>
periori trochle&aelig; religato, alteri <expan abbr="ver&ograve;">vero</expan> pondere <lb/>
appen&longs;o; tripla erit ponderis potentia. </s>
    </p>       
<p id="id.2.1.171.7.0.0.0" type="main">        
<s id="id.2.1.171.7.1.1.0"> Sit orbiculus, cuius centrum A, tro&shy;<lb/>
chle&aelig; <expan abbr="infern&egrave;">inferne</expan> affix&aelig;; &amp; &longs;it funis BCD <lb/>
EFG non &longs;olum huic orbiculo circumuo<lb/>
lutus, <expan abbr="ver&ugrave;m">verum</expan> etiam orbiculo trochle&aelig; &longs;u&shy;<lb/>
perioris, cuius centrum k; &longs;itq; funis in <lb/>
B &longs;uperiori trochle&aelig; religatus; &amp; in G &longs;it ap<lb/>
pen&longs;um pondus H; potentiaq; in L &longs;u&longs;ti<lb/>
neat pondus H.  </s>
     
<s id="id.2.1.171.7.1.1.0.a"> dico potentiam in L tri&shy;<lb/>
plam e&longs;&longs;e ponderis H.  </s>
     
<s id="id.2.1.171.7.1.1.0.b"> &longs;i enim du&aelig; e&longs;&longs;ent <lb/>
potenti&aelig; pondus H &longs;u&longs;tidentes, vna in <lb/>
K, altera in B, erunt vtr&aelig;q; &longs;imul tripl&aelig; <lb/>
<arrow.to.target n="note255"></arrow.to.target> ponderis H potentia enim in k dupla e&longs;t <lb/>
ponderis H, &amp; potentia in B ip&longs;i ponderi <lb/>
&aelig;qualis.  </s>
             
<s id="id.2.1.171.7.1.2.0"> &amp; quoniam &longs;ola potentia in L <lb/>
vtri&longs;q; &longs;cilicet potenti&aelig; in KB e&longs;t &aelig;qua&shy;<lb/>
lis.  </s>
             
<s id="id.2.1.171.7.1.3.0"> &longs;u&longs;tinet enim potentia in L; <expan abbr="t&ugrave;m">tum</expan> po&shy;<lb/>
tentiam in K, <expan abbr="t&ugrave;m">tum</expan> potentiam in B; idem <lb/>
<expan abbr="qu&eacute;">que</expan> efficit potentia in L, ac &longs;i du&aelig; e&longs;&longs;ent <lb/>
potenti&aelig;, vna in k, altera in B: Tri&shy;<lb/>
pla igitur erit potentia in L ponderis H. <lb/>
quod der&lt;*&gt;on&longs;trare o&lt;*&gt;ortebat. <arrow.to.target n="fig155"></arrow.to.target> </s>
    </p>       
<pb n="86"/>
       
<p id="id.2.1.171.9.0.0.0" type="main">        
<s id="id.2.1.171.9.1.1.0"> Si autem in L &longs;it potentia mouens pondus.  </s>
             
<s id="id.2.1.171.9.1.2.0"> di<lb/>
co &longs;patium ponderis moti triplum e&longs;&longs;e &longs;patii po&shy;<lb/>
tenti&aelig; mot&aelig;. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig155" place="text">       </figure>       
<p id="id.2.1.171.9.2.1.0" type="caption">        
<s id="id.2.1.171.9.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.172.1.0.0.0" type="margin">        
<s id="id.2.1.172.1.1.1.0"> <margin.target id="note255"></margin.target>15 <emph type="italics"/>Huius.  </s>
             
<s id="id.2.1.172.1.1.2.0"> In pr&aelig;cedenti.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.173.1.0.0.0" type="main">        
<s id="id.2.1.173.1.1.1.0"> Moueatur centrum or&shy;<lb/>
biculi K v&longs;q; ad M; cuius <lb/>
quidem motus &longs;patium <lb/>
mot&aelig; potenti&aelig; &longs;patio e&longs;t <arrow.to.target n="note256"></arrow.to.target><lb/>
&aelig;quale, &longs;icuti &longs;upra dictum <lb/>
e&longs;t: &amp; quando k erit in M, <lb/>
B erit in N; &amp; NB &aelig;qualis <lb/>
erit M k; &amp; dum k e&longs;t in M, <lb/>
&longs;it pondus H, hoc e&longs;t pun<lb/>
ctum G motum in O; &amp; per <lb/>
MK ducantur EF PQ ho<lb/>
rizonti &aelig;quidi&longs;tantes; erit <lb/>
vnaqu&aelig;q; EP BN FQ ip<lb/>
&longs;i KM &aelig;qualis.  </s>
             
<s id="id.2.1.173.1.1.2.0"> &amp; quoniam <lb/>
funis BCDEFG &aelig;qualis <lb/>
e&longs;t funi NCDPQO; <lb/>
idem enim e&longs;t funis; &amp; fu&shy;<lb/>
nis circa &longs;emicirculum ER <lb/>
F &aelig;qualis e&longs;t funi circa &longs;e&shy;<lb/>
micirculum PSQ: dem&shy;<lb/>
ptis igitur communibus <lb/>
BCDE, &amp; FO, erit OG <lb/>
tribus QF NB PE &longs;imul <lb/>
&longs;umptis &aelig;qualis.  </s>
             
<s id="id.2.1.173.1.1.3.0"> &longs;ed QF <lb/>
NB PE &longs;imul tripl&aelig; &longs;unt <lb/>
Mk, hoc e&longs;t &longs;patii poten&shy;<lb/>
ti&aelig; mot&aelig;; &longs;patium ergo <lb/>
GO ponderis H moti tri&shy;<lb/>
<arrow.to.target n="fig156"></arrow.to.target><lb/>
plum e&longs;t &longs;patii potenti&aelig; mot&aelig;. quod o&longs;tendere oportebat. </s>
             
<s id="id.2.1.173.1.1.4.0"> [quod o&longs;tendere oportebat.]  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig156" place="text">       </figure>       
<p id="id.2.1.173.1.2.1.0" type="caption">        
<s id="id.2.1.173.1.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.174.1.0.0.0" type="margin">        
<s id="id.2.1.174.1.1.1.0"> <margin.target id="note256"></margin.target><emph type="italics"/>In pr&aelig;cedenti.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.175.1.0.0.0" type="head">        
<pb/>
       
<s id="id.2.1.175.1.2.1.0"> PROPOSITIO XVIII. </s>
    </p>       
<p id="id.2.1.175.2.0.0.0" type="main">        
<s id="id.2.1.175.2.1.1.0"> Si vtriu&longs;q; duarum trochlearum binis orbicu<lb/>
lis, quarum altera <expan abbr="&longs;upern&egrave;">&longs;uperne</expan> <expan abbr="&agrave;">a</expan> potentia &longs;u&longs;tineatur, <lb/>
altera <expan abbr="ver&ograve;">vero</expan> <expan abbr="infern&egrave;">inferne</expan>, ibiq; annexa, collocata fue&shy;<lb/>
rit, funis circumnectatur; altero eius extremo <lb/>
alicubi, non autem &longs;uperiori trochle&aelig; religato, <lb/>
alteri <expan abbr="ver&ograve;">vero</expan> pondere appen&longs;o; quadrupla erit <lb/>
ponderis potentia. </s>
    </p>       
<p id="id.2.1.175.3.0.0.0" type="main">        
<s id="id.2.1.175.3.1.1.0"> Sit trochlea inferior, duos habens orbiculos, <lb/>
quorum centra AB; &longs;it <expan abbr="qu&eacute;">que</expan> trochlea &longs;uperior <lb/>
duos &longs;imiliter habens orbiculos, quorum cen&shy;<lb/>
tra CD; funi&longs;q; EFGHKLMNOP &longs;it cir&shy;<lb/>
ca omnes orbiculos reuolutus, qui &longs;it religatus <lb/>
in E; &amp; in P appendatur pondus Q; &longs;itq; po&shy;<lb/>
tentia in R.  </s>
     
<s id="id.2.1.175.3.1.1.0.a"> dico potentiam in R quadruplam <lb/>
e&longs;&longs;e ponderis <expan abbr="q.">que</expan> <expan abbr="C&ugrave;m">Cum</expan> enim &longs;i du&aelig; intelligan<lb/>
tur potenti&aelig;, vna in k, altera in D, potentia <lb/>
<arrow.to.target n="note257"></arrow.to.target> in k &longs;u&longs;tinens pondus Q fune k LMNOP &aelig;&shy;<lb/>
qualis erit ponderi; erunt du&aelig; &longs;imul potenti&aelig;, <lb/>
vna in D, altera in k, pondus Q &longs;u&longs;tinentes, <lb/>
tripl&aelig; eiu&longs;dem ponderis.  </s>
             
<s id="id.2.1.175.3.1.2.0"> Potentia <expan abbr="ver&ograve;">vero</expan> in C <lb/>
dupla e&longs;t potenti&aelig; in k, &amp; per con&longs;equens pon<lb/>
deris Q; idem enim e&longs;t, ac &longs;i in k appen&longs;um e&longs; <lb/>
<arrow.to.target n="note258"></arrow.to.target> &longs;et pondus &aelig;quale ponderi Q, cuius dupla e&longs;t <lb/>
potentia in C; du&aelig; igitur potenti&aelig; in DC qua&shy;<lb/>
drupl&aelig; &longs;unt ponderis <expan abbr="q.">que</expan> &amp; <expan abbr="c&ugrave;m">cum</expan> potentia in R <lb/>
orbiculis &longs;u&longs;tineat pondus Q, erit <expan abbr="pot&etilde;tia">potentia</expan> in R, <lb/>
ac &longs;i du&aelig; e&longs;&longs;ent potenti&aelig;, vna in D, altera in C, <lb/>
&amp; vtr&aelig;q; &longs;imul pondus Q &longs;u&longs;tinerent.  </s>
             
<s id="id.2.1.175.3.1.3.0"> ergo po&shy;<lb/>
tentia in R quadrupla e&longs;t ponderis <expan abbr="q.">que</expan> quod <lb/>
oport&lt;*&gt;bat demon&longs;trare. <arrow.to.target n="fig157"></arrow.to.target> </s>
     
<pb n="87"/>
       
<s id="id.2.1.175.3.3.1.0"> COROLLARIVM </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig157" place="text">       </figure>       
<p id="id.2.1.175.3.4.1.0" type="caption">        
<s id="id.2.1.175.3.4.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.176.1.0.0.0" type="margin">        
<s id="id.2.1.176.1.1.1.0"> <margin.target id="note257"></margin.target>16 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.176.1.1.2.0"> <margin.target id="note258"></margin.target>15 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.177.1.0.0.0" type="main">        
<s id="id.2.1.177.1.1.1.0"> Ex quo patet, &longs;i funis fuerit religatus in G, &amp; <lb/>
circa orbiculos, quorum centra &longs;unt BCD reuo&shy;<lb/>
lutus; potentiam in R pondus &longs;u&longs;tinentem &longs;imili&shy;<lb/>
ter ponderis Q quadruplam e&longs;&longs;e.  </s>
             
<s id="id.2.1.177.1.1.2.0"> orbiculus enim, <lb/>
cuius centrum A, nihil efficit. </s>
    </p>       
<p id="id.2.1.177.2.0.0.0" type="main">        
<s id="id.2.1.177.2.1.1.0"> Si autem in R &longs;it potentia mouens pondus.  </s>
             
<s id="id.2.1.177.2.1.2.0"> dico <lb/>
&longs;patium ponderis moti quadruplum e&longs;&longs;e &longs;patii <lb/>
potenti&aelig;. </s>
    </p>       
<p id="id.2.1.177.3.0.0.0" type="main">        
<s id="id.2.1.177.3.1.1.0"> Moueantur centra CD orbiculorum v&longs;q; ad <lb/>
ST; erunt ex &longs;uperius dictis CS DT &longs;patio <lb/>
potenti&aelig; &aelig;qualia; &amp; per CSDT ducantur Hk <lb/>
VX NO YZ horizonti &aelig;quidi&longs;tantes; &amp; <expan abbr="d&utilde;">dum</expan> <lb/>
centra CD &longs;unt in ST, &longs;it pondus Q, hoc e&longs;t <lb/>
punctum P motum in 9. &amp; quoniam funis EF <lb/>
GHKLMNOP &aelig;qualis e&longs;t funi EFGVX <lb/>
LMYZ 9; <expan abbr="c&ugrave;m">cum</expan> &longs;it idem funis: &amp; funes circa <lb/>
&longs;emicirculos NIO H <foreign lang="greek">a</foreign> k &longs;unt &aelig;quales funi&shy;<lb/>
bus, qui &longs;unt circa &longs;emicirculos <expan abbr="Y&lt;35&gt;Z">Y&lt;35&gt;Z</expan> V<foreign lang="greek">b</foreign>X; <lb/>
demptis igitur communibus EFGH kLMN <lb/>
&amp; O9; erit P9 ip&longs;is NY ZO VH <emph type="italics"/>X<emph.end type="italics"/>k &longs;i&shy;<lb/>
mul &longs;umptis &aelig;qualis.  </s>
             
<s id="id.2.1.177.3.1.2.0"> quatuor autem NY ZO <lb/>
VH Xk &longs;imul quadrupli &longs;unt DT, hoc e&longs;t <lb/>
&longs;patii potenti&aelig;; &longs;patium igitur P9 ponderis <lb/>
quadruplum e&longs;t &longs;patii potenti&aelig; quod demon<lb/>
&longs;trandum fuerat. <arrow.to.target n="fig158"></arrow.to.target> </s>
    </p>       
<pb/>
       
<p id="id.2.1.177.5.0.0.0" type="main">        
<s id="id.2.1.177.5.1.1.0"> Si autem funis &longs;it re&shy;<lb/>
ligatus in E trochle&aelig; &longs;u<lb/>
periori, &amp; potentia in R <lb/>
&longs;u&longs;tineat pondus Q; e&shy;<lb/>
rit potentia in R ponde<lb/>
ris Q quintupla.  </s>
             
<s id="id.2.1.177.5.1.2.0"> &amp; &longs;i in <lb/>
R &longs;it potentia mouens <lb/>
pondus; erit &longs;patium pon<lb/>
deris moti quintuplum <lb/>
&longs;patii potenti&aelig;.  </s>
             
<s id="id.2.1.177.5.1.3.0"> qu&aelig; om&shy;<lb/>
nia &longs;imili modo o&longs;ten&shy;<lb/>
dentur, &longs;icut in pr&aelig;ce&shy;<lb/>
dentibus demon&longs;tra&shy;<lb/>
tum e&longs;t. <arrow.to.target n="fig159"></arrow.to.target> </s>
    </p>       
<pb n="88"/>
       
<p id="id.2.1.177.7.0.0.0" type="main">        
<s id="id.2.1.177.7.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> potentia in R &longs;ub&longs;tineat pon&shy;<lb/>
dus Q trochlea tres orbiculos habente, <lb/>
quorum centra &longs;int ABC; &amp; &longs;it alia tro<lb/>
chlea <expan abbr="infern&egrave;">inferne</expan> affixa duos, vel tres orbicu&shy;<lb/>
los habens, quorum centra DEF; &longs;itq; <lb/>
funis circa omnes orbiculos reuolutus, &longs;i&shy;<lb/>
ue in G, &longs;iue in H religatus; &longs;imiliter <lb/>
o&longs;tendetur potentiam in R &longs;excuplam <lb/>
e&longs;&longs;e ponderis <expan abbr="q.">que</expan> Et &longs;i in R &longs;it potentia <lb/>
mouens pondus, o&longs;tendetur &longs;patium pon<lb/>
deris moti &longs;excuplum e&longs;&longs;e &longs;patii poten&shy;<lb/>
ti&aelig;. <arrow.to.target n="fig160"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.177.8.0.0.0" type="main">        
<s id="id.2.1.177.8.1.1.0"> Et &longs;i funis &longs;it religatus in K trochle&aelig; <lb/>
&longs;uperiori, &amp; in R &longs;it potentia pondus <lb/>
&longs;u&longs;tinens; &longs;imili modo o&longs;tendetur poten<lb/>
tiam in R &longs;eptuplam e&longs;&longs;e ponderis <expan abbr="q.">que&lt;/</expan> </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig158" place="text">       </figure>       
<p id="id.2.1.177.8.2.1.0" type="caption">        
<s id="id.2.1.177.8.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig159" place="text">       </figure>       
<p id="id.2.1.177.8.2.3.0" type="caption">        
<s id="id.2.1.177.8.2.3.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig160" place="text">       </figure>       
<p id="id.2.1.177.8.2.5.0" type="caption">        
<s id="id.2.1.177.8.2.5.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.177.9.0.0.0" type="main">        
<s id="id.2.1.177.9.1.1.0"> Et &longs;i in R &longs;it potentia mouens, o&longs;ten <lb/>
detur &longs;patium ponderis Q &longs;eptuplum e&longs;&longs;e <lb/>
&longs;patii potenti&aelig;.  </s>
             
<s id="id.2.1.177.9.1.2.0"> atq; ita in infinitum <lb/>
omnis potenti&aelig; ad pondus multiplex <lb/>
proportio inueniri poterit.  </s>
             
<s id="id.2.1.177.9.1.3.0"> &longs;emperq; o&shy;<lb/>
&longs;tendetur, ita e&longs;&longs;e pondus ad potentiam <lb/>
ip&longs;um &longs;u&longs;tinentem, &longs;icuti &longs;patium poten<lb/>
ti&aelig; pondus mouentis ad &longs;patium ponde&shy;<lb/>
ris moti. </s>
    </p>       
<p id="id.2.1.177.10.0.0.0" type="main">        
<s id="id.2.1.177.10.1.1.0"> Vectium autem ip&longs;orum orbiculorum <lb/>
motus in his fit hoc modo, videlicet vectes <lb/>
orbiculorum trochle&aelig; &longs;uperioris mouen<lb/>
tur, vti dictum e&longs;t in decima &longs;exta huius; <lb/>
hoc e&longs;t habent fulcimentum in extremita <lb/>
te, potentiam in medio, pondus in altera extremitate appen&longs;um.  </s>
             
<s id="id.2.1.177.10.1.2.0"> ve<lb/>
ctes <expan abbr="ver&ograve;">vero</expan> trochle&aelig; inferioris habent fulcimentum in medio, pon<lb/>
dus, &amp; potentiam in extremitatibus.  </s>
    </p>       
<p id="id.2.1.177.11.0.0.0" type="head">        
<pb/>
       
<s id="id.2.1.177.12.1.1.0"> COROLLARIVM </s>
    </p>       
<p id="id.2.1.177.13.0.0.0" type="main">        
<s id="id.2.1.177.13.1.1.0"> Manife&longs;tum e&longs;t in his, orbiculos trochle&aelig; &longs;u<lb/>
perioris efficere, vt pondus moueatur maiori <lb/>
potentia, <expan abbr="qu&agrave;m">quam</expan> &longs;it ip&longs;um pondus, &amp; per maius <lb/>
&longs;patium potenti&aelig; &longs;patio, &amp; per &aelig;quale tempo&shy;<lb/>
re minori; quod quidem orbiculi trochle&aelig; in&shy;<lb/>
ferioris non efficiunt. </s>
    </p>       
<p id="id.2.1.177.14.0.0.0" type="main">        
<s id="id.2.1.177.14.1.1.0"> Alio quoq; modo hanc potenti&aelig; ad pondus multiplicem propor<lb/>
tionem inuenire po&longs;&longs;umus.  </s>
    </p>       
<p id="id.2.1.177.15.0.0.0" type="head">        
<s id="id.2.1.177.15.1.1.0"> PROPOSITIO XVIIII. </s>
    </p>       
<p id="id.2.1.177.16.0.0.0" type="main">        
<s id="id.2.1.177.16.1.1.0"> Si vtriu&longs;q; duarum trochlearum &longs;ingulis orbi <lb/>
culis, quarum altera <expan abbr="&longs;upern&egrave;">&longs;uperne</expan> appen&longs;a, altera <expan abbr="ve&shy;r&ograve;">ve&shy;<lb/>
ro</expan> <expan abbr="infern&egrave;">inferne</expan> <expan abbr="&agrave;">a</expan> &longs;u&longs;tinente potentia rententa fuerit, <lb/>
funis circumuoluatur; altero eius extremo alicu<lb/>
bi religato, alteri autem pondere appen&longs;o; du&shy;<lb/>
pla erit ponderis potentia.  </s>
    </p>       
<pb n="89"/>
       
