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  <info>
    <author>Ceva, Giovanni</author>
    <title>Geometria motus</title>
    <date>1692</date>
    <place>Bologna</place>
    <translator/>
    <lang>la</lang>
    <cvs_file>cevag_geome_022_la_1692.xml</cvs_file>
    <cvs_version/>
    <locator>022.xml</locator>
  </info>
  <text>
    <front>
      <section>
        <pb xlink:href="022/01/001.jpg"/>
        <figure id="id.022.01.001.1.jpg" xlink:href="022/01/001/1.jpg"/>
        <pb xlink:href="022/01/002.jpg"/>
        <pb xlink:href="022/01/003.jpg"/>
        <p type="main">
          <s id="s.000001"><emph type="center"/>GEOMETRIA<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000002"><emph type="center"/>MOTUS<emph.end type="center"/></s>
        </p>
        <p type="main">
          <s id="s.000003"><emph type="center"/>OPVSCVLVM GEOMETRICVM<emph.end type="center"/></s>
        </p>
        <p type="main">
          <s id="s.000004"><emph type="center"/>A&apos;<emph.end type="center"/><!-- REMOVE S--><emph type="center"/><emph type="italics"/>IOANNE CEVA MEDIOLANENSI<emph.end type="italics"/><emph.end type="center"/></s>
        </p>
        <p type="main">
          <s id="s.000005"><emph type="center"/>In gratiam Aquarum excogitatum.<emph.end type="center"/></s>
        </p>
        <p type="main">
          <s id="s.000006"><emph type="center"/>CONTINET DVOS LIBROS<emph.end type="center"/></s>
        </p>
        <p type="main">
          <s id="s.000007"><emph type="center"/>Primum de Simplici Motu, <lb/>Alterum de Compo&longs;ito.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <figure id="id.022.01.003.1.jpg" xlink:href="022/01/003/1.jpg"/>
        <p type="main">
          <s id="s.000008"><emph type="center"/>BONONI&AElig;, M. DC. XCII.<emph.end type="center"/><lb/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000009"><emph type="center"/>Typis HH. </s>
          <s id="s.000010">Antonij Pi&longs;arij Superiorum permi&longs;&longs;u.<emph.end type="center"/></s>
        </p>
        <pb xlink:href="022/01/004.jpg"/>
        <pb xlink:href="022/01/005.jpg"/>
        <p type="main">
          <s id="s.000011"><emph type="center"/>SERENISSIMO<emph.end type="center"/></s>
        </p>
        <p type="main">
          <s id="s.000012"><emph type="center"/>MANTV&AElig; DUCI<emph.end type="center"/></s>
        </p>
        <p type="main">
          <s id="s.000013"><emph type="center"/>FERDINANDO <lb/>CAROLO.<emph.end type="center"/></s>
        </p>
        <p type="main">
          <s id="s.000014"><emph type="italics"/>ITerum, Sereni&longs;&longs;ime Princeps, tuis aduolutus <lb/>genibus opu&longs;culum exhibeo, in quo naturam motuum, pleniori <lb/>methodo, qu&agrave;m puto antea &longs;it actum, geometric&egrave; exequor. <lb/></s>
          <s id="s.000015">Nece&szlig;e habui h&aelig;c pr&aelig;mittere, qu&ograve; viam aperirem, &amp; quo&shy;<lb/>dammodo alueum &longs;ternerem aquarum doctrin&aelig;, quarum <lb/>argumentum vtili&longs;&longs;imum, &amp; profund&aelig; indaginis iam diu <lb/>meditor. </s>
          <s id="s.000016">Quam arduum &longs;it, &amp; per quas &longs;alebras eun&shy;<lb/>dum, vt nouum aliquid luce dignum &egrave; latebris natur&aelig; eruarur <lb/>vtinam Cel&longs;itudini tu&aelig; aliquis veritatum non vulgarium <lb/>indagator fidem faceret; &longs;cio equidem, &amp; laboris improbitas <lb/>tangeret benigni&longs;&longs;imum animum tuum, &amp; &longs;imul natur&aelig; inge&shy;<lb/>nium &longs;u&longs;piceres, qu&aelig; mentibus aliquorum vim inuentricem <lb/>in&longs;eruit, vt eorum iugi cogitatione humanis v&longs;ibus prouide-<emph.end type="italics"/><pb xlink:href="022/01/006.jpg"/><emph type="italics"/>ret. </s>
          <s id="s.000017">Et ver&ograve; &lpar;&longs;i in hoc genere de me quidquam confiteri decet&rpar; <lb/>ni&longs;i aduer&longs;&aelig; valetudinis experimento prudentior factus indo&shy;<lb/>lem meam huiu&longs;cemodi &longs;tudijs intemperanter addictam ali&shy;<lb/>quot ab hinc annis compe&longs;cui&szlig;em; nec non quotidie munus &agrave; <lb/>Cel&longs;itudine Tua &longs;ummo cum honore &amp; beneficentia demanda&shy;<lb/>tum &lpar;adeo vt hoc etiam nomine Te&longs;eruatorem meum appella&shy;<lb/>re po&longs;&longs;im&rpar; inde me reuoca&longs;&longs;et; eorum, credo equidem, ponderi, <lb/>a&longs;&longs;idu&aelig;que contemplationi &longs;uccumbere nece&longs;&longs;e erat. </s>
          <s id="s.000018">Vnde au&shy;<lb/>tem, Cel&longs;i&longs;&longs;ime dux, huic &longs;cienti&aelig; tanta vis, vt quos &longs;ibi &longs;emet <lb/>adiunxerit, nonni&longs;i altiori ratione queat a &longs;e ip&longs;a dimittere? <lb/></s>
          <s id="s.000019">An quod forta&longs;&longs;e vbi animus public&aelig; vtilitati de&longs;eruire c&aelig;pe&shy;<lb/>rit, veluti in natur&aelig; concilium admi&longs;&longs;us, &longs;ui quodammodo <lb/>oblitus, propriam humilioremque &longs;edem reui&longs;ere dedignetur; an <lb/>quia, c&ugrave;m inter c&aelig;teras &longs;cientias Geometria demon&longs;trationem, <lb/>hoc e&longs;t veritatem &longs;inceram, &amp; quandam primi veri particu&shy;<lb/>lam profiteatur, hinc ne&longs;cio quid diuinum habent &longs;ibi <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan>, <lb/>vnde nonni&longs;i Deo impellente, vbi nimirum officia, potiorque <lb/>ratio id po&longs;tulant, ab eius intuitu retrahatur. </s>