<p id="id.2.1.177.18.0.0.0" type="main">        
<s id="id.2.1.177.18.1.1.0"> Sit orbiculus trochle&aelig; <expan abbr="&longs;upern&egrave;">&longs;uperne</expan> appen&longs;&aelig;, cu <lb/>
ius centrum &longs;it A; &amp; BCD &longs;it trochle&aelig; infe<lb/>
rioris; &longs;it deinde funis EBC DFGHL reli&shy;<lb/>
gatus in E; &amp; in L &longs;it appen&longs;um pondus M; <lb/>
&longs;itq; potentia in N &longs;u&longs;tinens pondus M.  </s>
     
<s id="id.2.1.177.18.1.1.0.a"> <lb/>
dico potentiam in N duplam e&longs;&longs;e ponderis <lb/>
M.  </s>
     
<s id="id.2.1.177.18.1.1.0.b"> <expan abbr="C&ugrave;m">Cum</expan> enim &longs;upra o&longs;ten&longs;um &longs;it potentiam <lb/>
in L, qu&aelig; pondus, exempli gratia, O &longs;u&longs;ti&shy;<lb/>
neat  <arrow.to.target n="note259"></arrow.to.target> in N appen&longs;um, &longs;ubduplam e&longs;&longs;e eiu&longs;dem <lb/>
ponderis; potentia igitur in N ponderi O &aelig;&shy;<lb/>
qualis pondus M potenti&aelig; in L &aelig;quale &longs;u&longs;ti<lb/>
nebit; ponderi&longs;q; M dupla erit. quod demon <lb/>
&longs;trare oportebat. </s>
             
<s id="id.2.1.177.18.1.2.0"> [quod demon<lb/>
&longs;trare oportebat.]  </s>
     <lb/>
      </p>       
<p id="id.2.1.178.1.0.0.0" type="margin">        
<s id="id.2.1.178.1.1.1.0"> <margin.target id="note259"></margin.target>3 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.179.1.0.0.0" type="main">        

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure place="text">       </figure>       
<p id="id.2.1.179.1.1.1.0" type="caption">        
<s id="id.2.1.179.1.1.1.0.capt"> YYY </s>
     <lb/>
       
<s id="id.2.1.179.1.3.1.0"> ALITER. </s>
    </p>       
<p id="id.2.1.179.2.0.0.0" type="main">        
<s id="id.2.1.179.2.1.1.0"> Ii&longs;dem po&longs;itis.  </s>
             
<s id="id.2.1.179.2.1.2.0"> Quoniam potentia in F, <arrow.to.target n="note260"></arrow.to.target><lb/>
&longs;eu in D, quod idem e&longs;t, &aelig;qualis e&longs;t ponde<lb/>
ri M; &amp; BD e&longs;t vectis, cuius fulcimentum <lb/>
e&longs;t B, &amp; potentia in N e&longs;t, ac &longs;i e&longs;&longs;et in me&shy;<lb/>
dio vectis, &amp; pondus &aelig;quale ip&longs;i M, ac &longs;i e&longs;&shy;<lb/>
&longs;et in D propter funem FD; quod idem <lb/>
e&longs;t, ac &longs;i BCD e&longs;&longs;et orbiculus trochle&aelig; &longs;upe<lb/>
rioris, pondusq; appen&longs;um e&longs;&longs;et in fune DF, <lb/>
&longs;icut in decimaquinta, &amp; decima&longs;exta dictum e&longs;t; ergo potentia in <lb/>
N dupla e&longs;t ponderis M. quod erat o&longs;tendendum. </s>
    </p>       
<p id="id.2.1.180.1.0.0.0" type="margin">        
<s id="id.2.1.180.1.1.1.0"> <margin.target id="note260"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.181.1.0.0.0" type="main">        
<s id="id.2.1.181.1.1.1.0"> Si autem in N &longs;it potentia mouens pondus M, erit &longs;patium <lb/>
ponderis M duplum &longs;patii potenti&aelig; in N. quod ex duodecima <lb/>
huius manife&longs;tum e&longs;t; &longs;patium enim puncti L deor&longs;um ten&shy;<lb/>
dentis duplum e&longs;t &longs;pat^{1}i N &longs;ur&longs;um; erit igitur <expan abbr="&egrave;">e</expan> conuer&longs;o &longs;patium <lb/>
potenti&aelig; in N deor&longs;um tendentis dimidium &longs;aptii ponderis M &longs;ur<lb/>
&longs;um moti. </s>
    </p>       
<p id="id.2.1.181.2.0.0.0" type="main">        
<s id="id.2.1.181.2.1.1.0"> Sicut autem ex tertia, quinta, &longs;eptima huius, &amp;c.  </s>
             
<s id="id.2.1.181.2.1.2.0"> colligi po&longs;&longs;unt <lb/>
ponderis O rationes quotcunq; multiplices ip&longs;ius potenti&aelig; in L, <lb/>
<expan abbr="eod&etilde;">eodem</expan> quoq; modo o&longs;tendi poterunt potenti&aelig; in N pondus &longs;u&longs;tinen<lb/>
tis ponderis M quotcunq; multiplices.  </s>
             
<s id="id.2.1.181.2.1.3.0"> Atq; ita ex decimatertia 
<pb/>
decimaquarta rationes o&longs;ten <lb/>
dentur quotcunq; multiplices <lb/>
&longs;patii ponderis M ad &longs;patium <lb/>
potenti&aelig; mouentis in N con&longs;ti<lb/>
tut&aelig;. <arrow.to.target n="fig161"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.181.3.0.0.0" type="main">        
<s id="id.2.1.181.3.1.1.0"> Poterit quoq; ex decima&longs;e <lb/>
ptima decimaoctaua huius mul<lb/>
tiplex inueniri proportio, quam <lb/>
habet potentia pondus &longs;u&longs;ti<lb/>
nens ad ip&longs;um pondus; &longs;icut <lb/>
proportio potenti&aelig; in N ad pon<lb/>
dus M ex decimaquinta, &amp; deci <lb/>
ma&longs;exta o&longs;tendebatur: inuenie<lb/>
turq; ita e&longs;&longs;e pondus ad poten<lb/>
tiam pondus &longs;u&longs;tinentem, vt &longs;pa<lb/>
tium potenti&aelig; mouentis ad &longs;pa<lb/>
tium ponderis. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig161" place="text">       </figure>       
<p id="id.2.1.181.3.2.1.0" type="caption">        
<s id="id.2.1.181.3.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.181.4.0.0.0" type="main">        
<s id="id.2.1.181.4.1.1.0"> Vectium motus in his fit <lb/>
hoc modo, videlicet vectes or<lb/>
biculorum trochle&aelig; inferioris <lb/>
mouentur, vt vectis BD, qu&aelig; <lb/>
mouetur, ac &longs;i B e&longs;&longs;et fulcimen <lb/>
tum, &amp; pondus in D, &amp; poten<lb/>
tia in medio.  </s>
             
<s id="id.2.1.181.4.1.2.0"> Vectes <expan abbr="ver&ograve;">vero</expan> or<lb/>
biculorum trochle&aelig; &longs;uperioris mouentur, vt FH, cuius fulcimen <lb/>
tum e&longs;t in medio, pondus in H, &amp; potentia in F. </s>
    </p>       
<p id="id.2.1.181.5.0.0.0" type="head">        
<s id="id.2.1.181.5.1.1.0"> COROLLARIVM. </s>
    </p>       
<p id="id.2.1.181.6.0.0.0" type="main">        
<s id="id.2.1.181.6.1.1.0"> Ex hoc manife&longs;tum e&longs;t, orbiculos trochle&aelig; <lb/>
inferioris in his efficere, vt pondus maiori po&shy;
<pb n="90"/>
tentia moueatur, <expan abbr="qu&agrave;m">quam</expan> &longs;it ip&longs;um pondus, &amp; <lb/>
per maius &longs;patium &longs;patio potenti&aelig;, &amp; minori <lb/>
tempore per &aelig;quale.  </s>
             
<s id="id.2.1.181.6.1.2.0"> quod quidem orbiculi &longs;u<lb/>
perioris trochle&aelig; non efficiunt. </s>
    </p>       
<p id="id.2.1.181.7.0.0.0" type="main">        
<s id="id.2.1.181.7.1.1.0"> Cognitis proportionibus multiplicibus, iam ad &longs;uperparticu<lb/>
lares accedendum e&longs;t.  </s>
    </p>       
<p id="id.2.1.181.8.0.0.0" type="head">        
<s id="id.2.1.181.8.1.1.0"> PROPOSITIO XX. </s>
    </p>       
<p id="id.2.1.181.9.0.0.0" type="main">        
<s id="id.2.1.181.9.1.1.0"> Si vtriu&longs;q; duarum trochlearum &longs;ingulis or&shy;<lb/>
biculis, quarum altera <expan abbr="&longs;upern&egrave;">&longs;uperne</expan> <expan abbr="&agrave;">a</expan> potentia &longs;u&longs;ti&shy;<lb/>
neatur, altera <expan abbr="ver&ograve;">vero</expan> <expan abbr="infern&egrave;">inferne</expan>, ponderiq; alligata, <lb/>
<expan abbr="c&otilde;&longs;tituta">con&longs;tituta</expan> fuerit, funis reuoluatur; altero eius extre<lb/>
mo alicuibi, altero <expan abbr="ver&ograve;">vero</expan> inferiori trochle&aelig; reli<lb/>
gato; pondus potenti&aelig; &longs;e&longs;quialterum erit.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.181.11.0.0.0" type="main">        
<s id="id.2.1.181.11.1.1.0"> Sit ABC orbiculus <lb/>
trochle&aelig; &longs;uperioris, &amp; <lb/>
DEF trochle&aelig; inferio&shy;<lb/>
ris ponderi G alligat&aelig;; <lb/>
&longs;itq; funis HABCDE <lb/>
Fk circa orbiculos re&shy;<lb/>
uolutus, qui &longs;it religatus <lb/>
in K, &amp; in H trochle&aelig; <lb/>
inferiori; &longs;itq; potentia <lb/>
in L &longs;u&longs;tinens pondus <lb/>
G.  </s>
     
<s id="id.2.1.181.11.1.1.0.a"> dico pondus poten<lb/>
ti&aelig; &longs;e&longs;quialterum e&longs;&longs;e.  </s>
             
<s id="id.2.1.181.11.1.2.0"> <lb/>
<arrow.to.target n="note261"></arrow.to.target> Quoniam enim vterque <lb/>
funis CD AH tertiam <lb/>
&longs;u&longs;tinet partem ponde&shy;<lb/>
ris G, erit vnaqu&aelig;q; po<lb/>
tentia in DH &longs;ubtripla <lb/>
ponderis G; quibus &longs;i&shy;<lb/>
mul a&longs;&longs;umptis e&longs;t &aelig;qua&shy;<lb/>
<arrow.to.target n="fig162"></arrow.to.target><lb/>
<arrow.to.target n="note262"></arrow.to.target> lis potentia in L: potentia enim in L dupla e&longs;t potenti&aelig; in D, &amp; <lb/>
eius, qu&aelig; e&longs;t in H. quare potentia in L &longs;ub&longs;e&longs;quialtera e&longs;t ponde&shy;<lb/>
ris G.  </s>
     
<s id="id.2.1.181.11.1.2.0.a"> pondus ergo G ad pontentiam in L e&longs;t, vt tria ad duo; <lb/>
hoc e&longs;t &longs;e&longs;quialterum. quod demon&longs;trare oportebat. </s>
             
<s id="id.2.1.181.11.1.3.0"> [quod demon&longs;trare oportebat.]  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig162" place="text">       </figure>       
<p id="id.2.1.181.11.2.1.0" type="caption">        
<s id="id.2.1.181.11.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.182.1.0.0.0" type="margin">        
<s id="id.2.1.182.1.1.1.0"> <margin.target id="note261"></margin.target><emph type="italics"/>Cor.<emph.end type="italics"/> 5 <emph type="italics"/>huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.182.1.1.2.0"> <margin.target id="note262"></margin.target><emph type="italics"/>Ex.<emph.end type="italics"/> 15 <emph type="italics"/>huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.183.1.0.0.0" type="main">        
<pb n="91"/>
       
<s id="id.2.1.183.1.2.1.0"> Si autem in L &longs;it potentia mouens pondus.  </s>
             
<s id="id.2.1.183.1.2.2.0"> <lb/>
Dico &longs;patium potenti&aelig; &longs;patii ponderis &longs;e&longs;quial&shy;<lb/>
terum e&longs;&longs;e. </s>
    </p>       
<p id="id.2.1.183.2.0.0.0" type="main">        
<s id="id.2.1.183.2.1.1.0"> Ii&longs;dem po&longs;itis, perueniat orbi&shy;<lb/>
culus ABC v&longs;q; ad MNO, &amp; <lb/>
DEF ad PQR; &amp; H in S; &amp; <lb/>
pondus G v&longs;q; ad T.  </s>
     
<s id="id.2.1.183.2.1.1.0.a"> Et quoniam <lb/>
funis HABCDEFK e&longs;t &aelig;qualis <lb/>
funi SMNOPQRk, <expan abbr="c&ugrave;m">cum</expan> &longs;it <lb/>
idem funis; &amp; funes circa &longs;emicir<lb/>
culos ABC MNO &longs;unt inter &longs;e <lb/>
&longs;e &aelig;quales; qui <expan abbr="ver&ograve;">vero</expan> &longs;unt circa <lb/>
DEF PQR &longs;imiliter inter &longs;e &aelig;&shy;<lb/>
quales; Demptis igitur AS CP <lb/>
RK communibus, erunt duo CO <lb/>
MA tribus DP HS FR &aelig;qua&shy;<lb/>
les.  </s>
             
<s id="id.2.1.183.2.1.2.0"> &longs;ed vterq; CO AM &longs;eor&longs;um <lb/>
e&longs;t &aelig;qualis &longs;patio potenti&aelig; mot&aelig;.  </s>
             
<s id="id.2.1.183.2.1.3.0"> <lb/>
quare duo CO MA, &longs;imul &longs;patii <lb/>
potenti&aelig; dupli erunt: tre&longs;q; DP <lb/>
HS FR &longs;imul &longs;imili modo &longs;patii <lb/>
ponderis moti tripli erunt.  </s>
             
<s id="id.2.1.183.2.1.4.0"> dimidia <lb/>
<expan abbr="ver&ograve;">vero</expan> pars, hoc e&longs;t &longs;patium poten<lb/>
ti&aelig; mot&aelig; ad tertiam, ad &longs;patium <lb/>
&longs;cilicet ponderis moti ita &longs;e habet, <lb/>
vt duplum dimidii ad duplum ter&shy;<lb/>
tii; hoc e&longs;t, vt totum ad duas ter<lb/>
<arrow.to.target n="fig163"></arrow.to.target><lb/>
tias, quod e&longs;t vt tria ad duo.  </s>
             
<s id="id.2.1.183.2.1.5.0"> &longs;patium ergo potenti&aelig; in L &longs;pa&shy;<lb/>
tii ponderis G moti &longs;e&longs;quialterum e&longs;t. quod o&longs;tendere opor&shy;<lb/>
tebat. </s>
             
<s id="id.2.1.183.2.1.6.0"> [quod o&longs;tendere opor&shy;<lb/>
tebat.]  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig163" place="text">       </figure>       
<p id="id.2.1.183.2.2.1.0" type="caption">        
<s id="id.2.1.183.2.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.183.3.0.0.0" type="head">        
<pb/>
       
<s id="id.2.1.183.4.1.1.0"> PROPOSITIO XXI. </s>
    </p>       
<p id="id.2.1.183.5.0.0.0" type="main">        
<s id="id.2.1.183.5.1.1.0"> Si tribus duarum trochlearum orbiculis, qua <lb/>
rum altera vnius <expan abbr="tant&ugrave;m">tantum</expan> orbiculi <expan abbr="&longs;upern&egrave;">&longs;uperne</expan> <expan abbr="&agrave;">a</expan> po&shy;<lb/>
tentia &longs;u&longs;tineatur, altera <expan abbr="ver&ograve;">vero</expan> duorum <expan abbr="infern&egrave;">inferne</expan>, <lb/>
ponderiq; alligata, collocata fuerit, funis cir&shy;<lb/>
cumuoluatur; altero eius extremo alicubi, altero <lb/>
autem &longs;uperiori trochle&aelig; religato: pondus poten<lb/>
ti&aelig; &longs;e&longs;quitertium erit. </s>
    </p>       
<p id="id.2.1.183.6.0.0.0" type="main">        
<s id="id.2.1.183.6.1.1.0"> Sit pondus A trochle&aelig; inferiori alliga&shy;<lb/>
tum, qu&aelig; duos habeat orbiculos, quorum <lb/>
centra &longs;int BC; &longs;uperiorq; trochlea orbicu&shy;<lb/>
lum habeat, cuius centrum D; &amp; &longs;it funis <lb/>
EFGHkLMN circa omnes orbiculos re <lb/>
uolutus, qui religatus &longs;it in N, &amp; in E tro<lb/>
chle&aelig; &longs;uperiori; <expan abbr="&longs;itqu&eacute;">&longs;itque</expan> potentia in O <lb/>
&longs;u&longs;tinens pondus A.  </s>
     
<s id="id.2.1.183.6.1.1.0.a"> dico pondus po&shy;<lb/>
<arrow.to.target n="note263"></arrow.to.target> tenti&aelig; &longs;e&longs;quitertium e&longs;&longs;e.  </s>
             
<s id="id.2.1.183.6.1.2.0"> Quoniam enim <lb/>
vnu&longs;qui&longs;q; funis NM HG EF KL quar&shy;<lb/>
tam &longs;u&longs;tinent partem ponderis A, &amp; omnes <lb/>
&longs;imul totum &longs;u&longs;tinent pondus; tres HG <lb/>
EF kL &longs;imul tres &longs;u&longs;tinebunt partes pon&shy;<lb/>
deris A. quare pondus A ad hos omnes <lb/>
&longs;imul erit, vt quatuor ad tria: &amp; <expan abbr="c&ugrave;m">cum</expan> po&shy;<lb/>
tentia in O idem efficiat, quod HG EF kL <lb/>
&longs;imul efficiunt; omnes enim &longs;u&longs;tinet; erit po<lb/>
tentia in O tribus &longs;imul HG EF kL &aelig;&shy;<lb/>
qualis; &amp; ob id pondus A ad potentiam <lb/>
in O erit, vt quatuor ad tria; hoc e&longs;t &longs;e&longs;qui<lb/>
tertium. quod demon&longs;trare oportebat. <arrow.to.target n="fig164"></arrow.to.target> </s>
             
<s id="id.2.1.183.6.1.3.0"> [quod demon&longs;trare oportebat. <arrow.to.target n="fig164"></arrow.to.target>] </s>
    </p>       
<pb n="92"/>
       
<p id="id.2.1.183.8.0.0.0" type="main">        
<s id="id.2.1.183.8.1.1.0"> Si vero in O &longs;it potentia mouens pondus A.  </s>
     
<s id="id.2.1.183.8.1.1.0.a"> <lb/>
Dico &longs;patium potenti&aelig; in O decur&longs;um &longs;patii pon<lb/>
deris A moti &longs;e&longs;quitertium e&longs;&longs;e. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig164" place="text">       </figure>       
<p id="id.2.1.183.8.2.1.0" type="caption">        
<s id="id.2.1.183.8.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.184.1.0.0.0" type="margin">        
<s id="id.2.1.184.1.1.1.0"> <margin.target id="note263"></margin.target><emph type="italics"/>Cor.<emph.end type="italics"/> 1 <emph type="italics"/>&longs;eptimebuius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.185.1.0.0.0" type="main">        
<s id="id.2.1.185.1.1.1.0"> Ii&longs;dem po&longs;itis, &longs;it centrum B motum <lb/>
in P; &amp;C v&longs;q; ad Q; &amp; D in R; &amp; E in <lb/>
S eodem tempore: &amp; per centra ducantur <lb/>
ML 9Z FG TV Hk XY horizonti, <lb/>
&amp; inter &longs;e &longs;e &aelig;quidi&longs;tantes.  </s>
             
<s id="id.2.1.185.1.1.2.0"> Similiter, vt in <lb/>
pr&aelig;cedente o&longs;tendetur tres <emph type="italics"/>X<emph.end type="italics"/>H SE Yk <lb/>
quatuor TG VF ZL 9M &aelig;quales e&longs;&longs;e.  </s>
             
<s id="id.2.1.185.1.1.3.0"> &amp; <lb/>
quoniam tres XH SE Yk &longs;imul tripl&aelig; <lb/>
&longs;unt &longs;patii potenti&aelig;, quatuor <expan abbr="ver&ograve;">vero</expan> TG VF <lb/>
ZL 9M &longs;imul quadrupl&aelig; &longs;unt &longs;patii pon<lb/>
deris moti; erit &longs;patium potenti&aelig; ad &longs;pa&shy;<lb/>
tium ponderis, vt tertia pars ad quartam.  </s>
             
<s id="id.2.1.185.1.1.4.0"> <lb/>
&longs;ed tertia pars ad quartam e&longs;t, vt tres ter<lb/>
ti&aelig; ad tres quartas, hoc e&longs;t, vt totum ad <lb/>
tres quartas; quod e&longs;t, vt quatuor ad tria.  </s>
             
<s id="id.2.1.185.1.1.5.0"> <lb/>
&longs;patium ergo potenti&aelig; &longs;patii ponderis mo<lb/>
ti &longs;e&longs;quitertium e&longs;t. quod erat demon&shy;<lb/>
&longs;trandum. <arrow.to.target n="fig165"></arrow.to.target> </s>
             
<s id="id.2.1.185.1.1.6.0"> [quod erat demon&shy;<lb/>
&longs;trandum. <arrow.to.target n="fig165"></arrow.to.target>] </s>
    </p>       
<p id="id.2.1.185.2.0.0.0" type="main">        
<s id="id.2.1.185.2.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> funis in E per alium circumuol<lb/>
uatur orbiculum, qui deinde trochle&aelig; in <lb/>
feriori religetur; &longs;imiliter o&longs;tendetur pro <lb/>
portionem ponderis ad <expan abbr="potenti&atilde;">potentiam</expan> in O pon<lb/>
dus &longs;u&longs;tinentem &longs;e&longs;quiquartam e&longs;&longs;e.  </s>
             
<s id="id.2.1.185.2.1.2.0"> <expan abbr="qu&ograve;d">quod</expan> <lb/>
&longs;i in O &longs;it potentia mouens pondus, o&longs;ten <lb/>
detur &longs;patium potenti&aelig; &longs;patii ponderis &longs;e&longs;<lb/>
quiquartum e&longs;&longs;e.  </s>
             
<s id="id.2.1.185.2.1.3.0"> &amp; &longs;ic in infinitum proce<lb/>
dendo quamcunq; &longs;uperparticularem pro <lb/>
portionem ponderis ad potentiam inuenie<lb/>
mus; &longs;emperq; reperiemus, ita e&longs;&longs;e pondus <lb/>
ad potentiam pondus &longs;u&longs;tinentem, vt &longs;pa&shy;<lb/>
tium potenti&aelig; mouentis ad &longs;patium ponde&shy;<lb/>
ris moti.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig165" place="text">       </figure>       
<p id="id.2.1.185.2.2.1.0" type="caption">        
<s id="id.2.1.185.2.2.1.0.capt"> YYY </s>
    </p>       
<pb/>
       
<p id="id.2.1.185.4.0.0.0" type="main">        
<s id="id.2.1.185.4.1.1.0"> Motus <expan abbr="ver&ograve;">vero</expan> vectium fit hoc mo <lb/>
do, videlicet vectis ML fulci&shy;<lb/>
mentum e&longs;t M, <expan abbr="c&ugrave;m">cum</expan> funis &longs;it re <lb/>
ligatus in N, &amp; pondus in me&shy;<lb/>
dio, &amp; potentia in L. quia <expan abbr="ve&shy;r&ograve;">ve&shy;<lb/>
ro</expan> punctum L tendit &longs;ur&longs;um, quod <lb/>
<expan abbr="&agrave;">a</expan> fune KL mouetur, idcirco K &longs;ur&shy;<lb/>
&longs;um mouebitur, &amp; vectis HK ful<lb/>
cimentum erit H, pondus ac &longs;i e&longs;<lb/>
&longs;ent in k, &amp; potentia in medio; <lb/>
vectis autem FG fulcimentum <lb/>
erit G, pondus in medio; &amp; poten<lb/>
tia in F.  </s>
     
<s id="id.2.1.185.4.1.1.0.a"> punctum enim F &longs;ur&longs;um <lb/>
mouetur <expan abbr="&agrave;">a</expan> fune EF.  </s>
     
<s id="id.2.1.185.4.1.1.0.b"> Pr&aelig;terea <lb/>
G in orbiculo deor&longs;um tendit, <lb/>
quia H quoque in eius orbiculo <lb/>
deor&longs;um mouetur. <arrow.to.target n="fig166"></arrow.to.target> </s>
     
<pb n="93"/>
       
<s id="id.2.1.185.4.3.1.0"> PROPOSITIO XXII. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig166" place="text">       </figure>       
<p id="id.2.1.185.4.4.1.0" type="caption">        
<s id="id.2.1.185.4.4.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.185.5.0.0.0" type="main">        
<s id="id.2.1.185.5.1.1.0"> Si vtri&longs;que duarum trochlearum &longs;ingulis <lb/>
orbiculis, quarum altera <expan abbr="&longs;upern&egrave;">&longs;uperne</expan> <expan abbr="&agrave;">a</expan> potentia <lb/>
&longs;u&longs;tineatur, altera <expan abbr="ver&ograve;">vero</expan> <expan abbr="infern&egrave;">inferne</expan>, ponderiq; alli&shy;<lb/>
gata, collocata fuerit, circumducatur funis; al&shy;<lb/>
tero eius extremo alicubi, altero autem &longs;uperio<lb/>
ri trochle&aelig; religato.  </s>
             
<s id="id.2.1.185.5.1.2.0"> erit potentia ponderis &longs;e&longs;<lb/>
quialtera. </s>
    </p>       
<p id="id.2.1.185.6.0.0.0" type="main">        
<s id="id.2.1.185.6.1.1.0"> Sit orbiculus ABC trochle&aelig; ponderi D al <lb/>
ligat&aelig;; &amp; EFG trochle&aelig; &longs;uperioris, cuius <lb/>
centrum H; &longs;it deinde funis k ABCEFGL <lb/>
circa orbiculos reuolutus, &amp; religatus in L, &amp; <lb/>
in k trochle&aelig; &longs;uperiori; &longs;itq; potentia in M <lb/>
&longs;u&longs;tinens pondus D.  </s>
     
<s id="id.2.1.185.6.1.1.0.a"> dico potentiam ponde<lb/>
ris &longs;e&longs;quialteram e&longs;&longs;e.  </s>
             
<s id="id.2.1.185.6.1.2.0"> Quoniam enim poten<arrow.to.target n="note264"></arrow.to.target><lb/>
tia in E &longs;u&longs;tinens pondus D &longs;ubdupla e&longs;t pon<arrow.to.target n="note265"></arrow.to.target><lb/>
deris D, potenti&aelig; <expan abbr="ver&ograve;">vero</expan> in E dupla e&longs;t poten<arrow.to.target n="note266"></arrow.to.target><lb/>
tia in H; erit potentia in H ponderi D &aelig;qua <arrow.to.target n="note267"></arrow.to.target><lb/>
lis; &amp; <expan abbr="c&ugrave;m">cum</expan> potentia in K &longs;ubdupla &longs;it ponde <lb/>
ris D; erunt vtr&aelig;q; &longs;imul potenti&aelig; in H k &longs;e&longs;<lb/>
quialter&aelig; ponderis D.  </s>
     
<s id="id.2.1.185.6.1.2.0.a"> Itaq; <expan abbr="c&ugrave;m">cum</expan> potentia in <lb/>
M duabus potentiis in Hk &longs;imul &longs;umptis &longs;it <lb/>
&aelig;qualis, quemadmodum in &longs;uperioribus o&shy;<lb/>
&longs;ten&longs;um e&longs;t; erit potentia in M &longs;e&longs;quialtera <lb/>
ponderis D. quod oportebat demon&longs;trare. </s>
    </p>       
<p id="id.2.1.186.1.0.0.0" type="margin">        
<s id="id.2.1.186.1.1.1.0"> <margin.target id="note264"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.186.1.1.2.0"> <margin.target id="note265"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 15 <emph type="italics"/>huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.186.1.1.3.0"> <margin.target id="note266"></margin.target>2 <emph type="italics"/>Cor.<emph.end type="italics"/> </s>
             
<s id="id.2.1.186.1.1.4.0"> <margin.target id="note267"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.187.1.0.0.0" type="main">        
<s id="id.2.1.187.1.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> in M &longs;it potentia mouens pondus, <lb/>
&longs;imiliter vt in pr&aelig;cedentibus o&longs;tendetur, &longs;pa<lb/>
tium ponderis &longs;patii potenti&aelig; &longs;e&longs;quialterum <lb/>
e&longs;&longs;e. <arrow.to.target n="fig167"></arrow.to.target> </s>
    </p>       
<pb/>
       
<p id="id.2.1.187.3.0.0.0" type="main">        
<s id="id.2.1.187.3.1.1.0"> Et &longs;i funis in K per alium circumuoluatur <lb/>
orbiculum, cuius centrum &longs;it N; qui dein&shy;<lb/>
de trochle&aelig; inferiori religetur in O; &amp; po&shy;<lb/>
tentia in M &longs;u&longs;tineat pondus D.  </s>
     
<s id="id.2.1.187.3.1.1.0.a"> dico pro&shy;<lb/>
portionem potenti&aelig; ad pondus &longs;e&longs;quiter&shy;<lb/>
tiam e&longs;&longs;e. <arrow.to.target n="fig168"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.187.4.0.0.0" type="main">        
<s id="id.2.1.187.4.1.1.0"> Quoniam enim potentia in E &longs;u&longs;tinens <lb/>
<arrow.to.target n="note268"></arrow.to.target> pondus D fune ECB AKPO &longs;ubtripla e&longs;t <lb/>
<arrow.to.target n="note269"></arrow.to.target> ip&longs;ius D, ip&longs;ius autem E dupla e&longs;t potentia <lb/>
in H; erit potentia in H &longs;ub&longs;e&longs;quialtera pon<lb/>
deris D. &longs;imili quoq; modo quoniam po<lb/>
tentia in O_{3} qu&aelig; e&longs;t, ac &longs;i e&longs;&longs;et in centro or<lb/>
<arrow.to.target n="note270"></arrow.to.target> biculi ABC, &longs;ubtripla e&longs;t ponderis D; ip&shy;<lb/>
&longs;ius autem O dupla e&longs;t potentia in N; erit <lb/>
quoq; potentia in N &longs;ub&longs;e&longs;quialtera ponde&shy;<lb/>
ris D. quare du&aelig; &longs;imul potenti&aelig; in HN pon <lb/>
dus D &longs;uperant tertia parte, &longs;e &longs;e habentq; ad <lb/>
D in ratione &longs;e&longs;quitertia: &amp; <expan abbr="c&ugrave;m">cum</expan> potentia <lb/>
in M duabus &longs;it potentiis in HN &longs;imul &longs;um<lb/>
ptis &aelig;qualis, &longs;uperabit itidem potentia in <lb/>
M pondus D tertia parte.  </s>
             
<s id="id.2.1.187.4.1.2.0"> ergo proportio <lb/>
potenti&aelig; in M ad pondus D &longs;e&longs;quitertia <lb/>
e&longs;t. quod demon&longs;trare oportebat. </s>
             
<s id="id.2.1.187.4.1.3.0"> [quod demon&longs;trare oportebat.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig167" place="text">       </figure>       
<p id="id.2.1.187.4.2.1.0" type="caption">        
<s id="id.2.1.187.4.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig168" place="text">       </figure>       
<p id="id.2.1.187.4.2.3.0" type="caption">        
<s id="id.2.1.187.4.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.188.1.0.0.0" type="margin">        
<s id="id.2.1.188.1.1.1.0"> <margin.target id="note268"></margin.target>5 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.188.1.1.2.0"> <margin.target id="note269"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 15 <emph type="italics"/>huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.188.1.1.3.0"> <margin.target id="note270"></margin.target>3, 15,<emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.189.1.0.0.0" type="main">        
<s id="id.2.1.189.1.1.1.0"> Si autem in M &longs;it potentia mouens pon&shy;<lb/>
dus, &longs;imili modo o&longs;tendetur &longs;patium ponderis D &longs;patii potenti&aelig; in <lb/>
M &longs;e&longs;quitertium e&longs;&longs;e. </s>
    </p>       
<p id="id.2.1.189.2.0.0.0" type="main">        
<s id="id.2.1.189.2.1.1.0"> Et &longs;i funis in O per alium circumuoluatur <expan abbr="orbi&ccedil;ulum">orbiculum</expan>, qui tro&shy;<lb/>
chle&aelig; &longs;uperiori deinde religetur; eodem modo demon&longs;trabimus <lb/>
proportionem potenti&aelig; in M pondus &longs;u&longs;tinentis ad pondus &longs;e&longs;&shy;<lb/>
quiquartam e&longs;&longs;e.  </s>
             
<s id="id.2.1.189.2.1.2.0"> &amp; &longs;i in M &longs;it potentia mouens, &longs;imiliter o&longs;ten&shy;<lb/>
detur &longs;patium ponderis &longs;patii potenti&aelig; &longs;e&longs;quiquartum e&longs;&longs;e.  </s>
             
<s id="id.2.1.189.2.1.3.0"> pro&shy;<lb/>
cedendoq; hoc modo in infinitum quamcunq; proportionem <lb/>
potenti&aelig; ad pondus &longs;uperparticularem inueniemus; <expan abbr="&longs;emperqu&eacute;">&longs;emperque</expan> 
<pb n="94"/>
o&longs;tendemus potentiam pondus &longs;u&longs;tinentem ita e&longs;&longs;e ad pondus, <lb/>
vt &longs;patium ponderis ad &longs;patium potenti&aelig; pondus mouentis. </s>
    </p>       
<p id="id.2.1.189.3.0.0.0" type="main">        
<s id="id.2.1.189.3.1.1.0"> Motus <expan abbr="ver&ograve;">vero</expan> vectis EG e&longs;t, ac &longs;i G e&longs;&longs;et fulcimentum, <expan abbr="c&ugrave;m">cum</expan> <lb/>
funis &longs;it religatus in L; pondus ac &longs;i in E e&longs;&longs;et appen&longs;um, &amp; po&shy;<lb/>
tentia in medio.  </s>
             
<s id="id.2.1.189.3.1.2.0"> Vectis <expan abbr="ver&ograve;">vero</expan> CA fulcimentum e&longs;t A pondus in <lb/>
medio, &amp; potentia in C. &amp; K fulcimentum e&longs;t vectis Pk, pon&shy;<lb/>
dus in P, &amp; potentia in medio.  </s>
             
<s id="id.2.1.189.3.1.3.0"> qu&aelig; omnia &longs;icut in pr&aelig;ceden&shy;<lb/>
ti o&longs;tendentur.  </s>
    </p>       
<p id="id.2.1.189.4.0.0.0" type="head">        
<s id="id.2.1.189.4.1.1.0"> PROPOSITIO XXIII. </s>
    </p>       
<p id="id.2.1.189.5.0.0.0" type="main">        
<s id="id.2.1.189.5.1.1.0"> Si vtri&longs;q; duarum trochlearum &longs;ingulis or&shy;<lb/>
biculis, quarum altera <expan abbr="&longs;upern&egrave;">&longs;uperne</expan> <expan abbr="&agrave;">a</expan> potentia &longs;u&longs;ti&shy;<lb/>
neatur, altera <expan abbr="ver&ograve;">vero</expan> <expan abbr="infern&egrave;">inferne</expan>, ponderiq; alligata, <lb/>
<expan abbr="c&otilde;&longs;tituta">con&longs;tituta</expan> fuerit, circumferatur funis; vtroq; eius <lb/>
extremo alicuibi, non autem trochleis religato; <lb/>
&aelig;qualis erit ponderi potentia.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.189.7.0.0.0" type="main">        
<s id="id.2.1.189.7.1.1.0"> Sit orbiculus trochle&aelig; &longs;uperioris <lb/>
ABC, cuius centrum D; &amp; EFG <lb/>
trochle&aelig; ponderi H alligat&aelig;, cu&shy;<lb/>
ius centrum k; &amp; &longs;it funis LEF <lb/>
GABCM circa orbiculos reuo&shy;<lb/>
lutus, religatu&longs;q; in LM; &longs;itq; <lb/>
potentia in N &longs;u&longs;tinens pondus <lb/>
H.  </s>
     
<s id="id.2.1.189.7.1.1.0.a"> dico potentiam in N &aelig;qua<lb/>
lem e&longs;&longs;e ponderi H.  </s>
     
<s id="id.2.1.189.7.1.1.0.b"> Accipiatur <lb/>
quoduis punctum O in AG.  </s>
     
<s id="id.2.1.189.7.1.1.0.c"> &amp; <lb/>
quoniam &longs;i in O e&longs;&longs;et potentia &longs;u<lb/>
<arrow.to.target n="note271"></arrow.to.target> &longs;tinens pondus H, &longs;ubdupla e&longs;&longs;et <lb/>
<arrow.to.target n="note272"></arrow.to.target> ponderis H, &amp; potenti&aelig; in O <lb/>
dupla e&longs;t ea, qu&aelig; e&longs;t in D, &longs;iue <lb/>
(quod idem e&longs;t) in N; erit po<lb/>
tentia in N ponderi H &aelig;qualis. <lb/>
quod demon&longs;trare oportebat. <arrow.to.target n="fig169"></arrow.to.target> </s>
             
<s id="id.2.1.189.7.1.2.0"> [<lb/>
quod demon&longs;trare oportebat. <arrow.to.target n="fig169"></arrow.to.target>] </s>
    </p>       
<p id="id.2.1.189.8.0.0.0" type="main">        
<s id="id.2.1.189.8.1.1.0"> Et &longs;i in N &longs;it potentia mouens pondus.  </s>
             
<s id="id.2.1.189.8.1.2.0"> Dico <lb/>
&longs;patium potenti&aelig; in N &aelig;qualem e&longs;&longs;e &longs;patio pon<lb/>
deris H moti. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig169" place="text">       </figure>       
<p id="id.2.1.189.8.2.1.0" type="caption">        
<s id="id.2.1.189.8.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.190.1.0.0.0" type="margin">        
<s id="id.2.1.190.1.1.1.0"> <margin.target id="note271"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.190.1.1.2.0"> <margin.target id="note272"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 15 <emph type="italics"/>huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.191.1.0.0.0" type="main">        
<s id="id.2.1.191.1.1.1.0"> Quoniam enim &longs;patium puncti O moti, duplum e&longs;t, <expan abbr="t&ugrave;m">tum</expan> &longs;patii <lb/>
<arrow.to.target n="note273"></arrow.to.target> ponderis H moti, <expan abbr="t&ugrave;m">tum</expan> &longs;patii potenti&aelig; in N mot&aelig;; erit &longs;patium <lb/>
<arrow.to.target n="note274"></arrow.to.target> potenti&aelig; in N &longs;patio ponderis H &aelig;quale.  </s>
    </p>       
<p id="id.2.1.192.1.0.0.0" type="margin">        
<s id="id.2.1.192.1.1.1.0"> <margin.target id="note273"></margin.target>11 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.192.1.1.2.0"> <margin.target id="note274"></margin.target>16 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.193.1.0.0.0" type="head">        
<pb n="95"/>
       
<s id="id.2.1.193.1.2.1.0"> ALITER. </s>
    </p>       
<p id="id.2.1.193.2.0.0.0" type="main">        
<s id="id.2.1.193.2.1.1.0"> Ii&longs;dem po&longs;itis, transfera<lb/>
tur centrum orbiculi ABC <lb/>
v&longs;q; ad P; orbiculu&longs;q; po&longs;i<lb/>
tionem habeat QRS; dein<lb/>
de eodem tempore orbiculus <lb/>
EFG &longs;it in TVX, cuius cen<lb/>
trum &longs;it Y; &amp; pondus perue<lb/>
nerit in Z. ducantur per or<lb/>
biculorum centra line&aelig; GE <lb/>
TX AC QS horizonti &aelig;qui <lb/>
di&longs;tantes.  </s>
             
<s id="id.2.1.193.2.1.2.0"> &amp; &longs;icut in aliis <lb/>
demon&longs;tratum fuit, d uo fu&shy;<lb/>
nes AQ CS duobus XG <lb/>
TE &aelig;quales erunt; &longs;ed AQ <lb/>
CS &longs;imul dupli &longs;unt &longs;patii po<lb/>
tenti&aelig; mot&aelig;; &amp; duo XG TE <lb/>
&longs;imul &longs;unt &longs;imiliter dupli &longs;pa<lb/>
tii ponderis; erit igitur <expan abbr="&longs;pati&utilde;">&longs;patium</expan> <lb/>
potenti&aelig; &longs;patio ponderis &aelig;&shy;<lb/>
quale. quod demon&longs;trare o&shy;<lb/>
portebat. <arrow.to.target n="fig170"></arrow.to.target> </s>
             