          <s id="s.000020">Hoc equidem <lb/>puto; atque hinc diuina Geometria iure optimo a docti&longs;&longs;imis, &amp; <lb/>clari&longs;&longs;imis viris pa&longs;&longs;im nuncupatur. </s>
          <s id="s.000021">Quamobrem nemo non <lb/>eam &longs;u&longs;piciat, eiu&longs;que cultores oppid&ograve; diligat; ob eamque <expan abbr="caus&atilde;">causam</expan> <lb/>huic etiam qualicunque opu&longs;culo benign&egrave; annuas &longs;pero, adeo <lb/>vt iam Te in terris Dominum, Altorem, Seruatorem, Patro&shy;<lb/>numque appellare non dubitem, quam vna cum Cel&longs;i&longs;&longs;ima do&shy;<lb/>mo mihi, tot tibi nominibus deuincto, &longs;uperi vt &longs;eruent &longs;o&longs;pi&shy;<lb/>tentque, enix&egrave; oro, ac omnibus votis exopto.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000022"><emph type="italics"/>Sereni&longs;sim&aelig; Cel&longs;itudinis Tu&aelig;<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000023"><emph type="italics"/>Humillimus, &amp; Ob&longs;equenti&longs;&longs;imus Seruus<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000024">Ioannes Ceua. <!-- KEEP S--></s>
        </p>
      </section>
    </front>
    <body>
      <chap>
        <pb pagenum="1" xlink:href="022/01/007.jpg"/>
        <p type="main">
          <s id="s.000025"><emph type="center"/>GEOMETRIA<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000026"><emph type="center"/>MOTVS.<emph.end type="center"/></s>
        </p>
        <p type="main">
          <s id="s.000027"><emph type="center"/>DEF. I.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000028">CVrrat mobile ab A in D &longs;ecund&ugrave;m rectam <arrow.to.target n="marg1"/><lb/>AD, &amp; linea BHI &longs;it natur&aelig; illius, vt dedu&shy;<lb/>ctis ad AD perpendicularibus AB, CH, DI <lb/>ex punctis quibu&longs;cunque A, C, D; veloci&shy;<lb/>tatum gradus, quos mobile &longs;ortitur in ij&longs;&shy;<lb/>dem punctis A, C, D men&longs;urentur ab ip&longs;is <lb/>rectis AB, CH, CI. <!-- KEEP S--></s>
          <s id="s.000029">Figuram planam BADIHB apellabi&shy;<lb/>mus gene&longs;im motus ab A in D. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000030"><margin.target id="marg1"/><emph type="italics"/>Tab.<emph.end type="italics"/> 1. <emph type="italics"/>Fig.<emph.end type="italics"/> 1.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000031"><emph type="center"/>DEF. II.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000032">II&longs;dem manentibus, &longs;it etiam alia linea EFG talis natu&shy;<lb/><arrow.to.target n="marg2"/><lb/>r&aelig;, vt protractis rectis BA in E, HC in F, &amp; ID in G ha&shy;<lb/>beat DG ad CF eandem reciproc&egrave; rationem, quam HC <lb/>ad ID. </s>
          <s id="s.000033">Item &longs;it CF ad HE vt reciproc&egrave; BA ad HC, vo&shy;<lb/>cabimus figuram planam ADGIEA imaginem tempo&shy;<lb/>ris motus ab A in D iuxta gene&longs;im pr&aelig;dictam. </s>
        </p>
        <p type="margin">
          <s id="s.000034"><margin.target id="marg2"/><emph type="italics"/>Tab.<emph.end type="italics"/> 1. <emph type="italics"/>Fig.<emph.end type="italics"/> 2.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000035"><emph type="center"/>DEF. III.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000036">ADhuc po&longs;ita illa gene&longs;i, intelligatur linea PON eius <lb/><arrow.to.target n="marg3"/><lb/>natur&aelig;, vt &longs;i &longs;it KL ad LM vt tempus lationis ab A <lb/>in C ad tempus ab eodem C in D, habeat &longs;emper KP ad <lb/>LO eandem rationem, quam AB ad CH; &amp; LO ad NM <lb/>eandem, quam HC ad ID: Figuram planam PKMNOP <pb pagenum="2" xlink:href="022/01/008.jpg"/>vocabimus imaginem iuxta gene&longs;im BADI motus ab <lb/>A in D. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000037"><margin.target id="marg3"/><emph type="italics"/>Tab.<emph.end type="italics"/> 1. <emph type="italics"/>Fig.<emph.end type="italics"/> 3.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000038"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000039"><emph type="italics"/>Patet, cum motus &longs;unt &aelig;quabiles, gene&longs;es, &amp; imagines figu&shy;<lb/>ras e&szlig;e parallelogrammas.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000040"><emph type="center"/>DEF. IV.<emph.end type="center"/><lb/><arrow.to.target n="marg4"/><!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000041"><margin.target id="marg4"/><emph type="italics"/>Tab.<emph.end type="italics"/> 1. <emph type="italics"/>Fig.<emph.end type="italics"/> 4.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000042">SI &longs;int du&aelig; gene&longs;es, aut imagines ABCD, FEG, ita vt <lb/>cum gene&longs;es &longs;int, habeat AB ad FE eandem rationem, <lb/>quam velocitas in A ad velocitatem in F, &amp; cum imagines <lb/>velocitatum, quarum tempora AD, FG, velocitas, quam <lb/>habet mobile in&longs;tanti A ad velocitatem alterius mobilis <lb/>in&longs;tanti F, &longs;it vt AB ad FE, &amp; demum ip&longs;is figuris vt imagi&shy;<lb/>nibus temporum con&longs;ideratis habeat velocitas in A ad <lb/>velocitatem in F rationem eandem, quam AB ad FE, vo&shy;<lb/>cabuntur tum gene&longs;es ill&aelig;, cum imagines inter &longs;e homo&shy;<lb/>gene&aelig;. </s>
        </p>
        <p type="main">
          <s id="s.000043"><emph type="center"/>DEF. V.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000044">EAm planam Figuram, in qua duct&aelig; quotcunque <lb/>&etail;quidi&longs;tantes e&ograve; deinceps decre&longs;cunt, qu&ograve; ad idem <lb/>extremum propiores fiunt, acuminatam nuncupabimus. </s>
        </p>
        <p type="main">
          <s id="s.000045"><emph type="center"/>DEF. VI. AX. I.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000046">INter maximam, &amp; minimam eiu&longs;dem imaginis veloci&shy;<lb/>tatem cadit qu&aelig;dam media, qua tant&ugrave;m velocitate, &longs;i <lb/>conciperetur motus &aelig;quabilis, nihilomin&ugrave;s eodem tem&shy;<lb/>pore idem &longs;patium curreretur, ac iuxta imaginem propo&longs;i&shy;<lb/>tam: eam ergo mediam velocitatem dicimus propo&longs;it&aelig; <lb/>imaginis &aelig;quatricem. </s>
        </p>
        <pb pagenum="3" xlink:href="022/01/009.jpg"/>
        <p type="main">
          <s id="s.000047"><emph type="center"/>AX. II.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000048">SPatium iuxta imaginem velocitatum quamcunque <lb/>exactum, vel iuxta &aelig;quatricem imaginis e&longs;t maius eo <lb/>&longs;patio, quod curreretur eodem tempore minima eiu&longs;dem <lb/>imaginis velocitate; &longs;ed minus eo, quod velocitate ma&shy;<lb/>xima. </s>
        </p>
        <p type="main">
          <s id="s.000049"><emph type="center"/>AX. III.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000050">TEmpus, quo curritur &longs;patium iuxta quamlibet tem&shy;<lb/>poris imaginem, maius e&longs;t eo, quo idem &longs;patium <lb/>curreretur maxima velocitate, &longs;ed contra minus eo altero, <lb/>quo ip&longs;um &longs;patium minima velocitate exigeretur, earum <lb/>videlicet, qu&aelig; &longs;unt in gene&longs;i, aut imagine velocitatum pro&shy;<lb/>po&longs;iti motus, cuius nempe illa e&longs;t imago temporis. </s>
          <s id="s.000051">Fit er&shy;<lb/>go, vt tempus &aelig;quale ei, quo illud ip&longs;um &longs;patium currere&shy;<lb/>tur iuxta propo&longs;itam imaginem, &longs;it inter vtrumque dicto&shy;<lb/>rum temporum maximi, &amp; minimi. </s>
        </p>
        <p type="main">
          <s id="s.000052"><emph type="center"/>AX. IV.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000053">QV&aelig;cunque excogitetur figura plana, vel e&longs;t paralle&shy;<lb/>logrammum, vel acuminata figura, aut ex his com&shy;<lb/>po&longs;itum. </s>
          <s id="s.000054">Has tamen figuras inter binas volu&shy;<lb/>mus parallelas, ita vt vnum latus &longs;it ip&longs;as nectens normali&shy;<lb/>ter parallelas, quanquam etiam loco parallelarum po&longs;&longs;int <lb/>e&longs;&longs;e puncta, nemp&egrave; vbi de&longs;inunt in acuminatas pror&longs;us <lb/>figuras. </s>
        </p>
        <p type="main">
          <s id="s.000055"><emph type="center"/>PROP. I. THEOR. I.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000056">TEmpora, quibus duo motus complentur &longs;unt in ra&shy;<arrow.to.target n="marg5"/><lb/>tione imaginum homogenearum ip&longs;orum <expan abbr="tempor&utilde;">temporum</expan>. </s>
        </p>
        <pb pagenum="4" xlink:href="022/01/010.jpg"/>
        <p type="margin">
          <s id="s.000058"><margin.target id="marg5"/><emph type="italics"/>Tab.<emph.end type="italics"/> 1. <emph type="italics"/>Fig.<emph.end type="italics"/> 5.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000059">Motus &longs;int prim&ograve; &aelig;quabiles, curratque mobile &longs;patium <lb/>AB tempore, cuius imago CAB, curratur item ab alio mo&shy;<lb/>bili &longs;patium DE tempore, cuius imago DEF, &amp; &longs;int ip&longs;&aelig; <lb/><arrow.to.target n="marg6"/><lb/>temporum imagines inter&longs;e homogene&aelig;, &longs;cilicet FD ad <lb/>AC eandem habeat rationem, quam velocitas in A ad <lb/>velocitatem in D. Dico, tempus per AB ad id per DE e&longs;&shy;<lb/><arrow.to.target n="marg7"/><lb/>&longs;e vt figura ABC, ad DEF. <!-- KEEP S--></s>
          <s id="s.000060">Cum motus &aelig;quabiles &longs;int <lb/>erunt figur&aelig; dictarum imaginum rectangula, propterea il&shy;<lb/>lorum ratio componetur ex rationibus altitudinum AB ad <lb/><arrow.to.target n="marg8"/><lb/>DE, &amp; ba&longs;ium AC ad DF, ex ij&longs;dem ver&ograve; rationibus &longs;pa&shy;<lb/>tiorum &longs;cilicet, &amp; reciproca velocitatum &lpar;&longs;unt enim ima&shy;<lb/>gines inter &longs;e homogene&aelig;&rpar; nectitur etiam ratio temporum, <lb/>quibus <expan abbr="percurr&utilde;tur">percurruntur</expan> ip&longs;a &longs;patia AB, DE iuxta gene&longs;es ima&shy;<lb/>ginum ACB, DEF, ergo e&longs;t eadem ratio inter illa tempo&shy;<lb/>ra, ac inter imagines &longs;uas. <lb/><arrow.to.target n="marg9"/></s>
        </p>
        <p type="margin">