<s id="id.2.1.193.2.1.3.0"> [quod demon&longs;trare o&shy;<lb/>
portebat. <arrow.to.target n="fig170"></arrow.to.target>] </s>
    </p>       
<pb/>
       
<p id="id.2.1.193.4.0.0.0" type="main">        
<s id="id.2.1.193.4.1.1.0"> Quod etiam &longs;i vtraq; trochlea duos <lb/>
habuerit orbiculos, quorum centra <lb/>
&longs;int ABCD, funi&longs;q; per omnes cir<lb/>
cumuoluatur, qui in LM religetur; <lb/>
&longs;imiliter o&longs;tendetur potentiam in N <lb/>
&aelig;qualem e&longs;&longs;e ponderi H. vnaqu&aelig;q; <lb/>
enim potentia in EF &longs;u&longs;tinens pon&shy;<lb/>
dus &longs;ubquadrupla e&longs;t ponderis; &amp; po<lb/>
tenti&aelig; in CD dupl&aelig; &longs;unt earum, <lb/>
qu&aelig; &longs;unt in EF; erit vnaqu&aelig;q; po&shy;<lb/>
tentia in CD &longs;ubdupla ponderis H. <lb/>
quare potenti&aelig; in CD &longs;imul &longs;umpt&aelig; <lb/>
ponderi H erunt &aelig;quales.  </s>
             
<s id="id.2.1.193.4.1.2.0"> &amp; quo&shy;<lb/>
niam potentia in N duabus in CD <lb/>
pontentiis e&longs;t &aelig;qualis; erit potentia <lb/>
in N ponderi H, &aelig;qualis. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig170" place="text">       </figure>       
<p id="id.2.1.193.4.2.1.0" type="caption">        
<s id="id.2.1.193.4.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.193.5.0.0.0" type="main">        
<s id="id.2.1.193.5.1.1.0"> Et &longs;i in N &longs;it potentia mouens, &longs;i <lb/>
mili modo o&longs;tendetur, &longs;patium po&shy;<lb/>
tenti&aelig; &aelig;quale e&longs;&longs;e &longs;patio ponderis. </s>
    </p>       
<p id="id.2.1.193.6.0.0.0" type="main">        
<s id="id.2.1.193.6.1.1.0"> Si autem vtraq; trochlea tres, vel <lb/>
quatuor, vel quotcunq; habeat orbi&shy;<lb/>
culos; &longs;emper o&longs;tendetur <expan abbr="pot&etilde;tiam">potentiam</expan> in <lb/>
N &aelig;qualem e&longs;&longs;e ponderi H; &amp; &longs;pa<lb/>
tium potenti&aelig; pondus mouentis &aelig;&shy;<lb/>
quale e&longs;&longs;e &longs;patio ponderis moti. <arrow.to.target n="fig171"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.193.7.0.0.0" type="main">        
<s id="id.2.1.193.7.1.1.0"> Vectium autem motus hoc pacto &longs;e habent; orbiculorum qui <lb/>
dem trochle&aelig; &longs;uperioris, veluti AC in pr&aelig;cedenti figura fulcimen <lb/>
tum e&longs;t C, pondus <expan abbr="ver&ograve;">vero</expan> in A appen&longs;um, &amp; potentia in D medio.  </s>
             
<s id="id.2.1.193.7.1.2.0"> <lb/>
vectes autem orbiculorum trochle&aelig; inferioris ita mouentur, vt ip<lb/>
&longs;ius GE fulcimentum &longs;it E, pondus in medio appen&longs;um, &amp; po<lb/>
tentia in G. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig171" place="text">       </figure>       
<p id="id.2.1.193.7.2.1.0" type="caption">        
<s id="id.2.1.193.7.2.1.0.capt"> YYY </s>
    </p>       
<pb n="96"/>
       
<p id="id.2.1.193.9.0.0.0" type="head">        
<s id="id.2.1.193.9.1.1.0"> PROPOSITIO XXIIII. </s>
    </p>       
<p id="id.2.1.193.10.0.0.0" type="main">        
<s id="id.2.1.193.10.1.1.0"> Si tribus duarum trochlearum orbiculis, qua <lb/>
rum altera vnius dumtaxat orbiculi <expan abbr="&longs;upern&egrave;">&longs;uperne</expan> <expan abbr="&agrave;">a</expan> <lb/>
potentia &longs;u&longs;tineatur, altera <expan abbr="ver&ograve;">vero</expan> duorum <expan abbr="infer&shy;n&egrave;">infer&shy;<lb/>
ne</expan>, ponderiq, alligata fuerit con&longs;tituta, cir&shy;<lb/>
cundetur funis; vtroq; eius extremo alicubi, &longs;ed <lb/>
non &longs;uperiori trochle&aelig; religato: duplum erit <lb/>
pondus potenti&aelig;. </s>
    </p>       
<p id="id.2.1.193.11.0.0.0" type="main">        
<s id="id.2.1.193.11.1.1.0"> Sint AB centra orbiculorum <lb/>
trochle&aelig; ponderi C alligat&aelig;; D ve<lb/>
<expan abbr="r&ograve;">ro</expan> &longs;it centrum orbiculi trochle&aelig; &longs;u<lb/>
perioris; &longs;it deinde funis per om<lb/>
nes orbiculos circumuolutus, reli<lb/>
gatu&longs;q; in EF; &amp; &longs;it potentia in <lb/>
G &longs;u&longs;tinens pondus C.  </s>
     
<s id="id.2.1.193.11.1.1.0.a"> dico pon<lb/>
dus C duplum e&longs;&longs;e potenti&aelig; in G.  </s>
     
<s id="id.2.1.193.11.1.1.0.b"> <lb/>
Quoniam enim &longs;i in H k du&aelig; e&longs;&shy;<lb/>
&longs;ent potenti&aelig; pondus &longs;u&longs;tinentes <lb/>
duobus funibus orbiculis trochle&aelig; <lb/>
inferioris <expan abbr="tant&ugrave;m">tantum</expan> circumuolutis, e&longs;<lb/>
&longs;et vtiq; vtraq; potentia in k H &longs;ub <arrow.to.target n="note275"></arrow.to.target><lb/>
quadrupla ponderis C; &longs;ed poten&shy;<lb/>
tia in G &aelig;qualis e&longs;t potentiis in Hk <arrow.to.target n="note276"></arrow.to.target><lb/>
&longs;imul &longs;umptis; vniu&longs;cuiu&longs;q; enim <lb/>
potenti&aelig; in H, &amp; k dupla e&longs;t: erit <lb/>
potentia in G &longs;ubdupla ponderis <lb/>
C. pondus ergo potenti&aelig; duplum <lb/>
erit. quod demon&longs;trare opor&shy;<lb/>
tebat. <arrow.to.target n="fig172"></arrow.to.target> </s>
             
<s id="id.2.1.193.11.1.2.0"> [quod demon&longs;trare opor&shy;<lb/>
tebat. <arrow.to.target n="fig172"></arrow.to.target>] </s>
    </p>       
<pb/>
       
<p id="id.2.1.193.13.0.0.0" type="main">        
<s id="id.2.1.193.13.1.1.0"> Et &longs;i in G &longs;it potentia mouens pondus.  </s>
             
<s id="id.2.1.193.13.1.2.0"> Dico <lb/>
&longs;patium potenti&aelig; duplum e&longs;&longs;e &longs;patii ponderis. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig172" place="text">       </figure>       
<p id="id.2.1.193.13.2.1.0" type="caption">        
<s id="id.2.1.193.13.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.194.1.0.0.0" type="margin">        
<s id="id.2.1.194.1.1.1.0"> <margin.target id="note275"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 7 <emph type="italics"/>huius<emph.end type="italics"/> </s>
             
<s id="id.2.1.194.1.1.2.0"> <margin.target id="note276"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 15 <emph type="italics"/>huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.195.1.0.0.0" type="main">        
<s id="id.2.1.195.1.1.1.0"> Ii&longs;dem po&longs;itis, &longs;int <lb/>
moti orbiculi, &longs;imiliter <lb/>
demon&longs;trabitur ambos <lb/>
illos LM NO &aelig;quales <lb/>
e&longs;&longs;e quatuor PQ RS <lb/>
TV XY. &longs;ed LM NO <lb/>
&longs;imul dupli &longs;unt &longs;patii po<lb/>
tenti&aelig; in G mot&aelig;; &amp; <lb/>
quatuor PQ RS TV <lb/>
XY &longs;imul quadrupli &longs;unt <lb/>
&longs;patii ponderis moti.&longs;pa <lb/>
tium igitur potenti&aelig; ad <lb/>
&longs;patium ponderis e&longs;t tan<lb/>
quam &longs;ubduplum ad &longs;ub <lb/>
quadruplum.  </s>
             
<s id="id.2.1.195.1.1.2.0"> erit ergo <lb/>
potenti&aelig; &longs;patium pon&shy;<lb/>
deris &longs;patii duplum. <arrow.to.target n="fig173"></arrow.to.target> </s>
    </p>       
<pb n="97"/>
       
<p id="id.2.1.195.3.0.0.0" type="main">        
<s id="id.2.1.195.3.1.1.0"> Hinc autem con&longs;iderandum <lb/>
e&longs;t quomodo fiat motus; quia, <lb/>
<expan abbr="c&ugrave;m">cum</expan> funis &longs;it religatur in F, vectis <lb/>
NO in prima figura habebit ful&shy;<lb/>
cimentum O, pondus in medio, <lb/>
&amp; potentia in N. &longs;imiliter quo&shy;<lb/>
niam funis e&longs;t religatus in E, ve<lb/>
ctis PQ habebit <expan abbr="fulciment&utilde;">fulcimentum</expan> P, &amp; <lb/>
pondus in medio, &amp; potentia in <lb/>
<expan abbr="q.">que</expan> idcirco partes orbiculorum <lb/>
in N, &amp; Q &longs;ur&longs;um mouebuntur; <lb/>
orbiculi ergo non in eandem, &longs;ed <lb/>
in contrarias mouebuntur partes, <lb/>
videlicet vnus dextro&longs;um, alter&longs;i&shy;<lb/>
ni&longs;tror&longs;um.  </s>
             
<s id="id.2.1.195.3.1.2.0"> &amp; quoniam potenti&aelig; <lb/>
in NQ e&aelig;dem &longs;unt, qu&aelig; &longs;unt in <lb/>
LM; potenti&aelig; igitur in LM &aelig;&shy;<lb/>
quales &longs;ur&longs;um mouebuntur.  </s>
             
<s id="id.2.1.195.3.1.3.0"> ve<lb/>
ctis igitur LM in neutram moue<lb/>
bitur partem. quare neq; orbicu <lb/>
lus circumuertetur.  </s>
             
<s id="id.2.1.195.3.1.4.0"> [quare neq; orbicu<lb/>
lus circumuertetur.]  </s>
             
<s id="id.2.1.195.3.1.5.0"> Itaq; LM <lb/>
erit tanquam libra, cuius centrum <lb/>
D, <expan abbr="ponderaqu&eacute;">ponderaque</expan> appen&longs;a in LM <lb/>
&aelig;qualia quart&aelig; parti ponderis C; <lb/>
vnu&longs;qui&longs;q; enim funis LN MQ <lb/>
quartam &longs;u&longs;tinet partem ponderis C. mouebitur ergo totus orbi <lb/>
culus, cuius centrum D, &longs;ur&longs;um; &longs;ed non circumuertetur. <arrow.to.target n="fig174"></arrow.to.target> </s>
    </p>       
<pb/>
       
<p id="id.2.1.195.5.0.0.0" type="main">        
<s id="id.2.1.195.5.1.1.0"> Et &longs;i funis in F circa alios duos <lb/>
voluatur orbiculos, quorum cen&shy;<lb/>
tra &longs;int HK, qui deinde religetur <lb/>
in L; erit proportio ponderis ad <lb/>
potentiam &longs;e&longs;quialtera. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig173" place="text">       </figure>       
<p id="id.2.1.195.5.2.1.0" type="caption">        
<s id="id.2.1.195.5.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig174" place="text">       </figure>       
<p id="id.2.1.195.5.2.3.0" type="caption">        
<s id="id.2.1.195.5.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.195.6.0.0.0" type="main">        
<s id="id.2.1.195.6.1.1.0"> Si enim quatuor e&longs;&longs;ent potenti&aelig; <lb/>
<arrow.to.target n="note277"></arrow.to.target> in MNOI, e&longs;&longs;et vnaqu&aelig;q; &longs;ub&longs;e&longs;&shy;<lb/>
cupla ponderis C, quare quatuor <lb/>
&longs;imul potenti&aelig; in MNOI qua&shy;<lb/>
tuor &longs;ext&aelig; erunt ponderis C. &amp; <lb/>
quoniam du&aelig; &longs;imul potenti&aelig; in <lb/>
HD quatuor potentiis in MNOI <lb/>
&longs;unt &aelig;quales; &amp; potentia in G &aelig;&shy;<lb/>
qualis e&longs;t potentiis in DH: erit <lb/>
potentia in G quatuor &longs;imul po&shy;<lb/>
tentiis in MNOI &aelig;qualis; &amp; ob <lb/>
id quatuor &longs;ext&aelig; erit ponderis C.  </s>
     
<s id="id.2.1.195.6.1.1.0.a"> <lb/>
proportio igitur ponderis C ad po<lb/>
tentiam in G &longs;e&longs;quialtera e&longs;t. </s>
    </p>       
<p id="id.2.1.196.1.0.0.0" type="margin">        
<s id="id.2.1.196.1.1.1.0"> <margin.target id="note277"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 9 <emph type="italics"/>huius<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.197.1.0.0.0" type="main">        
<s id="id.2.1.197.1.1.1.0"> Et &longs;i in G &longs;it potentia mouens, <lb/>
&longs;imili modo o&longs;tendetur &longs;patium <lb/>
potenti&aelig; &longs;patii ponderis &longs;e&longs;quialte<lb/>
rum e&longs;&longs;e. <arrow.to.target n="fig175"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.197.2.0.0.0" type="main">        
<s id="id.2.1.197.2.1.1.0"> Et &longs;i funis in L adhuc circa duos <lb/>
alios orbiculos reuoluatur &longs;imi&shy;<lb/>
liter o&longs;tendetur proportionem <lb/>
ponderis ad potentiam &longs;e&longs;qui&shy;<lb/>
tertiam e&longs;&longs;e.  </s>
             
<s id="id.2.1.197.2.1.2.0"> <expan abbr="qu&ograve;d">quod</expan> &longs;i in G &longs;it <lb/>
potentia mouens, o&longs;tende&shy;<lb/>
tur &longs;patium potenti&aelig; &longs;patii ponde<lb/>
ris &longs;e&longs;quitertium e&longs;&longs;e, atq; ita dein&shy;<lb/>
ceps in infinitum procedendo, <lb/>
quamcunq; proportionem ponderis ad potentiam &longs;uperparticula<lb/>
rem inueniemus &longs;emperq; reperiemus ita e&longs;&longs;e pondus ad poten<lb/>
tiam pondus &longs;u&longs;tinentem, vt &longs;patium potenti&aelig; mouentis ad &longs;pa<lb/>
tium ponderis <expan abbr="&agrave;">a</expan> potentia moti.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig175" place="text">       </figure>       
<p id="id.2.1.197.2.2.1.0" type="caption">        
<s id="id.2.1.197.2.2.1.0.capt"> YYY </s>
    </p>       
<pb n="98"/>
       
<p id="id.2.1.197.4.0.0.0" type="main">        
<s id="id.2.1.197.4.1.1.0"> Motus vectium fit hoc modo, vectis YZ, <expan abbr="c&ugrave;m">cum</expan> funis &longs;it religatus <lb/>
in E, habet fulcimentum in Y, pondus in B medio appen&longs;um, &amp; <lb/>
potentia in Z. &amp; vectis PQ habet fulcimentum in P potentia in <lb/>
medio, &amp; pondus in <expan abbr="q.">que</expan> oportet enim orbiculos, quorum cen&shy;<lb/>
tra&longs;unt BD in eandem partem moueri, videlicet vt QZ &longs;ur&shy;<lb/>
&longs;um moueantur.  </s>
             
<s id="id.2.1.197.4.1.2.0"> &amp; quoniam funis religatus e&longs;t in L, erit T fulci <lb/>
mentum vectis ST, qui pondus habet in medio, &amp; potentia in <lb/>
S. &amp; quia S mouetur &longs;ur&longs;um, nece&longs;&longs;e e&longs;t etiam R &longs;ur&longs;um moue <lb/>
ri; &amp; ideo F erit fulcimentum vectis FR, &amp; pondus erit in R, <lb/>
&amp; potentia in medio.  </s>
             
<s id="id.2.1.197.4.1.3.0"> orbiculi igitur, quorum centra &longs;unt H k, <lb/>
in contrariam mouentur partem eorum, quorum centra &longs;unt BD: <lb/>
quare partes <expan abbr="orbiculor&utilde;">orbiculorum</expan> PF in orbiculis deor&longs;um <expan abbr="tend&etilde;t">tendent</expan>; videlicet <lb/>
ver&longs;us XV.  </s>
     
<s id="id.2.1.197.4.1.3.0.a"> vectis igitur VX in neutram partem mouebitur, <expan abbr="c&ugrave;m">cum</expan> <lb/>
P, &amp; F deor&longs;um moueantur; &amp; VX erit tanquam vectis, in cuius <lb/>
medio erit pondus appen&longs;um, &amp; in VX du&aelig; potenti&aelig; &aelig;quales <lb/>
&longs;ext&aelig; parti ponderis C. potenti&aelig; enim in MO hoc e&longs;t funes PV <lb/>
FX &longs;extam &longs;u&longs;tinent partem ponderis C. totus igitur orbiculus, <lb/>
cuius centrum A &longs;ur&longs;um <expan abbr="vn&agrave;">vna</expan> cum trochlea mouebitur; non au&shy;<lb/>
tem circumuertetur.  </s>
    </p>       
<p id="id.2.1.197.5.0.0.0" type="head">        
<s id="id.2.1.197.5.1.1.0"> PROPOSITIO XXV. </s>
    </p>       
<p id="id.2.1.197.6.0.0.0" type="main">        
<s id="id.2.1.197.6.1.1.0"> Si tribus duarum trochlearum orbiculis, <lb/>
quarum altera binis in&longs;ignita rotulis <expan abbr="&agrave;">a</expan> potentia <lb/>
<expan abbr="&longs;upern&egrave;">&longs;uperne</expan> detineatur; altera <expan abbr="ver&ograve;">vero</expan> vnius <expan abbr="tant&ugrave;m">tantum</expan> <lb/>
rotul&aelig; <expan abbr="infern&egrave;">inferne</expan> <expan abbr="c&otilde;&longs;tituta">con&longs;tituta</expan>, ac ponderi alligata fue<lb/>
rit, circumuoluatur funis; vtroq; eius extremo <lb/>
alicuibi, non autem inferiori trochle&aelig; religa&shy;<lb/>
to: dupla erit ponderis potentia.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.197.8.0.0.0" type="main">        
<s id="id.2.1.197.8.1.1.0"> Sit pondus A trochle&aelig; inferiori alligatum, <lb/>
qu&aelig; orbiculum habeat, cuius centrum &longs;it B; tro<lb/>
chlea <expan abbr="ver&ograve;">vero</expan> &longs;uperior duos orbiculos habeat, <lb/>
quorum centra &longs;int CD; &longs;itq; funis circa om<lb/>
nes orbiculos reuolutus, qui in EF &longs;it religatus; <lb/>
potentiaq; &longs;u&longs;tinens pondus &longs;it in G.  </s>
     
<s id="id.2.1.197.8.1.1.0.a"> dico po<lb/>
tentiam in G ponderis A duplam e&longs;&longs;e.  </s>
             
<s id="id.2.1.197.8.1.2.0"> &longs;i enim <lb/>
<arrow.to.target n="note278"></arrow.to.target> in H k du&aelig; e&longs;&longs;ent potenti&aelig; pondus &longs;u&longs;tinen<lb/>
<arrow.to.target n="note279"></arrow.to.target> tes, e&longs;&longs;et vtraq; &longs;ubdupla ponderis A; &longs;ed po<lb/>
<arrow.to.target n="note280"></arrow.to.target>tentia in D dupla e&longs;t potenti&aelig; in H, &amp; poten<lb/>
tia in C dupla potenti&aelig; in K; quare du&aelig; &longs;imul <lb/>
potenti&aelig; in CD vtriu&longs;q; &longs;imul potenti&aelig; in H k <lb/>
dupl&aelig; erunt.  </s>
             
<s id="id.2.1.197.8.1.3.0"> &longs;ed potenti&aelig; in H k ponderi A &longs;unt <lb/>
&aelig;quales, &amp; potenti&aelig; in CD ip&longs;i potenti&aelig; in G <lb/>
&longs;unt etiam &aelig;quales; potentia igitur in G ponde&shy;<lb/>
ris A dupla erit. quod oportebat demon&longs;trare. </s>
             