          <s id="s.000061"><margin.target id="marg6"/><emph type="italics"/>Def.<emph.end type="italics"/> 4. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000062"><margin.target id="marg7"/><emph type="italics"/>Cor. <!-- KEEP S--></s>
          <s id="s.000063">Def.<emph.end type="italics"/> 3. <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000064"><margin.target id="marg8"/><emph type="italics"/>Gal. <!-- KEEP S--></s>
          <s id="s.000065">pr. <!-- REMOVE S-->S de <lb/>motu &aelig;quab. <lb/></s>
          <s id="s.000066">Def.<emph.end type="italics"/> 4. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000067"><margin.target id="marg9"/><emph type="italics"/>Tab.<emph.end type="italics"/> 1. <emph type="italics"/>fig.<emph.end type="italics"/> 6. <lb/><emph type="italics"/>Def.<emph.end type="italics"/> 5. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000068">2. Sit motus vnus &aelig;quabilis, alter ver&ograve; quicunque; &longs;it <lb/>tamen imago huius temporis figura acuminata vt ALGE, <lb/>&amp; alterius temporis pr&aelig;dicti motus &aelig;quabilis, &longs;it HFM, </s>
        </p>
        <p type="main">
          <s id="s.000069"><arrow.to.target n="marg10"/><lb/>qu&aelig; rectangulum erit: Dico, imaginibus homogeneis exi&shy;<lb/>&longs;tentibus, fore inter has eandem rationem, ac homolog&egrave; <lb/>inter tempora decur&longs;uum ab A in E, &amp; ab F in M iuxt&atail; <lb/>gene&longs;es imaginum temporum propo&longs;itarum. </s>
          <s id="s.000070">Si enim non <lb/>e&longs;t ita, &longs;it qu&aelig;dam alia magnitudo Y, maior, vel minor <lb/>imagine acuminata ALGE, qu&aelig; ad imaginem FHM ha&shy;<lb/>beat eandem rationem, quam tempus per AE iuxta imagi&shy;<lb/>nem ALGE ad tempus per FM iuxta imaginem alteram <lb/>FHM; &longs;it ver&ograve; magnitudinis Y differentia ab imagine ma&shy;<lb/>gnitudo Z. <!-- KEEP S--></s>
          <s id="s.000071">Secetur AE bifariam in C, pariterque &longs;eg&shy;<lb/>menta AC, CE bifariam in B, D, &amp; &longs;ic vlteri&ugrave;s progredia&shy;<lb/>tur, donec, &longs;i compleatur rectangulum po&longs;tremum, &amp; ma&shy;<lb/>ximum DG, hoc minus exi&longs;tat quam Z. <!-- KEEP S--></s>
          <s id="s.000072">Tum ductis reli&shy;<lb/>quis &aelig;quidi&longs;tantibus CI, BK, &amp; &agrave; punctis N, I, K, I alijs <lb/>etiam &aelig;quidi&longs;tantibus rect&aelig; AE, efficiatur ip&longs;i ALGE cir&shy;<lb/>cum&longs;cripta figura, con&longs;tans ex rectangulis &aelig;quealtis AK <pb pagenum="5" xlink:href="022/01/011.jpg"/>BI, CN, DG, &amp; in&longs;cripta compo&longs;ita ex rectangulis inter &longs;e <lb/>pariter &aelig;quealtis BL, CR, DI, EN. <!-- KEEP S--></s>
          <s id="s.000073">Cum circum&longs;cript&atail; <lb/>figura differat ab in&longs;cripta exce&longs;&longs;u, quo rectangulum DG <lb/>&longs;uperat BL; &lpar;nam reliqua circum&longs;cripta AK, BI, CN, re&shy;<lb/>liquis in&longs;criptis &aelig;qualia &longs;unt&rpar; &longs;equitur, exce&longs;&longs;um illum e&longs;&longs;e <lb/>minorem magnitudine Z. <!-- KEEP S--></s>
          <s id="s.000074">Si ergo magnitudo Y ponatur <lb/>maior magnitudine ALGE pro exce&longs;&longs;u Z, maior etiam erit <lb/>circum&longs;cripta AK, BI, CN, DG. <!-- KEEP S--></s>
          <s id="s.000075">Qu&ograve;d &longs;i contr&agrave; Y intelli&shy;<lb/>gatur minor ip&longs;a ALGE ex defectu Z, erit quoque eadem <lb/>Y minor, qu&agrave;m in&longs;cripta figura BL, CK, DI, EN. <!-- KEEP S--></s>
          <s id="s.000076">Itaque <lb/>nunc, &longs;i fieri pote&longs;t, &longs;it Y maior magnitudine ALGE per ip&shy;<lb/>&longs;um exce&longs;&longs;um Z, &amp; intelligantur tot motus, quot &longs;unt re&shy;<lb/>ctangula in circum&longs;cripta figura, &longs;cilicet &longs;int ip&longs;i motus ab <lb/>A in B, &agrave; B in C, &agrave; C in D, &amp; &agrave; D in E &longs;ecundum deinceps, <lb/>temporum imagines AK, BI, CN, DG rectangula, qu&aelig; <lb/>&longs;int inter&longs;e, &amp; propo&longs;itis imaginibus homogene&aelig;, qui <lb/>motus erunt proptere&agrave; &aelig;quabiles. </s>
          <s id="s.000077">His po&longs;itis, tempus <lb/><arrow.to.target n="marg11"/><lb/>per FM iuxta imaginem MH ad tempus per AB iuxta ima&shy;<lb/>ginem rectangulum AK eandem habet rationem, quam re&shy;<lb/>ctangulum MH ad rectangulum AK, &longs;imiliter idem tem&shy;<lb/>pus per FM &longs;ecund&ugrave;m ip&longs;am imaginem rectangulum MH <lb/><arrow.to.target n="marg12"/><lb/>ad &longs;ingula reliqua tempora per BC, CD, DE imaginibus <lb/>deinceps rectangulis BI, CN, DG habet eandem rationem, <lb/>quam rectangulum MH ad &longs;ingula eodem ordine rectan&shy;<lb/>gula BI, CN, DG. <!-- KEEP S--></s>
          <s id="s.000078">Quo circa totidem rectangula ex MH, <lb/><arrow.to.target n="marg13"/><lb/>quot &longs;unt illa, ex quibus con&longs;tat circum&longs;cripta figura, ha&shy;<lb/>bebunt ad ea ip&longs;a circum&longs;cripta rectangula, &longs;eu ad eandem <lb/>circum&longs;criptam figuram AK, BI, CN, DG eandem ratio&shy;<lb/>nem, quam totidem tempora eiu&longs;dem imaginis MH ad &longs;i&shy;<lb/>mul tempora, quorum imagines &longs;unt illa ip&longs;a circum&longs;cripta <lb/>rectangula AK, BI, CN, DG. <!-- KEEP S--></s>
          <s id="s.000079">Quare etiam vnicum re&shy;<lb/>ctangulum MH ad circum&longs;criptam figuram AK, BI, CN, <lb/>DG erit in eadem ratione, in quo vnicum tempus per FM <lb/>iuxta imaginem MH ad omnia &longs;imul illa tempora iuxt&atail; <pb pagenum="6" xlink:href="022/01/012.jpg"/>imagines, qu&aelig; &longs;unt dicta circum&longs;cripta rectangula. </s>
          <s id="s.000080">Et <lb/>quoniam figura imaginis e&longs;t acuminata, habetque vi def. <lb/><!-- REMOVE S-->2. huius, applicatas, qu&aelig; &longs;unt in ratione reciproca veloci&shy;<lb/>tatum, quibus nempe mobile afficitur in punctis &longs;patij, &agrave; <lb/>quibus deducuntur ip&longs;&aelig; applicat&aelig;; hinc fit, vt earum ve&shy;<lb/>locitatum, quas mobile habet in decur&longs;u rect&aelig; AB, ea, qu&etail; <lb/>in A maxima &longs;it, &amp; qu&aelig; in B minima. </s>
          <s id="s.000081">Eodem modo iuxta <lb/>reliquas imagines BKIC, CIND, DNGE, qu&aelig; itidem acu&shy;<lb/>minat&aelig; &longs;unt, velocitates in fine decur&longs;uum C, D, E &lpar;&longs;unt <lb/>enim omnes vers&ugrave;s A acuminat&aelig;&rpar; minim&aelig; erunt, &amp; ma&shy;<lb/>xim&aelig; initio dictorum &longs;patiorum. </s>
          <s id="s.000082">Ideo tempora, qu&etail; im&shy;<lb/><arrow.to.target n="marg14"/><lb/>penduntur iuxta illas imagines, &longs;eu ip&longs;am <expan abbr="imagin&etilde;">imaginem</expan> ALGE, <lb/>cuius ill&aelig; &longs;unt omnes partes, minora erunt temporibus, <lb/>qu&aelig; decurrerent, &longs;i illi decur&longs;us forent &aelig;quabiles ex mini&shy;<lb/>mis illis velocitatibus exacti, vel quod in idem recidit, &longs;i <lb/>illi decur&longs;us e&longs;&longs;ent iuxta imagines rectangulorum circum&shy;<lb/>&longs;criptorum AK, BI, CN, DG; itaque rectangulum MH ad <lb/>figuram circum&longs;criptam AK, BI, CN, DG habebit mino&shy;<lb/>rem rationem, qu&agrave;m tempus per FM imagine MH ad tem&shy;<lb/>pus per AE imagine ALGE, &longs;eu qu&agrave;m rectangulum MH <lb/>habet ex hypothe&longs;i ad magnitudinem Y; igitur circum&longs;cri&shy;<lb/>pta figura, qu&aelig; pri&ugrave;s minor o&longs;ten&longs;a fuit magnitudine Y; <lb/>nunc maior concluditur; quod cum &longs;it ab&longs;urdum, &longs;equi&shy;<lb/>tur fals&ograve; nos po&longs;ui&longs;&longs;e magnitudinem Y maiorem; qu&agrave;&mtail; <lb/>ALGE. <!-- KEEP S--></s>
          <s id="s.000083">At &longs;i Y minor ponatur, <expan abbr="qu&atilde;">quam</expan> magnitudo ALGE de&shy;<lb/>fectu Z; in&longs;cripta, vt &longs;upra, figura con&longs;tante ex rectangulis <lb/>&aelig;qu&egrave; altis BL, CK, DI, EN, vt &longs;cilicet differentia ab ima&shy;<lb/>gine &longs;it minor magnitudine Z, liquebit, magnitudinem Y <lb/>minorem e&longs;&longs;e in&longs;cripta figura BL, CK, DI, EN; deind&etail; <lb/>procedendo vt &longs;upra, inueniemus rectangulum MH ad in&shy;<lb/>&longs;criptam figuram BL, CK, DI, EN in eadem ratione, i&ntail; <lb/>quo tempus per FM imagine MH ad omnia &longs;imul decur&shy;<lb/>&longs;uum tempora per AB, BC, CD, DE iuxta imagines re&shy;<lb/>ctangula in&longs;cripta BL, CH, DI, EN; H&aelig;c ver&ograve; tempor&atail; <pb pagenum="7" xlink:href="022/01/013.jpg"/>minora &longs;unt temporibus iuxta imagines ALKB, BKIC, <lb/>CIND, INGE &lpar;nam velocitates initio decur&longs;uum per <lb/>dictas rectas diximus e&longs;&longs;e maximas, &amp; quibus <expan abbr="con&longs;ider&atilde;-tur">con&longs;ideran&shy;<lb/>tur</expan> illi motus &aelig;quabiles &longs;ecund&ugrave;m imagines ip&longs;a illa re&shy;<lb/>ctangula in&longs;cripta&rpar; ergo rectangulum MH ad in&longs;cripta&mtail; <lb/>figuram BL, CK, DI, EN habebit maiorem rationem, <expan abbr="qu&atilde;">quam</expan> <lb/>tempus per FM iuxta imaginem MH ad tempora &longs;imul <lb/>imaginibus ALKB, BKIC, CIND, DNGE, &longs;iue ad tempus <lb/>iuxta imaginem ALGE ex illis compo&longs;itam. </s>
          <s id="s.000084">Ideoque re&shy;<lb/>ctangulum MH ad ip&longs;am in&longs;criptam figuram habebit ma&shy;<lb/>iorem rationem, qu&agrave;m ad magnitudinem Y, idcirco Y, qu&aelig; <lb/>minor o&longs;ten&longs;a fuit in&longs;cript&agrave; figura BL, CK, DI, EN, nunc <lb/>hac alia via maiorem inuenimus; ergo cum rur&longs;us hoc &longs;it <lb/>ab&longs;urdum, nece&longs;&longs;e e&longs;t magnitudinem Y neque minore&mtail; <lb/>e&longs;&longs;e magnitudine ALGE, propterea &aelig;quales inter &longs;e <expan abbr="er&utilde;t">erunt</expan>, <lb/>atque adeo tempus per FM imagine MN ad tempus per <lb/>AE imagine ALGE habebit eandem rationem, quam ima&shy;<lb/>go MH ad imaginem ALGE. <!-- KEEP S--></s>
          <s id="s.000085">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000086"><margin.target id="marg10"/><emph type="italics"/>Cor. <!-- KEEP S--></s>
          <s id="s.000087">Def.<emph.end type="italics"/> 3. <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000088"><margin.target id="marg11"/><emph type="italics"/>Cor. <!-- KEEP S--></s>