<s id="id.2.1.197.8.1.4.0"> [quod oportebat demon&longs;trare.] </s>
    </p>       
<p id="id.2.1.198.1.0.0.0" type="margin">        
<s id="id.2.1.198.1.1.1.0"> <margin.target id="note278"></margin.target>2. <emph type="italics"/>Cor.<emph.end type="italics"/> </s>
             
<s id="id.2.1.198.1.1.2.0"> <margin.target id="note279"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.198.1.1.3.0"> <margin.target id="note280"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 15 <emph type="italics"/>huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.199.1.0.0.0" type="main">        
<s id="id.2.1.199.1.1.1.0"> Si autem in G &longs;it potentia mouens pon&shy;<lb/>
dus, &longs;imiliter vt in pr&aelig;cedenti o&longs;tendetur &longs;pa<lb/>
tium ponderis &longs;patii potenti&aelig; duplum e&longs;&longs;e. <arrow.to.target n="fig176"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.199.2.0.0.0" type="main">        
<s id="id.2.1.199.2.1.1.0"> Hinc quoq; con&longs;iderandum e&longs;t vectem PQ <lb/>
non moueri, quia vectis LM habet fulcimen <lb/>
tum in L, potentia in medio, &amp; pondus in M.  </s>
     
<s id="id.2.1.199.2.1.1.0.a"> <lb/>
vectis autem NO habet fulcimentum in O, <lb/>
potentia in medio, &amp; pondus in N. quare M, &amp; N &longs;ur&longs;um mo<lb/>
uebuntur.  </s>
             
<s id="id.2.1.199.2.1.2.0"> in contrarias igitur partes orbiculi, quorum centra <lb/>
&longs;unt CD mouentur.  </s>
             
<s id="id.2.1.199.2.1.3.0"> idcirco vectis PQ in neutram partem mo<lb/>
uebitur; eritq;, ac &longs;i in medio e&longs;&longs;et appen&longs;um pondus, &amp; in PQ <lb/>
du&aelig; potenti&aelig; &aelig;quales dimidio ponderis A. vtraq; enim potentia <lb/>
in HK &longs;ubdupla e&longs;t ponderis A. totus igitur orbiculus, cuius <lb/>
centrum B &longs;ur&longs;um mouebitur, &longs;ed non circumuertetur.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig176" place="text">       </figure>       
<p id="id.2.1.199.2.2.1.0" type="caption">        
<s id="id.2.1.199.2.2.1.0.capt"> YYY </s>
    </p>       
<pb n="99"/>
       
<p id="id.2.1.199.4.0.0.0" type="main">        
<s id="id.2.1.199.4.1.1.0"> Et &longs;i funis in F duobus aliis adhuc circumuol&shy;<lb/>
uatur orbiculis, quorum centra &longs;int HK, qui de&shy;<lb/>
inde religetur in L; erit proportio potenti&aelig; in G <lb/>
ad pondus A &longs;e&longs;quialtera. </s>
    </p>       
<p id="id.2.1.199.5.0.0.0" type="main">        
<s id="id.2.1.199.5.1.1.0"> Si enim in MNOP quatuor e&longs;&longs;ent poten<lb/>
ti&aelig; pondus &longs;u&longs;tinentes, vnaqu&aelig;q; &longs;ubquadru<arrow.to.target n="note281"></arrow.to.target><lb/>
pla e&longs;&longs;et ponderis A: &longs;ed <expan abbr="c&ugrave;m">cum</expan> potentia in k <arrow.to.target n="note282"></arrow.to.target><lb/>
&longs;it dupla potenti&aelig; in N; erit potentia in k <lb/>
ponderis A &longs;ubdupla.  </s>
             
<s id="id.2.1.199.5.1.2.0"> &amp; quoniam potentia <lb/>
in D duabus in MO potentiis e&longs;t &aelig;qualis; erit <lb/>
quoq; potentia in D ponderis A &longs;ubdupla.  </s>
             
<s id="id.2.1.199.5.1.3.0"> <lb/>
<expan abbr="c&ugrave;m">cum</expan> autem adhuc potentia in C potenti&aelig; in P <lb/>
&longs;it dupla, erit &longs;imiliter <expan abbr="pot&etilde;tia">potentia</expan> in C ponderis A <lb/>
&longs;ubdupla.  </s>
             
<s id="id.2.1.199.5.1.4.0"> tres igitur potenti&aelig; in CD k tribus <lb/>
medietatibus ponderis A &longs;unt &aelig;quales.  </s>
             
<s id="id.2.1.199.5.1.5.0"> quo&shy;<lb/>
niam autem potentia in G potentiis in CDK <lb/>
e&longs;t &aelig;qualis, erit potentia in G tribus medie&shy;<lb/>
tatibus ponderis A &aelig;qualis.  </s>
             
<s id="id.2.1.199.5.1.6.0"> Proportio igi&shy;<lb/>
tur potenti&aelig; ad pondus &longs;e&longs;quialtera e&longs;t. </s>
    </p>       
<p id="id.2.1.200.1.0.0.0" type="margin">        
<s id="id.2.1.200.1.1.1.0"> <margin.target id="note281"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 7 <emph type="italics"/>huius<emph.end type="italics"/> </s>
             
<s id="id.2.1.200.1.1.2.0"> <margin.target id="note282"></margin.target>15 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.201.1.0.0.0" type="main">        
<s id="id.2.1.201.1.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> in G &longs;it potentia mouens, erit &longs;pa<lb/>
tium ponderis &longs;patii potenti&aelig; &longs;e&longs;quialterum. <arrow.to.target n="fig177"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.201.2.0.0.0" type="main">        
<s id="id.2.1.201.2.1.1.0"> Et &longs;i funis in L adhuc circa duos alios or<lb/>
biculos reuoluatur, &longs;imiliter o&longs;tendetur pro&shy;<lb/>
portionem potenti&aelig; ad pondus &longs;e&longs;quitertiam <lb/>
e&longs;&longs;e.  </s>
             
<s id="id.2.1.201.2.1.2.0"> &amp; &longs;ic in infinitum omnes proportiones <lb/>
potenti&aelig; ad pondus &longs;uperparticulares inue&shy;<lb/>
niemus.  </s>
             
<s id="id.2.1.201.2.1.3.0"> o&longs;tendemu&longs;q; potentiam pondus <lb/>
&longs;u&longs;tinentem ad pondus ita e&longs;&longs;e, vt &longs;patium <lb/>
ponderis moti ad <expan abbr="&longs;pat&igrave;um">&longs;patium</expan> potenti&aelig; pondus <lb/>
mouentis.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig177" place="text">       </figure>       
<p id="id.2.1.201.2.2.1.0" type="caption">        
<s id="id.2.1.201.2.2.1.0.capt"> YYY </s>
    </p>       
<pb/>
       
<p id="id.2.1.201.4.0.0.0" type="main">        
<s id="id.2.1.201.4.1.1.0"> Motus vectium fiet hoc <lb/>
modo, videlicet Q erit ful<lb/>
cimentum vectis QR, po&shy;<lb/>
tentia in medio, pondus <lb/>
in R; &amp; vectis Z 9 fulci <lb/>
mentum erit Z, pondus in <lb/>
medio, potentiaq; in 9. &longs;i <lb/>
militer X erit fulcimentum <lb/>
vectis VX, potentia in me <lb/>
dio, pondus in V.  </s>
     
<s id="id.2.1.201.4.1.1.0.a"> &amp; quo<lb/>
niam V &longs;ur&longs;um mouetur, Y <lb/>
quoq; &longs;ur&longs;um mouebitur; <lb/>
&amp; vectis YF fulcimentum <lb/>
erit F: quare F, &amp; Z in orbi <lb/>
culis deor&longs;um mouebun&shy;<lb/>
tur.  </s>
             
<s id="id.2.1.201.4.1.2.0"> &amp; ob id vectis ST in <lb/>
neutram mouebitur par&shy;<lb/>
tem; &amp; ST erit tamquam <lb/>
libra, cuius centrum D, &amp; <lb/>
pondera in ST &aelig;qualia <lb/>
quart&aelig; parti ponderis A. <lb/>
vnu&longs;qui&longs;q; enim funis SZ <lb/>
TF quartam &longs;u&longs;tinet par&shy;<lb/>
tem ponderis A. orbicu&shy;<lb/>
lus ergo, cuius centrum D, <lb/>
&longs;ur&longs;um mouebitur; non au<lb/>
tem circumuertetur. <arrow.to.target n="fig178"></arrow.to.target> </s>
    </p>       
<pb n="100"/>
       
<p id="id.2.1.201.6.0.0.0" type="main">        
<s id="id.2.1.201.6.1.1.0"> Hactenus proportiones ponderis ad potentiam multiplices, <lb/>
&amp; &longs;ubmultiplices; deinde &longs;uperparticulares, <expan abbr="&longs;ub&longs;uperparticu&shy;lare&longs;qu&eacute;">&longs;ub&longs;uperparticu&shy;<lb/>
lare&longs;que</expan> declarat&aelig; fuerunt: nunc autem reliquum e&longs;t, vt propor&shy;<lb/>
tiones inter pondus, &amp; potentiam &longs;uperpartientes, &amp; multi&shy;<lb/>
plices &longs;uperparticulares, <expan abbr="multiplicesqu&eacute;">multiplicesque</expan> &longs;uperpartientes mani&shy;<lb/>
fe&longs;tentur.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig178" place="text">       </figure>       
<p id="id.2.1.201.6.2.1.0" type="caption">        
<s id="id.2.1.201.6.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.201.7.0.0.0" type="head">        
<s id="id.2.1.201.7.1.1.0"> PROPOSITIO XXVI. </s>
     <lb/>
       
<s id="id.2.1.201.7.3.1.0"> PROBLEMA. </s>
    </p>       
<p id="id.2.1.201.8.0.0.0" type="main">        
<s id="id.2.1.201.8.1.1.0"> Si proportionem &longs;uperpartientem inuenire <lb/>
volumus, quemadmodum &longs;i proportio, quam <lb/>
habet pondus ad potentiam pondus &longs;u&longs;tinen&shy;<lb/>
tem fuerit &longs;uperbipartiens, &longs;icut quinque ad <lb/>
tria.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.201.10.0.0.0" type="main">        
<s id="id.2.1.201.10.1.1.0"> <arrow.to.target n="note283"></arrow.to.target> Exponatur potentia in A pondus B &longs;u&longs;ti<lb/>
nens, proportionemq; habeat pondus B ad <lb/>
potentiam in A, vt quinq; ad vnum; hoc e&longs;t, <lb/>
&longs;it potentia in A &longs;ubquintupla ponderis B: de&shy;<lb/>
inde eodem fune circa alios orbiculos reuo&shy;<lb/>
<arrow.to.target n="note284"></arrow.to.target> luto inueniatur potentia in C, qu&aelig; tripla &longs;it <lb/>
potenti&aelig; in A.  </s>
     
<s id="id.2.1.201.10.1.1.0.a"> &amp; quoniam pondus B ad po<lb/>
tentiam in A e&longs;t, vt quinq; ad vnum; &amp; <lb/>
potentia in A ad potentiam in C e&longs;t, vt vnum <lb/>
ad tria; erit pondus B ad potentiam in C, vt <lb/>
quinq; ad tria; hoc e&longs;t &longs;uperbipartiens. </s>
    </p>       
<p id="id.2.1.202.1.0.0.0" type="margin">        
<s id="id.2.1.202.1.1.1.0"> <margin.target id="note283"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 9 <emph type="italics"/>huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.202.1.1.2.0"> <margin.target id="note284"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 17 <emph type="italics"/>huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.203.1.0.0.0" type="main">        
<s id="id.2.1.203.1.1.1.0"> Et hoc modo omnes proportiones ponde<lb/>
ris ad potentiam &longs;uperpartientes inuenientur; <lb/>
vt &longs;i &longs;upertripartientem quis inuenire volue&shy;<lb/>
rit; eodem incedat ordine; fiat &longs;cilicet poten<lb/>
tia in A &longs;u&longs;tinens pondus B &longs;ub&longs;eptupla ip&shy;<lb/>
&longs;ius ponderis B; deinde fiat potentia in C ip&shy;<lb/>
&longs;ius A quadrupla; erit pondus B ad poten&shy;<lb/>
tiam in C, vt &longs;eptem ad quatuor: <expan abbr="v&iacute;delicet">videlicet</expan> <lb/>
&longs;upertripartiens. </s>
    </p>       
<p id="id.2.1.203.2.0.0.0" type="main">        
<s id="id.2.1.203.2.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> in C &longs;it potentia mo&shy;<lb/>
uens pondus erit &longs;patium <expan abbr="pot&etilde;ti&aelig;">potenti&aelig;</expan> <lb/>
&longs;patii ponderis &longs;uperbipartiens. <arrow.to.target n="fig179"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.203.3.0.0.0" type="main">        
<s id="id.2.1.203.3.1.1.0"> <arrow.to.target n="note285"></arrow.to.target> Spatium enim potenti&aelig; in C tertia pars <lb/>
e&longs;t &longs;patii potenti&aelig; in A, ita videlicet &longs;e habent, <lb/>
vt quinq; ad quindecim; &amp; &longs;patium potenti&aelig; <lb/>
<arrow.to.target n="note286"></arrow.to.target> in A quintuplum e&longs;t &longs;patii ponderis B, hoc <lb/>
e&longs;t, vt quindecim ad tria; erit igitur &longs;patium <lb/>
potenti&aelig; in C ad &longs;patium ponderis B, vt <lb/>
quinq; ad tria; videlicet &longs;uperbipartiens.  </s>
             
<s id="id.2.1.203.3.1.2.0"> &amp; &longs;emper o&longs;tendemus, ita <lb/>
e&longs;&longs;e &longs;patium potenti&aelig; mouentis ad &longs;patium ponderis; vt pondus <lb/>
ad potentiam pondus &longs;u&longs;tinentem. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig179" place="text">       </figure>       
<p id="id.2.1.203.3.2.1.0" type="caption">        
<s id="id.2.1.203.3.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.204.1.0.0.0" type="margin">        
<s id="id.2.1.204.1.1.1.0"> <margin.target id="note285"></margin.target>17 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.204.1.1.2.0"> <margin.target id="note286"></margin.target>14 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.205.1.0.0.0" type="main">        
<s id="id.2.1.205.1.1.1.0"> Similiq; pror&longs;us ratione proportionem potenti&aelig; ad pondus &longs;u&shy;
<pb n="101"/>
perpartientem inueniemus.  </s>
             
<s id="id.2.1.205.1.1.2.0"> &longs;i enim C e&longs;&longs;et inferius, &amp; in ip&longs;o <lb/>
appen&longs;um e&longs;&longs;et pondus; B <expan abbr="ver&ograve;">vero</expan> &longs;uperius, in quo e&longs;&longs;et potentia pon<lb/>
dus in C &longs;u&longs;tinens, e&longs;&longs;et potentia in B &longs;uperbipartiens ponderis <lb/>
in C appen&longs;i: <expan abbr="c&ugrave;m">cum</expan> B ad A &longs;it, vtquinq; ad vnum; A <expan abbr="ver&ograve;">vero</expan> ad <arrow.to.target n="note287"></arrow.to.target><lb/>
C, vt vnum ad tria. <arrow.to.target n="note288"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.205.2.0.0.0" type="main">        
<s id="id.2.1.205.2.1.1.0"> Si autem multiplicem &longs;uperparticularem in&shy;<lb/>
uenire voluerimus; vt proportio, quam habet <lb/>
pondus ad potentiam pondus &longs;u&longs;tinentem, &longs;it <lb/>
duplex &longs;e&longs;quialtera, vt quinq; ad duo. </s>
    </p>       
<p id="id.2.1.206.1.0.0.0" type="margin">        
<s id="id.2.1.206.1.1.1.0"> <margin.target id="note287"></margin.target>18 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.206.1.1.2.0"> <margin.target id="note288"></margin.target>5 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.207.1.0.0.0" type="main">        
<s id="id.2.1.207.1.1.1.0"> Eodem modo, quo &longs;uperpartientes inuenimus, has quo&shy;<lb/>
que omnes multiplices &longs;uperparticulares reperiemus.  </s>
             
<s id="id.2.1.207.1.1.2.0"> vt fiat <arrow.to.target n="note289"></arrow.to.target><lb/>
pondus B ad potentiam in A, vt quinq; ad vnum; potentia ve <arrow.to.target n="note290"></arrow.to.target><expan abbr="r&ograve;"><lb/>
ro</expan> in C ad potentiam in A, vt duo ad vnum; quod fiet, &longs;i fu&shy;<lb/>
nis &longs;it religatus in D, non autem trochle&aelig; &longs;uperiori, vel in F: erit <lb/>
pondus B ad potentiam in C, vt quinq; ad duo; hoc e&longs;t duplum <lb/>
&longs;e&longs;quialterum. </s>
    </p>       
<p id="id.2.1.208.1.0.0.0" type="margin">        
<s id="id.2.1.208.1.1.1.0"> <margin.target id="note289"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 9 <emph type="italics"/>huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.208.1.1.2.0"> <margin.target id="note290"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 15, 16, <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.209.1.0.0.0" type="main">        
<s id="id.2.1.209.1.1.1.0"> Et <expan abbr="&egrave;">e</expan> conuer&longs;o proportionem potenti&aelig; ad pondus multiplicem <lb/>
&longs;uperparticularem inueniemus; &amp; vt in reliquis o&longs;tendetur, ita e&longs; <lb/>
&longs;e &longs;patium potenti&aelig; mouentis ad &longs;patium ponderis, vt pondus <lb/>
ad potentiam pondus &longs;u&longs;tinentem. </s>
    </p>       
<p id="id.2.1.209.2.0.0.0" type="main">        
<s id="id.2.1.209.2.1.1.0"> Omnem quoq; multiplicem &longs;uperpartientem <lb/>
eodem modo inueniemus; vt &longs;i proportio, quam <lb/>
habet pondus ad potentiam, &longs;it duplex &longs;uperbi <lb/>
partiens, vt octo ad tria. </s>
    </p>       
<p id="id.2.1.209.3.0.0.0" type="main">        
<s id="id.2.1.209.3.1.1.0"> Fiat potentia in A pondus B &longs;u&longs;tinens &longs;uboctupla ponderis B; <arrow.to.target n="note291"></arrow.to.target><lb/>
&amp; potentia in C potenti&aelig; in A &longs;it tripla; erit pondus B ad po<lb/>
tentiam in C, vt octo ad tria.  </s>
             
<s id="id.2.1.209.3.1.2.0"> &amp; <expan abbr="&egrave;">e</expan> conuer&longs;o omnem potenti&aelig; ad 
<pb/>
pondus proportionem multipticem &longs;uperpartientem in ueniemus.  </s>
             
<s id="id.2.1.209.3.1.3.0"> <lb/>
&amp; vt in c&aelig;teris reperiemus ita e&longs;&longs;e pondus ad potentiam pondus <lb/>
&longs;u&longs;tinentem, vt &longs;patium potenti&aelig; mouentis ad &longs;patium pon&shy;<lb/>
deris. </s>
    </p>       
<p id="id.2.1.210.1.0.0.0" type="margin">        
<s id="id.2.1.210.1.1.1.0"> <margin.target id="note291"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 9 <emph type="italics"/>huius Ex<emph.end type="italics"/> 17 <emph type="italics"/>huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.211.1.0.0.0" type="main">        
<s id="id.2.1.211.1.1.1.0"> Notandum autem e&longs;t, <expan abbr="qu&ograve;d">quod</expan> <expan abbr="c&ugrave;m">cum</expan> in pr&aelig;cedentibus demo&longs;tratio <lb/>
nibus &longs;&aelig;pius dictum fuerit, potentiam pondus &longs;u&longs;tinentem ip&longs;ius <lb/>
ponderis duplam e&longs;&longs;e, vel triplam, &amp; huiu&longs;modi; vt in decima&shy;<lb/>
quinta huius o&longs;ten&longs;um e&longs;t; quia tamen potentia non &longs;olum pon<lb/>
dus, <expan abbr="ver&ugrave;m">verum</expan> etiam trochleam &longs;u&longs;tinet; idcirco maioris <expan abbr="long&egrave;">longe</expan> vir&shy;<lb/>
tutis, maiori&longs;q; ip&longs;i ponderi proportionis con&longs;tituenda videtur <lb/>
ip&longs;a potentia.  </s>
             
<s id="id.2.1.211.1.1.2.0"> quod quidem verum e&longs;t, &longs;i etiam trochle&aelig; graui<lb/>
tatem con&longs;iderare voluerimus.  </s>
             