          <s id="s.000089">Def.<emph.end type="italics"/> 3. <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000090"><margin.target id="marg12"/><emph type="italics"/>Ex prami&szlig;&atail; <lb/>parte.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000091"><margin.target id="marg13"/><emph type="italics"/>Euang. <!-- REMOVE S-->Tor&shy;<lb/>ric. <!-- REMOVE S-->lem.<emph.end type="italics"/> 18. <emph type="italics"/>in <lb/>libro de dim. <lb/></s>
          <s id="s.000092">parabol&aelig;.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000093"><margin.target id="marg14"/><emph type="italics"/>Ax.<emph.end type="italics"/> 3. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000094">3. Imagines propo&longs;it&aelig; &longs;int du&aelig; acuminat&aelig;. </s>
          <s id="s.000095">Dico ni&shy;<lb/><arrow.to.target n="marg15"/><lb/>hilominus, tempora iuxta illas imagines per AE, HI e&longs;&longs;e vt <lb/>ip&longs;&aelig; imagines ALGE ad HIK, qu&aelig; &longs;int inter &longs;e homoge&shy;<lb/>ne&aelig; vt &longs;emper &longs;upponetur. </s>
          <s id="s.000096">Nam &longs;i intelligatur alius mo&shy;<lb/>tus per MF iuxta imaginem rectangulum MFN, qui &aelig;qua&shy;<lb/><arrow.to.target n="marg16"/><lb/>bilis erit, manife&longs;tum e&longs;t ex &longs;ecundo ca&longs;u, tempus per AE <lb/>iuxta imaginem ALGE ad tempus per FM iuxta <expan abbr="imagin&etilde;">imaginem</expan> <lb/>rectangulum MH, habere eandem rationem, quam imago <lb/>ALGE ad imaginem rectangulum MH; &amp; &longs;imiliter tem&shy;<lb/>pus per FM imagine rectangulum MN ad tempus per HI <lb/>iuxta imaginem HKI habet eandem rationem, quam ima&shy;<lb/>go NM ad imaginem HKI, ergo ex &aelig;quali tempus per AE <lb/>ad tempus per HI &longs;ecund&ugrave;m imagines propo&longs;itas erit vt <lb/>imago ip&longs;a ALGE ad imaginem HKI. </s>
          <s id="s.000097">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000098"><margin.target id="marg15"/><emph type="italics"/>Tab.<emph.end type="italics"/> 1. <emph type="italics"/>Fig.<emph.end type="italics"/> 7.</s>
        </p>
        <p type="margin">
          <s id="s.000099"><margin.target id="marg16"/><emph type="italics"/>Cor. <!-- KEEP S--></s>
          <s id="s.000100">Def.<emph.end type="italics"/> 3 <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000101">4. Demum imagines &longs;int qu&aelig;cunque, mod&ograve; &longs;int ho&shy;<lb/><arrow.to.target n="marg17"/><lb/>mogene&aelig;, ADFB, GHKL: Dico rur&longs;us inter &longs;e e&longs;&longs;e vt tem-<pb pagenum="8" xlink:href="022/01/014.jpg"/>pora per AB, AK iuxta ip&longs;a imagines. </s>
          <s id="s.000102">Vel enim h&aelig; ima&shy;<lb/>gines &longs;unt &longs;implices, hoc e&longs;t tant&ugrave;m parallelogramm&etail;, aut <lb/>tant&ugrave;m acuminat&aelig;, &amp; tunc &longs;upra o&longs;tendimus propo&longs;itum, <lb/>quemadmodum etiam &longs;i vna acuminata, altera parallelo&shy;<lb/>gramma; vel non &longs;unt huiu&longs;modi &amp; componentur ex illis. <lb/><arrow.to.target n="marg18"/><lb/>Sint ergo in imagine ADFB partes ab &aelig;quidi&longs;tantibus di&shy;<lb/>&longs;tinct&aelig; ADEN, OFB acuminat&aelig; &amp; NEFO paralellogram-<lb/><arrow.to.target n="marg19"/><lb/>mum, erunt h&aelig; procul dubio inter &longs;e, totique imagini ho&shy;<lb/>mogene&aelig;; &longs;int pariter in alia imagine partes GHCM, <lb/>MCKL, per &aelig;quidi&longs;tantem MC di&longs;tinct&aelig; inter &longs;e acumi&shy;<lb/><arrow.to.target n="marg20"/><lb/>nat&aelig;, qu&aelig; itidem inter &longs;e, &amp; imagini, cuius &longs;unt partes, ho&shy;<lb/>mogene&aelig; erunt. </s>
          <s id="s.000103">His acceptis, quoniam tempus per AN <lb/><arrow.to.target n="marg21"/><lb/>iuxta imaginem ADEN acuminatam ad tempus per HC <lb/>iuxta aliam imaginem item acuminatam HGMC, habet <lb/>eandem rationem, ac imago ADEN ad <expan abbr="imagin&etilde;">imaginem</expan> GHCM. <lb/>&longs;imiliter tempus per HC iuxta imaginem GHCM ad tem&shy;<lb/>pus per CK iuxta imaginem acuminatam MCKL e&longs;t vt <lb/>illa ad hanc imaginem; componendo, inde per conuer&longs;io&shy;<lb/>nem rationis, &amp; conuertendo, tempus per HC &longs;ecund&ugrave;m <lb/>imaginem GHCM ad tempora &longs;imul per HC, CK, <expan abbr="quor&utilde;">quorum</expan> <lb/>imagines GHCM, MCKL, hoc e&longs;t ad tempus per HK iux&shy;<lb/>ta imaginem GHKL habebit <expan abbr="e&atilde;dem">eandem</expan> rationem, quam ima&shy;<lb/>go GHCM ad imaginem GHCL; &amp; ideo ex &aelig;quali tem&shy;<lb/>pus per AN, cuius imago ADEN, ad tempus per HK, iux&shy;<lb/>ta imaginem GHKL, erit in eadem ratione, in qua e&longs;t ima&shy;<lb/>go ADEN ad imaginem GHKL. <!-- KEEP S--></s>
          <s id="s.000104">Pr&aelig;terea tempus per <lb/>AN iuxta imaginem ADEN ad idem ip&longs;um tempus habet <lb/>eandem rationem, quam imago ADEN ad eandem ip&longs;am; <lb/>tempus per NO iuxta imaginem rectangulum NEPO ad <lb/><arrow.to.target n="marg22"/><lb/>tempus pr&aelig;dictum per AN e&longs;t in eadem ratione <expan abbr="imagin&utilde;">imaginum</expan> <lb/>NEPO ad ADEN, &amp; &longs;imiliter tempus per OB iuxta ima&shy;<lb/>ginem OPFB habet ad tempus per AN eandem rationem, <lb/>ac imago OPFB ad imaginem &longs;&aelig;p&egrave; dictam ADEN; <expan abbr="itaq;">itaque</expan> ex <lb/>lem. 18. Toric. <!-- REMOVE S-->in lib. 

de dim: parabol&aelig;, erunt tria <expan abbr="t&etilde;pora">tempora</expan> per <pb pagenum="9" xlink:href="022/01/015.jpg"/>AN, NO, OB iuxta imagines deinceps ADEN, NEPO, <lb/>OPFB, hoc e&longs;t erit tempus per AB iuxta imaginem ADFB <lb/>ad &longs;imul tria tempora per AN iuxta eandem imaginem <lb/>ADEN, vt imago ADFB ad triplum imaginis ADEN, &amp; <lb/>cum tria &aelig;qualia tempora per AN ad vnicum ex illis &longs;it <lb/>vt triplum imaginis ADEN ad vnicam imaginem; &longs;equi&shy;<lb/>tur ex &aelig;quali tempus per AB ad tempus per AN iuxt&atail; <lb/>imaginem ADEN habere eandem rationem, quam imago <lb/>ADFB ad imaginem ADEN: &amp; o&longs;ten&longs;um fuit tempus per <lb/>AN iuxta imaginem ADEN ad tempus per HK iuxta <lb/>imaginem GHKL habere eandem rationem, quam imago <lb/>ADEN ad imaginem GHKL, ergo rur&longs;us, &amp; tandem ex <lb/>&aelig;quali, tempus per AB iuxta imaginem ADFB ad <expan abbr="t&etilde;pus">tempus</expan> <lb/>per HK iuxta imaginem GHKL habebit eandem <expan abbr="ration&etilde;">rationem</expan>, <lb/>quam imago ADFB ad imaginem GHKL. <!-- KEEP S--></s>
          <s id="s.000106">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000107"><margin.target id="marg17"/><emph type="italics"/>Tab.<emph.end type="italics"/> 1 <emph type="italics"/>Fig. 9<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000108"><margin.target id="marg18"/><emph type="italics"/>Ax.<emph.end type="italics"/> 4. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000109"><margin.target id="marg19"/><emph type="italics"/>Def.<emph.end type="italics"/> 4. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000110"><margin.target id="marg20"/><emph type="italics"/>Def:<emph.end type="italics"/> 4. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000111"><margin.target id="marg21"/><emph type="italics"/>Ex tertia <lb/>parte huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000112"><margin.target id="marg22"/><emph type="italics"/>Ex<emph.end type="italics"/> 2. <emph type="italics"/>part&etail; <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000113"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000114"><emph type="italics"/>Hinc colligitur, &longs;i prima magnitudo ad &longs;ecundam fuerit vt <lb/>tertia ad quartam, item alia prima ad aliam &longs;ecundam vt <lb/>alia tertia ad aliam quartam, &amp; &longs;ic vlteri&ugrave;s quoad vi&longs;u&mtail; <lb/>fuerit, &longs;int pr&aelig;terea omnes prim&aelig;, item omnes terti&aelig; inter&longs;e <lb/>&aelig;quales, con&longs;tat, inquam, primarum vnam ad omnes &longs;ecun&shy;<lb/>das habere eandem rationem, ac vna tertiarum ad omnes <lb/>quartas.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000115"><emph type="center"/>PROP. II. THEOR. II.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000116">Spatia, qu&aelig; curruntur iuxta qua&longs;cunque homogeneas <lb/><expan abbr="velocitat&utilde;">velocitatum</expan> imagines, &longs;unt inter&longs;e, vt e&aelig;dem ill&aelig; ima&shy;<lb/>gines. </s>
          <s id="s.000117">Sint prim&ugrave;m motus &aelig;quabiles, curraturque &longs;pa&shy;<lb/><arrow.to.target n="marg23"/><lb/>tium AB iuxta imaginem velocitatum, qu&aelig; rectangulum <lb/>erit ILMK, &longs;patium ver&ograve; DE tran&longs;igatur iuxta imagine&mtail; <lb/>pr&aelig;dict&aelig; homogeneam rectangulum FHNG &lpar;nam erunt <pb pagenum="10" xlink:href="022/01/016.jpg"/>homogene&aelig; ip&longs;&aelig; imagines, &longs;i vt ex Def. <!-- REMOVE S-->4. huius IL ad HF <lb/>erit vt velocitas in&longs;tanti I ad velocitatem mobilis in&longs;tanti <lb/>F&rpar; Dico &longs;patium AB ad DE e&longs;&longs;e vt imago rectangulu&mtail; <lb/>ILMK ad imaginem rectangulum FHNG. </s>
          <s id="s.000118">Componuntur <lb/>ip&longs;a illa rectangula ex ratione altitudinum IK ad FG, &amp; ex <lb/>ea ba&longs;ium IL ad FH; ver&ugrave;m ex ij&longs;dem, ea nempe <expan abbr="tempor&utilde;">temporum</expan> <lb/><arrow.to.target n="marg24"/><lb/>IK ad FG, atque ea velocitatum IL ad FH componitur <lb/>etiam ratio &longs;patiorum AB ad DE, ergo ip&longs;a &longs;patia erunt vt <lb/>propo&longs;it&etail; imagines. <lb/><arrow.to.target n="marg25"/></s>
        </p>
        <p type="margin">
          <s id="s.000119"><margin.target id="marg23"/><emph type="italics"/>Tab.<emph.end type="italics"/> 1. <emph type="italics"/>Fig.<emph.end type="italics"/> 9. <lb/><emph type="italics"/>Cor. <!-- KEEP S--></s>
          <s id="s.000120">Dif.<emph.end type="italics"/> 3. <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000121"><margin.target id="marg24"/><emph type="italics"/>Gil. <!-- REMOVE S-->de motu <lb/>&aelig;quabili.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000122"><margin.target id="marg25"/><emph type="italics"/>Tab.<emph.end type="italics"/> 1. <emph type="italics"/>fig<emph.end type="italics"/> 10.</s>
        </p>
        <p type="main">
          <s id="s.000123">2. Sint nunc motus iuxta imagines, quarum altera acu&shy;<lb/>minata, altera rectangulum &longs;it. </s>
          <s id="s.000124">Dico rur&longs;us &longs;patium AB, <lb/>quod curritur iuxta imaginem ABCD ad &longs;patium DE, <lb/>quod curritur iuxta alteram imaginem, e&longs;&longs;e vt imago <lb/>ABCD ad imaginem PHNG. <!-- KEEP S--></s>
          <s id="s.000125">Ni&longs;i ita &longs;it, erit alia magni&shy;<lb/>tudo Y maior, vel minor imagine ABCD, qu&aelig; quidem ad <lb/>alteram imaginem HPGN habebit eandem rationem, <expan abbr="qu&atilde;">quam</expan> <lb/>&longs;patium AB ad DE. <!-- KEEP S--></s>
          <s id="s.000126">Sit prim&ugrave;m maior exce&longs;&longs;u Z. Cir&shy;<lb/>cum&longs;cribatur; vt egimus in &longs;ecunda parte prim&aelig; huius, fi&shy;<lb/>gura imagini ABCD con&longs;tans ex rectangulis &aelig;qu&egrave; altis, <lb/>excedatque imaginem ABCD exce&longs;&longs;u minori, quam Z; &longs;it <lb/>ergo circum&longs;cripta illa AE, HF, IG, KG, quam prim&ograve; fa&shy;<lb/>cil&egrave; o&longs;tendemus minorem magnitudine Y; nam h&aelig;c exce&longs;&shy;<lb/>&longs;u magis di&longs;tat ab imagine, qu&agrave;m circum&longs;cripta illa. </s>