<s id="id.2.1.211.1.1.3.0"> &longs;ed quoniam inter potentiam, &amp; <lb/>
pondus proportionem qu&aelig;rimus: ideo hanc trochle&aelig; grauitatem <lb/>
ommi&longs;imus, quam &longs;iquis etiam con&longs;iderare voluerit, vim ip&longs;i po&shy;<lb/>
tenti&aelig; &aelig;qualem trochle&aelig; addere poterit.  </s>
             
<s id="id.2.1.211.1.1.4.0"> Quod ip&longs;um etiam in <lb/>
fune ob&longs;eruari poterit.  </s>
             
<s id="id.2.1.211.1.1.5.0"> &amp; &longs;icut hoc in decimaquinta con&longs;ideraui<lb/>
mus, idem quoq; in reliquis aliis con&longs;iderare poterimus.  </s>
    </p>       
<pb n="97"/>
       
<p id="id.2.1.211.3.0.0.0" type="main">        
<s id="id.2.1.211.3.1.1.0"> Noui&longs;&longs;e etiam oportet, <expan abbr="qu&ograve;d">quod</expan> &longs;icuti proportio <lb/>
nes omnes inter potentiam, &amp; pondus vnico <lb/>
fune inuent&aelig; fuerunt; ita etiam pluribus funi&shy;<lb/>
bus, <expan abbr="trochlei&longs;qu&eacute;">trochlei&longs;que</expan> e&aelig;dem inueniri poterunt.  </s>
             
<s id="id.2.1.211.3.1.2.0"> vt <lb/>
&longs;i multiplicem &longs;uperparticularem proportionem <lb/>
pluribus funibus inuenire voluerimus, veluti &longs;i <lb/>
proportio, quam habet pondus ad potentiam <lb/>
pondus &longs;u&longs;tinentem, fuerit duplex &longs;e&longs;quialtera, vt <lb/>
quinq; ad duo; oportet hanc proportionem ex <lb/>
pluribus componere.  </s>
             
<s id="id.2.1.211.3.1.3.0"> vt (exempli gratia) ex pro&shy;<lb/>
portione &longs;e&longs;quiquarta, vt <expan abbr="quinqu&eacute;">quinque</expan> ad quatuor, <lb/>
&amp; ex dupla, vt quatuor ad duo.  </s>
             
<s id="id.2.1.211.3.1.4.0"> exponatur igitur po<arrow.to.target n="note292"></arrow.to.target><lb/>
tentia in A pondus B &longs;u&longs;tinens, ad quam pondus <lb/>
<expan abbr="proportion&etilde;">proportionem</expan> habeat &longs;e&longs;quiquartam, vt quinq; ad <lb/>
quatuor: deinde alio fune inueniatur <expan abbr="pot&etilde;tia">potentia</expan> in C,<arrow.to.target n="note293"></arrow.to.target><lb/>
cuius dupla &longs;it potentia in A.  </s>
     
<s id="id.2.1.211.3.1.4.0.a"> &amp; <expan abbr="quoni&atilde;">quoniam</expan> B ad A e&longs;t, <lb/>
vt quinq; ad quatuor; &amp; A ad C, vt quatuor ad <lb/>
duo; erit pondus B ad potentiam in C, vt quin<lb/>
que ad duo; hoc e&longs;t proportionem habebit du&shy;<lb/>
plicem &longs;e&longs;quialteram. <arrow.to.target n="fig180"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.211.4.0.0.0" type="main">        
<s id="id.2.1.211.4.1.1.0"> Et notandum e&longs;t hanc quoq; <expan abbr="proportion&etilde;">proportionem</expan> inue<lb/>
niri po&longs;&longs;e, &longs;i proportionem quinq; ad duo ex pluri<lb/>
bus componamus, vt quinq; ad quindecim &amp; quin<lb/>
decim ad viginti &amp; viginti ad duo.  </s>
             
<s id="id.2.1.211.4.1.2.0"> Et hoc modo <lb/>
non &longs;olum omnem aliam proportionem inuenie<lb/>
mus, &longs;ed quamcunq, multis, <expan abbr="infinitisqu&eacute;">infinitisque</expan> mo&shy;<lb/>
dis comperiemus.  </s>
             
<s id="id.2.1.211.4.1.3.0"> omnis enim proportio ex infi&shy;<lb/>
nitis proportionibus componi pote&longs;t.  </s>
             
<s id="id.2.1.211.4.1.4.0"> vt patet <lb/>
in commentario E utocii in quartam propo&longs;itio&shy;<lb/>
nem &longs;ecundi libri Archimedis de &longs;phera, &amp; cy&shy;<lb/>
lindro. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig180" place="text">       </figure>       
<p id="id.2.1.211.4.2.1.0" type="caption">        
<s id="id.2.1.211.4.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.212.1.0.0.0" type="margin">        
<s id="id.2.1.212.1.1.1.0"> <margin.target id="note292"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 21 <emph type="italics"/>huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.212.1.1.2.0"> <margin.target id="note293"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 2 <emph type="italics"/>huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.213.1.0.0.0" type="main">        
<s id="id.2.1.213.1.1.1.0"> Po&longs;&longs;umus quoq; pluribus funibus, trochleis <lb/>
<expan abbr="ver&ograve;">vero</expan> inferioribus <expan abbr="tant&ugrave;m">tantum</expan>, vel &longs;uperioribus vti.  </s>
    </p>       
<pb/>
       
<p id="id.2.1.213.3.0.0.0" type="main">        
<s id="id.2.1.213.3.1.1.0"> Sit pondus A, cui alligata &longs;it trochlea <lb/>
orbiculum habens, cuius centrum B; <lb/>
religetur funis in C, qui circa orbiculum <lb/>
reuoluatur, funi&longs;q; perueniat in D: erit <lb/>
<arrow.to.target n="note294"></arrow.to.target> potentia in D &longs;u&longs;tinens pondus A &longs;ub&shy;<lb/>
dupla ponderis A.  </s>
     
<s id="id.2.1.213.3.1.1.0.a"> deinde funis in D <lb/>
alteri trochle&aelig; religetur, &amp; circa huius <lb/>
trochle&aelig; orbiculum alius reuoluatur fu <lb/>
nis, qui religetur in E, &amp; perueniat in <lb/>
<arrow.to.target n="note295"></arrow.to.target> F; erit potentia in F &longs;ubdupla eius, <lb/>
quod &longs;u&longs;tinet <expan abbr="pot&etilde;tia">potentia</expan> in D: e&longs;tenim ac&longs;i <lb/>
D dimidium ponderis A &longs;u&longs;tineret &longs;i <lb/>
ne trochlea; quare potentia in F &longs;ubqua&shy;<lb/>
drupla erit ponderis A. &amp; &longs;i adhuc fu <lb/>
nis in F alteri trochle&aelig; religetur, &amp; <lb/>
per eius orbiculum circumuoluatur a&shy;<lb/>
lius funis, qui religetur in G, &amp; per <lb/>
ueniat in H; erit potentia in H &longs;ub <lb/>
dupla potenti&aelig; in F.  </s>
     
<s id="id.2.1.213.3.1.1.0.b"> ergo potentia in <lb/>
H &longs;uboctupla erit ponderis A. &amp; &longs;ic <lb/>
in infinitum &longs;emper &longs;ubduplam poten<lb/>
tiam <expan abbr="pr&aelig;ced&etilde;tis">pr&aelig;cedentis</expan> potenti&aelig; inueniemus. <arrow.to.target n="fig181"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.213.4.0.0.0" type="main">        
<s id="id.2.1.213.4.1.1.0"> Et &longs;i in H &longs;it potentia mouens, erit <lb/>
&longs;patium potenti&aelig; &longs;patii ponderis octu<lb/>
<arrow.to.target n="note296"></arrow.to.target> plum.  </s>
             
<s id="id.2.1.213.4.1.2.0"> &longs;patium enim D duplum e&longs;t &longs;pa<lb/>
tii ponderis A, &amp; &longs;patium F &longs;patii D <lb/>
duplum; erit &longs;patium F &longs;patii ponde<lb/>
ris A quadruplum.  </s>
             
<s id="id.2.1.213.4.1.3.0"> &longs;imiliter quoniam <lb/>
&longs;patium potenti&aelig; in H <expan abbr="dupl&utilde;">duplum</expan> e&longs;t &longs;patii <lb/>
F, erit &longs;patium potenti&aelig; in H &longs;patii <lb/>
ponderis A octuplum.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig181" place="text">       </figure>       
<p id="id.2.1.213.4.2.1.0" type="caption">        
<s id="id.2.1.213.4.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.214.1.0.0.0" type="margin">        
<s id="id.2.1.214.1.1.1.0"> <margin.target id="note294"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.214.1.1.2.0"> <margin.target id="note295"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.214.1.1.3.0"> <margin.target id="note296"></margin.target>11 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.215.1.0.0.0" type="main">        
<pb n="103"/>
       
<s id="id.2.1.215.1.2.1.0"> Sit deinde pondus A funi alliga&shy;<lb/>
tum, qui orbiculo trochle&aelig; &longs;uperio<lb/>
ris &longs;it circumuolutus, &amp; religatus in <arrow.to.target n="note297"></arrow.to.target><lb/>
B; &longs;itq; potentia in C &longs;u&longs;tinens pon<lb/>
dus A: erit potentia in C ponderis A <lb/>
dupla, deinde C alteri funi religetur, <lb/>
qui per alterius trochle&aelig; orbicu<lb/>
lum circumuoluatur, &amp; religetur <lb/>
in D; erit potentia in E dupla po<arrow.to.target n="note298"></arrow.to.target><lb/>
tenti&aelig; in C.  </s>
     
<s id="id.2.1.215.1.2.1.0.a"> Quare potentia in E <lb/>
quadrupla erit ponderis A.  </s>
     
<s id="id.2.1.215.1.2.1.0.b"> &amp; &longs;i ad <lb/>
huc E alteri funi religetur, qui etiam <lb/>
circa orbiculum alterius trochle&aelig; re<lb/>
uoluatur, &amp; religetur in F; erit poten<lb/>
tia in G dupla potenti&aelig; in E.  </s>
     
<s id="id.2.1.215.1.2.1.0.c"> ergo <lb/>
potentia in G octupla erit ponderis <lb/>
A. &amp; &longs;ic in infinitum &longs;emper pr&aelig; <lb/>
cedentis potenti&aelig; potentiam du&shy;<lb/>
plam inueniemus. <arrow.to.target n="fig182"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.215.2.0.0.0" type="main">        
<s id="id.2.1.215.2.1.1.0"> Si autem in G &longs;it potentia mo&shy;<lb/>
uens,  <arrow.to.target n="note299"></arrow.to.target> erit &longs;patium ponderis octu&shy;<lb/>
plum &longs;patii potenti&aelig; in G. &longs;patium <lb/>
enim ponderis A duplum e&longs;t &longs;patii <lb/>
potenti&aelig; in C, &amp; C duplum e&longs;t &longs;patii <lb/>
ip&longs;ius E; quare &longs;patium ponderis <lb/>
A &longs;patii potenti&aelig; in E quadruplum <lb/>
erit.  </s>
             
<s id="id.2.1.215.2.1.2.0"> &longs;imiliter quoniam &longs;patium E <lb/>
duplum e&longs;t &longs;patii potenti&aelig; in G; erit ergo &longs;patium ponderis A <lb/>
octuplum &longs;patii potenti&aelig; in G. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig182" place="text">       </figure>       
<p id="id.2.1.215.2.2.1.0" type="caption">        
<s id="id.2.1.215.2.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.216.1.0.0.0" type="margin">        
<s id="id.2.1.216.1.1.1.0"> <margin.target id="note297"></margin.target>15 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.216.1.1.2.0"> <margin.target id="note298"></margin.target><emph type="italics"/>Ex e adem.<emph.end type="italics"/> </s>
             
<s id="id.2.1.216.1.1.3.0"> <margin.target id="note299"></margin.target>16 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.217.1.0.0.0" type="head">        
<pb/>
       
<s id="id.2.1.217.1.2.1.0"> COROLLARIVM. </s>
    </p>       
<p id="id.2.1.217.2.0.0.0" type="main">        
<s id="id.2.1.217.2.1.1.0"> Ex his manife&longs;tum e&longs;t maiorem &longs;emper ha&shy;<lb/>
bere proportionem &longs;patium potenti&aelig; mouen&shy;<lb/>
tis ad &longs;patium ponderis moti, <expan abbr="qu&agrave;m">quam</expan> pondus <lb/>
ad eandem potentiam. </s>
    </p>       
<p id="id.2.1.217.3.0.0.0" type="main">        
<s id="id.2.1.217.3.1.1.0"> Hoc autem ex iis, qu&aelig; in corollario quart&aelig; huius de vecte dicta <lb/>
&longs;unt, patet.  </s>
    </p>       
<p id="id.2.1.217.4.0.0.0" type="head">        
<s id="id.2.1.217.4.1.1.0"> PROPOSITIO XXVII. </s>
     <lb/>
       
<s id="id.2.1.217.4.3.1.0"> PROBLEMA. </s>
    </p>       
<p id="id.2.1.217.5.0.0.0" type="main">        
<s id="id.2.1.217.5.1.1.0"> Datum pondus <expan abbr="&agrave;">a</expan> data potentia trochleis <lb/>
moueri. </s>
    </p>       
<p id="id.2.1.217.6.0.0.0" type="main">        
<s id="id.2.1.217.6.1.1.0"> Data potentia, vel e&longs;t maior, vel &aelig;qualis, vel minor dato <lb/>
pondere.  </s>
    </p>       
<pb n="104"/>
       
<p id="id.2.1.217.8.0.0.0" type="main">        
<s id="id.2.1.217.8.1.1.0"> Et &longs;i e&longs;t maior, tunc poten&shy;<lb/>
tia, vel ab&longs;q; alio in&longs;trumento, <lb/>
vel fune circa orbiculum trochle&aelig; <lb/>
&longs;ur&longs;um appen&longs;&aelig; reuoluto datum <lb/>
pondus mouebit.  </s>
             
<s id="id.2.1.217.8.1.2.0"> Minor enim po<arrow.to.target n="note300"></arrow.to.target><lb/>
tentia; <expan abbr="qu&agrave;m">quam</expan> data, ponderi&aelig;que&shy;<lb/>
ponderat, data ergo mouebit.  </s>
             
<s id="id.2.1.217.8.1.3.0"> <lb/>
Quod idem fieri pote&longs;t iuxta om&shy;<lb/>
nes propo&longs;itiones, quibus poten&shy;<lb/>
tia pondus &longs;u&longs;tinens, vel &aelig;qualis, <lb/>
vel minor pondere o&longs;ten&longs;a e&longs;t. <arrow.to.target n="fig183"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.217.9.0.0.0" type="main">        
<s id="id.2.1.217.9.1.1.0"> Si autem &aelig;qualis, <lb/>
pondus mouebit fune <lb/>
per orbiculum trochle&aelig; <lb/>
ponderi alligat&aelig; circum <lb/>
uoluto.  </s>
             
<s id="id.2.1.217.9.1.2.0"> potentia enim <arrow.to.target n="note301"></arrow.to.target><lb/>
&longs;u&longs;tinens pondus &longs;ubdu<lb/>
pla e&longs;t ponderis, poten<lb/>
tia igitur ponderi &aelig;qua <lb/>
lis datum pondus mo&shy;<lb/>
uebit.  </s>
             
<s id="id.2.1.217.9.1.3.0"> Quod etiam <expan abbr="&longs;e&shy;cund&ugrave;m">&longs;e&shy;<lb/>
cundum</expan> propo&longs;itiones, <lb/>
quibus potentiam pon<lb/>
dere minorem e&longs;&longs;e o&shy;<lb/>
&longs;ten&longs;um e&longs;t, fieri po&shy;<lb/>
te&longs;t. <arrow.to.target n="fig184"></arrow.to.target> </s>
    </p>       
<pb/>
       
<p id="id.2.1.217.11.0.0.0" type="main">        
<s id="id.2.1.217.11.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> minor, &longs;it datum pondus <lb/>
vt &longs;exaginta, potentia <expan abbr="ver&ograve;">vero</expan> mouens <lb/>
<arrow.to.target n="note302"></arrow.to.target> data &longs;it tredecim.  </s>
             
<s id="id.2.1.217.11.1.2.0"> inueniatur poten&shy;<lb/>
tia in A &longs;u&longs;tinens pondus B, qu&aelig; pon<lb/>
deris B &longs;it &longs;ubquintupla.  </s>
             
<s id="id.2.1.217.11.1.3.0"> &amp; quoniam <lb/>
potentia in A pondus &longs;u&longs;tinens e&longs;t <lb/>
vt duodecim; maior igitur poten&shy;<lb/>
tia, <expan abbr="qu&agrave;m">quam</expan> duodecim in A pondus <lb/>
B mouebit.  </s>
             
<s id="id.2.1.217.11.1.4.0"> Quare potentia vt tre&shy;<lb/>
decim in A pondus B mouebit. quod. <lb/>
facere oportebat. </s>
             
<s id="id.2.1.217.11.1.5.0"> [quod.]  </s>
             
<s id="id.2.1.217.11.1.6.0"> [<lb/>
facere oportebat.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig183" place="text">       </figure>       
<p id="id.2.1.217.11.2.1.0" type="caption">        
<s id="id.2.1.217.11.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig184" place="text">       </figure>       
<p id="id.2.1.217.11.2.3.0" type="caption">        
<s id="id.2.1.217.11.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.218.1.0.0.0" type="margin">        
<s id="id.2.1.218.1.1.1.0"> <margin.target id="note300"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 1 <emph type="italics"/>huius<emph.end type="italics"/> </s>
             
<s id="id.2.1.218.1.1.2.0"> <margin.target id="note301"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.218.1.1.3.0"> <margin.target id="note302"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 9 <emph type="italics"/>huius<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.219.1.0.0.0" type="main">        
<s id="id.2.1.219.1.1.1.0"> <expan abbr="Animaduertend&utilde;">Animaduertendum</expan> quoq; e&longs;t in mo <lb/>
uendis ponderibus, potentiam ali&shy;<lb/>
quando for&longs;itan melius mouere mo <lb/>
uendo &longs;e deor&longs;um, <expan abbr="qu&agrave;m">quam</expan> mouendo <lb/>
&longs;e &longs;ur&longs;um.  </s>
             
<s id="id.2.1.219.1.1.2.0"> vt circumuoluatur adhuc <lb/>
funis per alium trochle&aelig; &longs;uperioris <lb/>
orbiculum, cuius centrum C, funi&longs;q; <lb/>
<arrow.to.target n="note303"></arrow.to.target> perueniat in D; erit <expan abbr="pot&etilde;tia">potentia</expan> in D &longs;u&longs;ti<lb/>
<expan abbr="n&etilde;s">nens</expan> <expan abbr="p&otilde;dus">pondus</expan> B &longs;imiliter duodecim, <expan abbr="qu&etilde;">quem</expan> <lb/>
admodum erat in A.  </s>
     
<s id="id.2.1.219.1.1.2.0.a"> Ideo poten&shy;<lb/>
tia vt tredecim in D pondus B mo&shy;<lb/>
uebit.  </s>
             
<s id="id.2.1.219.1.1.3.0"> &amp; quia mouet &longs;e deor&longs;um, <lb/>
forta&longs;&longs;e trahet facilius, <expan abbr="qu&agrave;m">quam</expan> in A; <lb/>
atq; tempus e&longs;t idem, &longs;icut etiam <lb/>
erat in A. <arrow.to.target n="fig185"></arrow.to.target> </s>
    </p>       
<pb n="105"/>
       
<p id="id.2.1.219.3.0.0.0" type="head">        
<s id="id.2.1.219.3.1.1.0"> PROPOSITIO XXVIII. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig185" place="text">       </figure>       
<p id="id.2.1.219.3.2.1.0" type="caption">        
<s id="id.2.1.219.3.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.220.1.0.0.0" type="margin">        
<s id="id.2.1.220.1.1.1.0"> <margin.target id="note303"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 5 <emph type="italics"/>Huius<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.221.1.0.0.0" type="head">        <lb/>
       
<s id="id.2.1.221.1.2.1.0"> PROBLEMA. </s>
    </p>       
<p id="id.2.1.221.2.0.0.0" type="main">        
<s id="id.2.1.221.2.1.1.0"> Propo&longs;itum &longs;it nobis efficere, potentiam pon<lb/>
dus mouentem, &amp; pondus per data &longs;patia &longs;ibi in <lb/>
uicem longitudine commen&longs;urabilia moueri. </s>
    </p>       
<p id="id.2.1.221.3.0.0.0" type="main">        
<s id="id.2.1.221.3.1.1.0"> Sit datum &longs;patium potenti&aelig;, vt tria, <arrow.to.target n="note304"></arrow.to.target><lb/>
ponderis <expan abbr="ver&ograve;">vero</expan>, vt quatuor.  </s>
             
<s id="id.2.1.221.3.1.2.0"> inueniatur po<lb/>
tentia in A pondus B &longs;u&longs;tinens, qu&aelig; pon<lb/>
deris &longs;it &longs;e&longs;quitertia, vt quatuor ad <expan abbr="tr&igrave;a">tria</expan>.  </s>
             
<s id="id.2.1.221.3.1.3.0"> &longs;i <lb/>
igitur in A &longs;it potentia mouens pondus; <arrow.to.target n="note305"></arrow.to.target><lb/>
erit &longs;patium ponderis &longs;patii potenti&aelig; &longs;e&longs;&shy;<lb/>
quitertium, vt quatuor ad tria. quod face <lb/>
re oportebat. <arrow.to.target n="fig186"></arrow.to.target> </s>
             
<s id="id.2.1.221.3.1.4.0"> [quod face <lb/>
re oportebat. <arrow.to.target n="fig186"></arrow.to.target>] </s>
    </p>       
<p id="id.2.1.221.4.0.0.0" type="main">        
<s id="id.2.1.221.4.1.1.0"> Hoc autem &amp; ex iis, qu&aelig; dicta &longs;unt in <lb/>
vige&longs;ima &longs;ecunda, &amp; in vige&longs;imaquinta <lb/>
huius efficere po&longs;&longs;umus &longs;olo fune.  </s>
             
<s id="id.2.1.221.4.1.2.0"> <expan abbr="Qu&ograve;d">Quod</expan> &longs;i <lb/>
pluribus funibus id efficere voluerimus, <lb/>
non &longs;olum multis, &longs;ed infinitis modis hoc <lb/>
efficere poterimus, vt &longs;upra dictum e&longs;t. <arrow.to.target n="note306"></arrow.to.target><lb/>
Quare hoc affirmare po&longs;&longs;umus, quod qui&shy;<lb/>
dem mirum e&longs;&longs;e videtur: videlicet.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig186" place="text">       </figure>       
<p id="id.2.1.221.4.2.1.0" type="caption">        
<s id="id.2.1.221.4.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.222.1.0.0.0" type="margin">        
<s id="id.2.1.222.1.1.1.0"> <margin.target id="note304"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 22 <emph type="italics"/>huius.<emph.end type="italics"/> </s>
             
<s id="id.2.1.222.1.1.2.0"> <margin.target id="note305"></margin.target><emph type="italics"/>Ex eadem.<emph.end type="italics"/> </s>
             
<s id="id.2.1.222.1.1.3.0"> <margin.target id="note306"></margin.target><emph type="italics"/>In<emph.end type="italics"/> 26 <emph type="italics"/>huius.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.223.1.0.0.0" type="head">        
<pb/>
       
<s id="id.2.1.223.1.2.1.0"> COROLLARIVM. I. </s>
    </p>       
<p id="id.2.1.223.2.0.0.0" type="main">        
<s id="id.2.1.223.2.1.1.0"> Ex his manife&longs;tum e&longs;&longs;e, Quamlibet datam in <lb/>
numeris proportionem inter pondus, &amp; poten<lb/>
tiam; &amp; inter &longs;patium ponderis moti, &amp; &longs;patium <lb/>
potenti&aelig; mot&aelig;; infinitis modis trochleis inueni&shy;<lb/>
ri po&longs;&longs;e.  </s>
    </p>       
<p id="id.2.1.223.3.0.0.0" type="head">        
<s id="id.2.1.223.3.1.1.0"> COROLLARIVM II. </s>
    </p>       
<p id="id.2.1.223.4.0.0.0" type="main">        
<s id="id.2.1.223.4.1.1.0"> Ex dictis etiam manife&longs;tum e&longs;t, <expan abbr="qu&ograve;">quo</expan> pondus <lb/>
facilius mouetur, <expan abbr="e&ograve;">eo</expan> quoq; tempus maius e&longs;&longs;e; <lb/>
<expan abbr="qu&ograve;">quo</expan> <expan abbr="ver&ograve;">vero</expan> difficilius, <expan abbr="e&ograve;">eo</expan> minus e&longs;&longs;e. &amp; <expan abbr="&egrave;">e</expan> con&shy;<lb/>
uer&longs;o. </s>
             
<s id="id.2.1.223.4.1.2.0"> [&amp; <expan abbr="&egrave;">e</expan> con&shy;<lb/>
uer&longs;o.] </s>
    </p>       
      </chap>
      <pb n="106"/>
      <chap>
<p id="id.2.1.223.5.0.0.0" type="head">        
<s id="id.2.1.223.6.1.1.0"> DE AXE IN <lb/>
PERITROCHIO. </s>
     <lb/>

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure place="text">       </figure>       
<p id="id.2.1.223.6.3.1.0" type="caption">        
<s id="id.2.1.223.6.3.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.223.7.0.0.0" type="main">        
<s id="id.2.1.223.7.1.1.0"> Fabricam, &amp; <expan abbr="c&otilde;&longs;tructionem">con&longs;tructionem</expan> hu&shy;<lb/>
ius in&longs;trumenti Pappus in octauo <lb/>
mathematicarum collectionum <lb/>
libro docet; axemq; vocat AB, <lb/>
tympanum <expan abbr="ver&ograve;">vero</expan> CD circa idem <lb/>
centrum; &amp; &longs;cytalas in foramini&shy;<lb/>
bus tympani EF GH &amp; c.  </s>
             
<s id="id.2.1.223.7.1.2.0"> ita vt potentia, 
<pb/>
<arrow.to.target n="fig187"></arrow.to.target><lb/>
qu&aelig; &longs;emper in &longs;cytalis e&longs;t, vt in F, dum circum&shy;<lb/>
uertit tympanum, &amp; axem, &longs;ur&longs;um moueat pon&shy;<lb/>
dus K axi appen&longs;um fune LM circa axem reuo<lb/>
luto.  </s>
             
<s id="id.2.1.223.7.1.3.0"> Nobis igitur re&longs;tat, vt o&longs;tendamus, cur ma&shy;<lb/>
gna pondera ab exigua virtute, <expan abbr="quou&egrave;">quoue</expan> etiam mo <lb/>
do hoc in&longs;trumento moueantur; temporis quin <lb/>
etiam, &longs;patiiq; mouentis inuicem potenti&aelig;, ac <lb/>
moti ponderis rationem aperiamus; huiu&longs;modi&shy;<lb/>
que in&longs;trumenti v&longs;um ad vectem reducamus.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig187" place="text">       </figure>       
<p id="id.2.1.223.7.2.1.0" type="caption">        
<s id="id.2.1.223.7.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.223.8.0.0.0" type="head">        
<pb n="107"/>
       
<s id="id.2.1.223.9.1.1.0"> PROPOSITIO I. </s>
    </p>       
<p id="id.2.1.223.10.0.0.0" type="main">        
<s id="id.2.1.223.10.1.1.0"> Potentia pondus &longs;u&longs;tinens axe in peritrochio <lb/>
ad pondus eandem habet proportionem, quam <lb/>
&longs;emidiameter axis ad &longs;emidiametrum tympani <lb/>
<expan abbr="vn&aacute;">vna</expan> cum &longs;cytala. <arrow.to.target n="fig188"></arrow.to.target> </s>
    </p>       
<p id="id.2.1.223.11.0.0.0" type="main">        
<s id="id.2.1.223.11.1.1.0"> Sit diameter axis AB, cuius centrum C; &longs;it diameter tympani <lb/>
DCE circa idem centrum; &longs;intq; AB DE in eadem recta linea; <lb/>
&longs;int deinde &longs;cytal&aelig; in foraminibus tympani DF GH &amp; c.inter &longs;e &longs;e <lb/>
&aelig;quales, atq; <expan abbr="&aelig;qu&egrave;">&aelig;que</expan> di&longs;tantes; &longs;itq; FE horizonti &aelig;quidi&longs;tans; 
<pb/>
<arrow.to.target n="fig189"></arrow.to.target><lb/>
pondus autem K in fune BL circa axem volubili &longs;it appen&longs;um.  </s>
             
<s id="id.2.1.223.11.1.2.0"> &amp; <lb/>
potentia in F &longs;u&longs;tineat pondus K.  </s>
     
<s id="id.2.1.223.11.1.2.0.a"> Dico potentiam in F ad pondus <lb/>
k ita &longs;e habere, vt CB ad CF. fiat vt CF ad CB, ita pondus <lb/>
k ad aliud M, quod appendatur in F.  </s>
     
<s id="id.2.1.223.11.1.2.0.b"> &amp; quoniam pondera M k <lb/>
appen&longs;a &longs;unt in FB; erit FB tanquam vectis, &longs;iue libra; quia ve <lb/>
<expan abbr="r&ograve;">ro</expan> Ce&longs;t punctum immobile, circa quod axis, tympanusq; reuol&shy;<lb/>
uuntur; erit C fulcimentum vectis FB; vellibr&aelig; centrum.  </s>
             
<s id="id.2.1.223.11.1.3.0"> <expan abbr="c&ugrave;m">cum</expan> <lb/>
<arrow.to.target n="note307"></arrow.to.target> autem it a &longs;it CF ad CB, vt k ad M, pondera k M &aelig;queponde&shy;<lb/>
rabunt.  </s>
             
<s id="id.2.1.223.11.1.4.0"> Potentia igitur in F &longs;u&longs;tinens pondus k, ne deor&longs;um ver&shy;<lb/>
gat, ponderi K &aelig;queponderabit; ip&longs;iq; M &aelig;qualis erit.  </s>
             
<s id="id.2.1.223.11.1.5.0"> idem enim <lb/>
pr&aelig;&longs;tat potentia, quod pondus M.  </s>
     
<s id="id.2.1.223.11.1.5.0.a"> pondus igitur K ad poten<lb/>
<arrow.to.target n="note308"></arrow.to.target>tiam in F erit, vt CF ad CB; &amp; conuertendo, potentia ad <lb/>
pondus erit, vt CB ad CF, hoc e&longs;t, &longs;emidiameter axis ad &longs;emi
<pb n="108"/>
diametrum tympani <expan abbr="vn&agrave;">vna</expan> cum &longs;cytala DF.  </s>
     
<s id="id.2.1.223.11.1.5.0.b"> Similiter etiam o&longs;ten&shy;<lb/>
detur, &longs;i potentia pondus &longs;u&longs;tinens fuerit in <expan abbr="q.">que</expan> tunc enim &longs;u&longs;ti&shy;<lb/>
neret vecte CQ; &amp; ad pondus eam haberet proportionem, quam <arrow.to.target n="note309"></arrow.to.target><lb/>
habet CB ad <expan abbr="Cq.">Cque</expan> Videlicet &longs;emidiameter axis ad &longs;emidiame&shy;<lb/>
trum tympani <expan abbr="vn&aacute;">vna</expan> cum &longs;cytala <expan abbr="Eq.">Eque</expan> quod demon&longs;trare opor&shy;<lb/>
tebat.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig188" place="text">       </figure>       
<p id="id.2.1.223.11.2.1.0" type="caption">        
<s id="id.2.1.223.11.2.1.0.capt"> YYY </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig189" place="text">       </figure>       
<p id="id.2.1.223.11.2.3.0" type="caption">        
<s id="id.2.1.223.11.2.3.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.224.1.0.0.0" type="margin">        
<s id="id.2.1.224.1.1.1.0"> <margin.target id="note307"></margin.target>6. <emph type="italics"/>Primi Archim. de &aelig;quepon.<emph.end type="italics"/> </s>
             
<s id="id.2.1.224.1.1.2.0"> [de &aelig;quepon.<emph.end type="italics"/>] </s>
             
<s id="id.2.1.224.1.1.3.0"> <margin.target id="note308"></margin.target><emph type="italics"/>Cor.<emph.end type="italics"/> 4. <emph type="italics"/>quinti.<emph.end type="italics"/> </s>
             
<s id="id.2.1.224.1.1.4.0"> <margin.target id="note309"></margin.target>2 <emph type="italics"/>Huuius. de vecte.<emph.end type="italics"/> </s>
             
<s id="id.2.1.224.1.1.5.0"> [de vecte.<emph.end type="italics"/>] </s>
    </p>       
<p id="id.2.1.225.1.0.0.0" type="head">        
<s id="id.2.1.225.1.1.1.0"> COROLLARIVM. </s>
    </p>       
<p id="id.2.1.225.2.0.0.0" type="main">        
<s id="id.2.1.225.2.1.1.0"> Manife&longs;tum e&longs;t potentiam &longs;emper minorem <lb/>
e&longs;&longs;e pondere. </s>
    </p>       
<p id="id.2.1.225.3.0.0.0" type="main">        
<s id="id.2.1.225.3.1.1.0"> Semidiameter enim axis &longs;emper &longs;emidiametro tympani mi&shy;<lb/>
nor e&longs;t.  </s>
             
<s id="id.2.1.225.3.1.2.0"> &amp; potentia <expan abbr="e&ograve;">eo</expan> minor e&longs;t pondere, <expan abbr="qu&ograve;">quo</expan> &longs;emidiameter axis <lb/>
minor e&longs;t &longs;emidiametro tympani <expan abbr="vn&aacute;">vna</expan> cum &longs;cytala.  </s>
             
<s id="id.2.1.225.3.1.3.0"> quare <expan abbr="qu&ograve;">quo</expan> lon<lb/>
gior e&longs;t CF, vel CQ; &amp; <expan abbr="qu&ograve;">quo</expan> breuior e&longs;t CB, minor adhuc &longs;em<lb/>
per potentia in F, vel in Q pondus k &longs;u&longs;tinebit.  </s>
             
<s id="id.2.1.225.3.1.4.0"> <expan abbr="qu&ograve;">quo</expan> enim minor <lb/>
e&longs;t CB, <expan abbr="e&ograve;">eo</expan> minorem habebit proportionem &longs;emidiameter axis <lb/>
ad &longs;emidiametrum tympani <expan abbr="vn&aacute;">vna</expan> cum &longs;cytala. </s>
    </p>       
<p id="id.2.1.225.4.0.0.0" type="main">        
<s id="id.2.1.225.4.1.1.0"> Hoc autem loco con&longs;iderandum occurrit, <expan abbr="qu&ograve;d">quod</expan> &longs;i in alia &longs;cyta&shy;<lb/>
la appendatur pondus, vt in T, &longs;u&longs;tinens pondus k; it a <expan abbr="nemp&egrave;">nempe</expan>, vt <lb/>
pondus in T appen&longs;um, pondusq; k circa axem con&longs;titutum <lb/>
maneant; erit pondus in T grauius pondere M in F appen&longs;o.  </s>
             
<s id="id.2.1.225.4.1.2.0"> <lb/>
iungatur enim TB, &amp; <expan abbr="&agrave;">a</expan> puncto C horizonti perpendicularis du&shy;<lb/>
catur CI, qu&aelig; lineam TB &longs;ecet in I; tandemq; connectatur <lb/>
TC, qu&aelig; &aelig;qualis erit CF.  </s>
     
<s id="id.2.1.225.4.1.2.0.a"> Quoniam autem pondera appen&longs;a <lb/>
&longs;unt in TB, <expan abbr="perind&egrave;">perinde</expan> &longs;e &longs;e habebunt, ac &longs;i in punctis TB ip&longs;orum <lb/>
centra grauitatum haberent; vt antca dictum e&longs;t.  </s>
             
<s id="id.2.1.225.4.1.3.0"> &amp; quia ma&shy;<lb/>
nent, erit punctum I (ex prima huius de libra) amborum &longs;imul <lb/>
grauitatis centrum; <expan abbr="c&ugrave;m">cum</expan> &longs;it CI horizonti perpendicularis.  </s>
             
<s id="id.2.1.225.4.1.4.0"> &longs;ed <lb/>
quoniam angulus BCI e&longs;t rectus, erit BIC acutus, lineaq; BI <arrow.to.target n="note310"></arrow.to.target><lb/>
ip&longs;a BC maior erit.  </s>
             
<s id="id.2.1.225.4.1.5.0"> quare angulus CIT erit obtu&longs;us; atq; <arrow.to.target n="note311"></arrow.to.target><lb/>
ideo line^{a} CT ip&longs;a T^{I} maior erit.  </s>
             
<s id="id.2.1.225.4.1.6.0"> <expan abbr="C&ugrave;m">Cum</expan> autem CT maior &longs;it <lb/>
TI, &amp; IB maior BC; maiorem habebit proportionem TC ad <lb/>
CB, <expan abbr="qu&agrave;m">quam</expan> TI ad IB; &amp; conuertendo, minorem habebit pro&shy;
<pb/>
<arrow.to.target n="fig190"></arrow.to.target><lb/>
portionem BC ad CT, hoc e&longs;t ad CF, <expan abbr="qu&agrave;m">quam</expan> BI ad IT; vt ex <lb/>
vige&longs;ima &longs;exta quinti elementorum (iuxta Commandini editio&shy;<lb/>
nem) patet.  </s>
             
<s id="id.2.1.225.4.1.7.0"> Quoniam <expan abbr="ver&ograve;">vero</expan> punctum I e&longs;t ponderum in TB <lb/>
<arrow.to.target n="note312"></arrow.to.target> exi&longs;tentium centrum grauitatis; erit pondus in T ad pondus in B, <lb/>
vt BI ad IT.  </s>
     
<s id="id.2.1.225.4.1.7.0.a"> pondus <expan abbr="ver&ograve;">vero</expan> in F ad idem pondus in B e&longs;t, vt BC <lb/>
ad CF; maiorem igitur proportionem habebit pondus in T ad <lb/>
pondus in B, <expan abbr="qu&agrave;m">quam</expan> pondus in F ad idem pondus in B.  </s>
     
<s id="id.2.1.225.4.1.7.0.b"> ergo <lb/>
<arrow.to.target n="note313"></arrow.to.target> grauius erit pondus in T, <expan abbr="qu&agrave;m">quam</expan> pondus in F. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig190" place="text">       </figure>       
<p id="id.2.1.225.4.2.1.0" type="caption">        
<s id="id.2.1.225.4.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.226.1.0.0.0" type="margin">        
<s id="id.2.1.226.1.1.1.0"> <margin.target id="note310"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 19 <emph type="italics"/>primi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.226.1.1.2.0"> <margin.target id="note311"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 13 <emph type="italics"/>primi.<emph.end type="italics"/> </s>
             
<s id="id.2.1.226.1.1.3.0"> <margin.target id="note312"></margin.target>6. <emph type="italics"/>Primi Archim. de &aelig;quepon.<emph.end type="italics"/> </s>
             
<s id="id.2.1.226.1.1.4.0"> [de &aelig;quepon.<emph.end type="italics"/>] </s>
             
<s id="id.2.1.226.1.1.5.0"> <margin.target id="note313"></margin.target>10. <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.227.1.0.0.0" type="main">        
<s id="id.2.1.227.1.1.1.0"> Si <expan abbr="ver&ograve;">vero</expan> loco ponderis in T animata potentia &longs;u&longs;tinens pon&shy;<lb/>
dus k con&longs;tituatur; qu&aelig; ita degrauet &longs;e, ac &longs;i in centrum mundi <lb/>
tendere vellet; quemadmodum &longs;uapte natura efficit pondus in T <lb/>
appen&longs;um; erit h&aelig;c eadem ponderi in T appen&longs;o &aelig;qualis; alio&shy;<lb/>
quin non &longs;u&longs;tineret; qu&aelig; quidem ip&longs;a potentia in F collocata ma
<pb n="109"/>
ior erit.  </s>
             
<s id="id.2.1.227.1.1.2.0"> &longs;icuti enim &longs;e &longs;e habet pondus in T ad pondus in F, ita <lb/>
&amp; potentia in T ad potentiam in F; <expan abbr="c&ugrave;m">cum</expan> potenti&aelig; &longs;int ponderi&shy;<lb/>
bus &aelig;quales.  </s>
             
<s id="id.2.1.227.1.1.3.0"> <expan abbr="ver&ugrave;m">verum</expan> &longs;i vnaqu&aelig;q; potentia &longs;eor&longs;um &longs;umpta, <expan abbr="t&agrave;m">tam</expan> <lb/>
in T, <expan abbr="qu&agrave;m">quam</expan> in F &longs;u&longs;tinens pondus <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> <expan abbr="circ&utilde;ferentiam">circunferentiam</expan> THFN <lb/>
moueri &longs;e vellet, veluti apprehen&longs;a manu &longs;cytala; tunc eademmet <lb/>
potentia, vel in F, vel in T con&longs;tituta idem pondus k &longs;u&longs;tinere po<lb/>
terit; <expan abbr="c&ugrave;m">cum</expan> &longs;emper in cuiu&longs;cunq; extremitate &longs;cytal&aelig; ponatur, ab <lb/>
eodem centro C &aelig;quidi&longs;tans fuerit, ac &longs;ecundum eandem circum<lb/>
ferentiam ab eodem centro &aelig;qualiter &longs;emper di&longs;tantem perpen&longs;io<lb/>
nem habeat.  </s>
             
<s id="id.2.1.227.1.1.4.0"> neq; enim (&longs;icuti pondus) proprio nutu magis in <lb/>
centrum ferri exoptat, qu&lt;*&gt;m circulariter moueri; <expan abbr="c&ugrave;m">cum</expan> vtrunq;, &longs;eu <lb/>
quemlibet alium motum nullo pror&longs;us re&longs;piciat di&longs;crimine.  </s>
             
<s id="id.2.1.227.1.1.5.0"> pro&shy;<lb/>
pterea non eodem modo res &longs;e &longs;e habet, &longs;iue pondera, &longs;iue <expan abbr="an&iacute;mat&aelig;">animat&aelig;</expan> <lb/>
potenti&aelig; ii&longs;dem locis eodem munere abeundo fuerint con&longs;titut&aelig;. </s>
    </p>       
<p id="id.2.1.227.2.0.0.0" type="main">        
<s id="id.2.1.227.2.1.1.0"> Potentia autem mouet pondus vecte FB, videlicet dum po<lb/>
tentia in F circumuertit tympanum, circumuertit etiam axem; &amp; <lb/>
FB fit tamquam vectis, cuius fulcimentum C, potentia mouens <lb/>
in F, &amp; podus in B appen&longs;um.  </s>
             
<s id="id.2.1.227.2.1.2.0"> &amp; dum punctum F peruenit in N; <lb/>
punctum H erit in F, &amp; punctum B erit in O; ita vt ducta NO <lb/>
tran&longs;eat per C; eodemq; tempore pondus k motum erit in P, ita <lb/>
vt OBP &longs;it &aelig;qualis ip&longs;i BL, <expan abbr="c&ugrave;m">cum</expan> &longs;it idem funis. </s>
    </p>       
<p id="id.2.1.227.3.0.0.0" type="main">        
<s id="id.2.1.227.3.1.1.0"> Deinde ex quarta huius de vecte <expan abbr="facil&egrave;">facile</expan> eliciemus &longs;patium po&shy;<lb/>
tenti&aelig; mouentis ad &longs;patium ponderis moti ita e&longs;&longs;e, vt &longs;emidiame<lb/>
ter tympani <expan abbr="c&ugrave;m">cum</expan> &longs;cytala ad &longs;emidiametrum axis, hoc e&longs;t, vt CF <lb/>
ad CB, <expan abbr="c&ugrave;m">cum</expan> circumferentia FN ad BO, &longs;it vt CF ad CB.  </s>
     
<s id="id.2.1.227.3.1.1.0.a"> &amp; quo<arrow.to.target n="note314"></arrow.to.target><lb/>
niam BL, e&longs;t &aelig;qualis OBP, dempta communi BP, erit OB ip<lb/>
&longs;i PL &aelig;qualis.  </s>
             
<s id="id.2.1.227.3.1.2.0"> quare FN &longs;patium potenti&aelig; ad PL &longs;patium pon&shy;<lb/>
deris erit, vt CF ad CB, videlicet &longs;emidiameter tympani <expan abbr="c&ugrave;m">cum</expan> <lb/>
&longs;cytala ad &longs;emidiametrum axis.  </s>
             
<s id="id.2.1.227.3.1.3.0"> Quod idem o&longs;tendetur, poten&shy;<lb/>
tia vel in Q, vel in qualibet alia &longs;cytala exi&longs;tente, vt in S. <expan abbr="c&ugrave;m">cum</expan> <lb/>
enim &longs;cytal&aelig; &longs;int &longs;ibi inuicem &aelig;quales, atq; &aelig;qualiter di&longs;tantes; <lb/>
vbicunq; &longs;it potentia &aelig;quali mota velocitate &longs;emper &aelig;quali tem&shy;<lb/>
pore &aelig;quale &longs;patium pertran&longs;ibit, hoc e&longs;t ex Q in R, vel ex Sin T <lb/>
eodem tempore mouebitur, <expan abbr="qu&ograve;">quo</expan> ex F in N.  </s>
     
<s id="id.2.1.227.3.1.3.0.a"> &longs;ed <expan abbr="qu&ograve;">quo</expan> tempore po<lb/>
tentia ex F in N mouetur, eodemmet pror&longs;us pondus k ex L in <lb/>
P quoq; mouetur; vbicunq; igitur &longs;it potentia, erit &longs;patium poten&shy;
<pb/>
<arrow.to.target n="fig191"></arrow.to.target><lb/>
ti&aelig; ad &longs;patium ponderis moti, vt CF ad CB, hoc e&longs;t &longs;emidia&shy;<lb/>
meter tympani cum &longs;cytala, ad &longs;emidiametrum axis.  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig191" place="text">       </figure>       
<p id="id.2.1.227.3.2.1.0" type="caption">        
<s id="id.2.1.227.3.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.228.1.0.0.0" type="margin">        
<s id="id.2.1.228.1.1.1.0"> <margin.target id="note314"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 4 <emph type="italics"/>huius de vecte.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.229.1.0.0.0" type="head">        
<s id="id.2.1.229.1.1.1.0"> COROLLARIVM. I. </s>
    </p>       
<p id="id.2.1.229.2.0.0.0" type="main">        
<s id="id.2.1.229.2.1.1.0"> Ex his manife&longs;tum e&longs;t, ita e&longs;&longs;e pondus ad po&shy;<lb/>
tentiam pondus &longs;u&longs;tinentem, vt &longs;patium poten&shy;<lb/>
ti&aelig; mouentis ad &longs;patium ponderis moti.  </s>
    </p>       
<p id="id.2.1.229.3.0.0.0" type="head">        
<pb n="110"/>
       
<s id="id.2.1.229.4.1.1.0"> COROLLARIVM II. </s>
    </p>       
<p id="id.2.1.229.5.0.0.0" type="main">        
<s id="id.2.1.229.5.1.1.0"> Manife&longs;tum e&longs;t etiam, maiorem &longs;emper ha&shy;<lb/>
bere proportionem &longs;patium potenti&aelig; mouentis <lb/>
ad &longs;patium ponderis moti, <expan abbr="qu&agrave;m">quam</expan> pondus ad ean<lb/>
dem potentiam. </s>
    </p>       
<p id="id.2.1.229.6.0.0.0" type="main">        
<s id="id.2.1.229.6.1.1.0"> Pr&aelig;terea <expan abbr="qu&ograve;">quo</expan> circulus FHN circa &longs;cytalas e&longs;t maior, <expan abbr="e&ograve;">eo</expan> quoq; <lb/>
in pondere mouendo maius &longs;umetur tempus; dummodo potentia <lb/>
&aelig;quali moueatur velocitate.  </s>
             
<s id="id.2.1.229.6.1.2.0"> tempu&longs;q; <expan abbr="e&ograve;">eo</expan> maius erit, <expan abbr="qu&ograve;">quo</expan> diame<lb/>
ter vnius diametro alterius e&longs;t maior.  </s>
             
<s id="id.2.1.229.6.1.3.0"> circulorum enim circumfe-<arrow.to.target n="note315"></arrow.to.target><lb/>
renti&aelig; ita &longs;e habent, vt diametri.  </s>
             
<s id="id.2.1.229.6.1.4.0"> <expan abbr="C&ugrave;m">Cum</expan> vero ex trige&longs;ima &longs;exta <lb/>
quarti libri Pappi Mathematicarum collectionum, duorum in&aelig; <lb/>
qualium circulorum &aelig;quales circumferentias inuenire po&longs;simus; <lb/>
ideo tempus quoq; portionum circulorum in&aelig;qualium hoc modo <lb/>
inueniemus.  </s>
             
<s id="id.2.1.229.6.1.5.0"> <expan abbr="&egrave;">e</expan> conuer&longs;o autem, <expan abbr="qu&ograve;">quo</expan> maior erit axis circumferen<lb/>
tia citius pondus &longs;ur&longs;um mouebitur.  </s>
             
<s id="id.2.1.229.6.1.6.0"> maior enim pars funis BL <lb/>
in vna circumuer&longs;ione completa circa circulum ABO reuoluitur, <lb/>
<expan abbr="qu&agrave;m">quam</expan> &longs;i minor e&longs;&longs;et; <expan abbr="c&ugrave;m">cum</expan> funis circumuolutus &longs;it circumferen&shy;<lb/>
ti&aelig; circuli &aelig;qualis, circa quem reuoluitur.  </s>
    </p>       
<p id="id.2.1.230.1.0.0.0" type="margin">        
<s id="id.2.1.230.1.1.1.0"> <margin.target id="note315"></margin.target>23 <emph type="italics"/>Octaui libri Pappi.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.231.1.0.0.0" type="head">        
<s id="id.2.1.231.1.1.1.0"> COROLLAR VM. </s>
    </p>       
<p id="id.2.1.231.2.0.0.0" type="main">        
<s id="id.2.1.231.2.1.1.0"> Ex his manife&longs;tum e&longs;t, <expan abbr="qu&ograve;">quo</expan> facilius pondus mo<lb/>
uetur, tempus quoq; <expan abbr="e&ograve;">eo</expan> maius e&longs;&longs;e; &amp; <expan abbr="qu&ograve;">quo</expan> dif&shy;<lb/>
ficilius, <expan abbr="e&ograve;">eo</expan> tempus minus e&longs;&longs;e. &amp; <expan abbr="&egrave;">e</expan> conuer&longs;o. </s>
             
<s id="id.2.1.231.2.1.2.0"> [&amp; <expan abbr="&egrave;">e</expan> conuer&longs;o.]  </s>
    </p>       
<pb/>
       
<p id="id.2.1.231.4.0.0.0" type="head">        
<s id="id.2.1.231.4.1.1.0"> PROPOSITIO II. </s>
     <lb/>
       
<s id="id.2.1.231.4.3.1.0"> PROBLEMA. </s>
    </p>       
<p id="id.2.1.231.5.0.0.0" type="main">        
<s id="id.2.1.231.5.1.1.0"> Datum pondus <expan abbr="&agrave;">a</expan> data potentia axe in peritro&shy;<lb/>
chio moueri. </s>
    </p>       
<p id="id.2.1.231.6.0.0.0" type="main">        
<s id="id.2.1.231.6.1.1.0"> Sit datum pondus &longs;exagin<lb/>
ta; potentia <expan abbr="ver&ograve;">vero</expan> vt decem.  </s>
             
<s id="id.2.1.231.6.1.2.0"> <lb/>
exponatur qu&aelig;dam recta li&shy;<lb/>
nea AB, qu&aelig; diuidatur in C, <lb/>
ita vt AC ad CB eandem <lb/>
<arrow.to.target n="fig192"></arrow.to.target><lb/>
habeat proportionem, quam &longs;exaginta ad decem.  </s>
             
<s id="id.2.1.231.6.1.3.0"> &amp; &longs;i CB axis <lb/>
&longs;emidiameter e&longs;&longs;et, &amp; CA &longs;emidiameter tympani <expan abbr="c&ugrave;m">cum</expan> &longs;cytalis; <lb/>
<arrow.to.target n="note316"></arrow.to.target> patet potentiam vt decem in A ponderi &longs;exaginta in B &aelig;quepon<lb/>
derare.  </s>
             
<s id="id.2.1.231.6.1.4.0"> Accipiatur autem inter BC quoduis punctum D; fiatq; <lb/>
BD &longs;emidiameter axis, &amp; DA &longs;emidiameter tympani <expan abbr="c&ugrave;m">cum</expan> &longs;cy&shy;<lb/>
talis; ponaturq; pondus &longs;exaginta in B fune circa axem, &amp; potentia <lb/>
<arrow.to.target n="note317"></arrow.to.target> <emph type="italics"/>in A.  </s>
     
<s id="id.2.1.231.6.1.4.0.a"> Quoniam enim AD ad DB maiorem habet proportio&shy;<lb/>
nem, quam AC ad CB; maiorem habebit proportionem AD ad <lb/>
DB, quam pondus &longs;exaginta in B appen&longs;um ad potentiam vt decem<emph.end type="italics"/><lb/>
<arrow.to.target n="note318"></arrow.to.target> in A.  </s>
     
<s id="id.2.1.231.6.1.4.0.b"> Quare potentia in A pondus &longs;exaginta axe imperitro&shy;<lb/>
chio mouebit, cuius axis &longs;emidiameter e&longs;t BD, &amp; DA &longs;emidia<lb/>
meter tympani <expan abbr="c&ugrave;m">cum</expan> &longs;cytalis. quod erat faciendum. </s>
             
<s id="id.2.1.231.6.1.5.0"> [quod erat faciendum.]  </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig192" place="text">       </figure>       
<p id="id.2.1.231.6.2.1.0" type="caption">        
<s id="id.2.1.231.6.2.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.232.1.0.0.0" type="margin">        
<s id="id.2.1.232.1.1.1.0"> <margin.target id="note316"></margin.target><emph type="italics"/>Per pr&aelig;cedentem.<emph.end type="italics"/> </s>
             
<s id="id.2.1.232.1.1.2.0"> <margin.target id="note317"></margin.target><emph type="italics"/>Lemma in primi huius de vecte.<emph.end type="italics"/> </s>
             
<s id="id.2.1.232.1.1.3.0"> <margin.target id="note318"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 11 <emph type="italics"/>huius de vecte.<emph.end type="italics"/> </s>
    </p>       
<p id="id.2.1.233.1.0.0.0" type="head">        
<pb n="111"/>
       
<s id="id.2.1.233.1.2.1.0"> ALITER. </s>
     <lb/>
       
<s id="id.2.1.233.1.4.1.0"> <expan abbr="Organic&egrave;">Organice</expan> <expan abbr="ver&ograve;">vero</expan> melius erit hoc pacto. </s>
    </p>       
<p id="id.2.1.233.2.0.0.0" type="main">        
<s id="id.2.1.233.2.1.1.0"> Exponatur axis, cuius <lb/>
diameter &longs;it BD, &amp; cen&shy;<lb/>
trum C, quem quidem <lb/>
axem maiorem, vel mino <lb/>
rem con&longs;tituemus, veluti <lb/>
<arrow.to.target n="fig193"></arrow.to.target><lb/>
magnitudo, ponderi&longs;q; grauitas po&longs;tulat.  </s>
             
<s id="id.2.1.233.2.1.2.0"> producatur deinde BD <lb/>
v&longs;q; ad A: fiatq; BC ad CA, vt decem ad &longs;exaginta.  </s>
             
<s id="id.2.1.233.2.1.3.0"> &amp; &longs;i CA tym<lb/>
pani <expan abbr="c&ugrave;m">cum</expan> &longs;cytalis &longs;emidiameter e&longs;&longs;et, potentia decem in A ponde<lb/>
ri &longs;exaginta in B &aelig;queponderaret.  </s>
             
<s id="id.2.1.233.2.1.4.0"> producatur <expan abbr="ver&ograve;">vero</expan> BA ex parte <lb/>
A, &amp; in hac producta linea quoduis accipiatur punctum E; fiatq; <lb/>
CE &longs;emidiameter tympani <expan abbr="c&ugrave;m">cum</expan> &longs;cytalis; ponaturq; potentia vt <lb/>
decem in E; habebit EC ad CB maiorem proportionem, <expan abbr="qu&agrave;m">quam</expan> <lb/>
pondus &longs;exaginta in B ad potentiam vt decem in E.  </s>
     
<s id="id.2.1.233.2.1.4.0.a"> potentia igi&shy;<lb/>
tur vt decem in E mouebit pondus &longs;exaginta in B appen&longs;um fune <lb/>
circa axem, cuius &longs;emidiameter e&longs;t CB, &amp; CE &longs;emidiameter tym<lb/>
pani <expan abbr="c&ugrave;m">cum</expan> &longs;cytalis. quod facere oportebat. </s>
             
<s id="id.2.1.233.2.1.5.0"> [quod facere oportebat.] </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig193" place="text">       </figure>       
<p id="id.2.1.233.2.2.1.0" type="caption">        
<s id="id.2.1.233.2.2.1.0.capt"> YYY </s>
    </p>       
<pb/>
       
<p id="id.2.1.233.4.0.0.0" type="main">        
<s id="id.2.1.233.4.1.1.0"> Sub hoc facultatis genere &longs;unt ergat&aelig;, &longs;uccu&shy;<lb/>
l&aelig;, terebr&aelig;, tympana cum &longs;uis axibus, &longs;iue dentata, <lb/>
&longs;iue non; &amp; &longs;imilia. </s>
    </p>       
<p id="id.2.1.233.5.0.0.0" type="main">        
<s id="id.2.1.233.5.1.1.0"> Terebra <expan abbr="ver&ograve;">vero</expan> habet etiam ne&longs;cioquid cochle&aelig;; dum enim mo&shy;<lb/>
uet pondus, &longs;cilicet dum perforat, ex &longs;ua <expan abbr="fer&egrave;">fere</expan> natura &longs;emper vlte&shy;<lb/>
rius progreditur<emph type="italics"/>:<emph.end type="italics"/> habet enim <expan abbr="fer&egrave;">fere</expan> helices tamquam circa conum <lb/>
de&longs;criptas.  </s>
             
<s id="id.2.1.233.5.1.2.0"> quoniam autem verticem habet acutum, ad cunei quoq; <lb/>
rationem <expan abbr="commod&egrave;">commode</expan> referri poterit. <arrow.to.target n="fig194"></arrow.to.target> </s>
     

        </p>
      </chap>
<pb n="112"/>
      <chap>
        <p id="id.2.1.233.5.0.0.0.a" type="head">
<s id="id.2.1.233.5.3.1.0"> DE CVNEO. </s>
     

<s> ZZZ head of figure ZZZ </s>
    </p>       <figure id="fig194" place="text">       </figure>       
<p id="id.2.1.233.5.4.1.0" type="caption">        
<s id="id.2.1.233.5.4.1.0.capt"> YYY </s>
    </p>       
<p id="id.2.1.233.6.0.0.0" type="main">        
<s id="id.2.1.233.6.1.1.0"> Aristoteles in qu&aelig;&longs;tioni&shy;<lb/>
bus Mechanicis qu&aelig;&longs;tione deci&shy;<lb/>
ma&longs;eptima a&longs;&longs;erit, cuneum &longs;cin&shy;<lb/>
dendo ponderi duorum vicem <lb/>
pror&longs;us gerere vectium &longs;ibi inui&shy;<lb/>
cem contrariorum hoc niodo. </s>
    </p>       
<p id="id.2.1.233.7.0.0.0" type="main">        
<s id="id.2.1.233.7.1.1.0"> Sit cuneus ABC, cu<lb/>
ius vertex B, &amp; &longs;it AB <lb/>
&aelig;qualis BC; quod au<lb/>
tem &longs;cindendum e&longs;t, <lb/>
&longs;it DEFG; &longs;itq; pars <lb/>
cunei HB k intra DE <lb/>
FG, &amp; HB &aelig;qualis <lb/>
&longs;it ip&longs;i Bk.  </s>
             
<s id="id.2.1.233.7.1.2.0"> percutiatur <lb/>
(vt fieri &longs;olet) cuneus <lb/>
in AC, dum cuneus in <lb/>
AC percutitur, AB fit <lb/>
vectis, cuius fulcimen <lb/>
tum e&longs;t H, &amp; pondus in <lb/>
B.  </s>
     
<s id="id.2.1.233.7.1.2.0.a"> eodemq; modo CB <lb/>
fit vectis, cuius fulci&shy;<lb/>
<arrow.to.target n="fig195"></arrow.to.target><lb/>
mentum e&longs;t K, &amp; pondus &longs;imiliter in B.  </s>
     
<s id="id.2.1.233.7.1.2.0.b"> &longs;ed dum percutitur cu&shy;<lb/>
neus, maiori adhuc ip&longs;ius portione ip&longs;um DEFG ingreditur, <lb/>
<expan abbr="qu&agrave;m">quam</expan> prius e&longs;&longs;et: &longs;it autem portio h&aelig;c MBL; &longs;itq; M B ip&longs;i BL <lb/>
&aelig;qualis.  </s>
             
<s id="id.2.1.233.7.1.3.0"> &amp; <expan abbr="c&ugrave;m">cum</expan> MB BI. &longs;int ip&longs;is HB BK maiores; erit ML maior 
<pb/>