          <s id="s.000127">Pr&aelig;&shy;<lb/>terea &longs;i intelligantur tot motus &aelig;quabiles, quot &longs;unt <expan abbr="rect&atilde;-gula">rectan&shy;<lb/>gula</expan> circum&longs;cripta, ij nempe, qui fierent temporibus AH, <lb/>HI, IK, KD iuxta deinceps imagines ip&longs;a rectangula AE, <lb/>HF, IG, KC inter&longs;e, &amp; propo&longs;itis imaginibus homogeneas, <lb/>velocitates, quibus ijdem motus con&longs;iderarentur, forent <lb/>HE, IF, KG, DC, nimirum maxim&aelig; imaginum ABEH, <lb/>HEFI, IFGK, KGCD; Cumque ita &longs;it, longiora &longs;patia cur&shy;</s>
        </p>
        <p type="main">
          <s id="s.000128"><arrow.to.target n="marg26"/><lb/>rerentur iuxta imagines rectangula circum&longs;cripta, quam <lb/>ij&longs;dem temporibus, imaginibu&longs;que po&longs;tremis, hoc e&longs;t <expan abbr="qu&atilde;">quam</expan> <lb/>tempore AD iuxta imaginem ABCD; obidque &longs;patium <lb/>AB ad DE, &longs;eu magnitudo Y ad imaginem HPGN habe-<pb pagenum="11" xlink:href="022/01/017.jpg"/>bit minorem rationem, qu&agrave;m omnes ill&aelig; &longs;imul imagines, <lb/><arrow.to.target n="marg27"/><lb/>&longs;eu quam circum&longs;cripta figura AE, HF, IG, KC ad ean&shy;<lb/>dem imaginem HPGN; quare Y, qu&aelig; pri&ugrave;s o&longs;ten&longs;a fuit <lb/>maior, nunc reperitur minor eadem circum&longs;cripta, quod <lb/>cum fieri nequeat, impo&longs;&longs;ibile etiam e&longs;t magnitudinem Y <lb/>maiorem e&longs;&longs;e magnitudine imaginis ABCD. <!-- KEEP S--></s>
          <s id="s.000129">Sit ergo mi&shy;<lb/>nor, &longs;i etiam fieri pote&longs;t, &amp; defectus ip&longs;ius Y &longs;upra ABCD <lb/>&longs;it Z. <!-- KEEP S--></s>
          <s id="s.000130">In&longs;cribatur imagini figura ex rectangulis &aelig;quealtis, vt <lb/>nempe deficiat ab imagine defectu minori Z; &longs;ic enim ip&longs;a <lb/>in&longs;cripta, qu&aelig; &longs;it AB, IE, KF, DG erit magnitudine pro&shy;<lb/>pinquior imagini ABCD, qu&agrave;m Y, ideoque Y minor erit <lb/>dicta in&longs;cripta figura. </s>
          <s id="s.000131">Deinde, quoniam, &longs;i ponantur mo&shy;<lb/>tus &aelig;quabiles, quorum imagines rect angula in&longs;cripta HB, <lb/>IE, KF, DG, qu&aelig;que inter &longs;e, &amp; propo&longs;itis imaginibus &longs;int <lb/>homogene&aelig;; velocitates, quibus efficerentur dicti motus, <lb/>e&longs;&longs;ent AB, IE, KF, DG, minim&aelig; &longs;cilicet imaginum ABEH <lb/>HEFI, IFGK. KGCD, &amp; ideo &longs;patia, qu&aelig; percurrerentur <lb/>temporibus HA, HI, IK, KD imaginibus illis, maiora e&longs;&shy;<lb/><arrow.to.target n="marg28"/><lb/>&longs;ent, qu&agrave;m qu&aelig; ij&longs;dem temporibus tran&longs;igerentur iuxt&atail; <lb/>imagines pr&aelig;dictas rectangula circum&longs;cripta, hinc fit vt <lb/>&longs;patium AB ad DE, &longs;eu magnitudo Y ad imagine HPGN <lb/>habeat maiorem rationem, qu&agrave;m in&longs;cripta figura ad ean&shy;<lb/>dem imaginem HPGN; quare Y, qu&aelig; minor erat in&longs;cripta <lb/>figura, mod&ograve; re&longs;ultat maior, non ergo Y minor e&longs;&longs;e pote&longs;t <lb/>imagine ABCD, &longs;ed neque maior vt o&longs;tendimus, ergo &longs;pa&shy;<lb/>tium AB ad DE erit, vt imago ABCD ad imaginem <lb/>PHNG. <!-- KEEP S--></s>
          <s id="s.000132">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000133"><margin.target id="marg26"/><emph type="italics"/>Ax.<emph.end type="italics"/> 2. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000134"><margin.target id="marg27"/><emph type="italics"/>Cor. <!-- KEEP S--></s>
          <s id="s.000135">pr.<emph.end type="italics"/> 1. <emph type="italics"/>hu&shy;<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000136"><margin.target id="marg28"/><emph type="italics"/>Ex.<emph.end type="italics"/> 2 <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000137">3. &amp; 4. Si ver&ograve; imagines acuminat&aelig; &longs;int, aut demum <lb/>qu&aelig; cumque, eodem prors&ugrave;s modo, quo prima propo&longs;itio&shy;<lb/>ne, o&longs;tendemus hoc etiam propo&longs;itum, ergo patet omne <lb/>intentum. </s>
        </p>
        <pb pagenum="12" xlink:href="022/01/018.jpg"/>
        <p type="main">
          <s id="s.000138"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000139"><emph type="italics"/>Cum prors&ugrave;s geometric&egrave; o&longs;tenderimus &longs;uperiores duas pro&shy;<lb/>po&longs;itiones, vtili&longs;&longs;imum e&longs;t ob&longs;eruare, quomodo liceat vti tem&shy;<lb/>poris in&longs;tantibus, non vt punctis prors&ugrave;s geometricis, &longs;ed vt <lb/>quantitatibus dicam minoribus quibu&longs;cunque datis. </s>
          <s id="s.000140">Hinc <lb/>oritur indiui&longs;ibilium methodus, qu&aelig; intelligentiam affert <lb/>faciliorem, ac &longs;i rigori geometrico penitus in&longs;i&longs;teremus, quam&shy;<lb/>quam e&aelig; tamen difficiliores Geometras mihi magis decer&etail; <lb/>videantur.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000141"><emph type="center"/>PROP. III. THEOR. III.<emph.end type="center"/><lb/><arrow.to.target n="marg29"/><!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000142"><margin.target id="marg29"/><emph type="italics"/>Tab.<emph.end type="italics"/> 2, <emph type="italics"/>Fig.<emph.end type="italics"/> 1.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000143">SPatia, qu&aelig; curruntur iuxta quaslibet homogeneas ve&shy;<lb/>locitatum imagines, nectuntur ex rationibus tempo&shy;<lb/>rum, ac &aelig;quatricum. </s>
        </p>
        <p type="main">
          <s id="s.000144">Velocitates &aelig;quatrices duorum motuum, quorum ima&shy;<lb/>gines velocitatum &longs;int ABCD, EFHI ponantur AG, EL. <lb/><!-- KEEP S--></s>
          <s id="s.000145">Dico &longs;patia, &longs;eu ip&longs;as imagines componi ex ratione tem&shy;<lb/>porum AD ad EI; &amp; ex ea &aelig;quatricum AE ad EL. <!-- KEEP S--></s>
          <s id="s.000146">Nam <lb/>&longs;i motus, qui e&longs;t iuxta imaginem ABCD per&longs;eueret velo&shy;<lb/>citate AG, e&longs;&longs;et quidem &aelig;quabilis, idemque &longs;patium illa </s>
        </p>
        <p type="main">
          <s id="s.000147"><arrow.to.target n="marg30"/><lb/>velocitate, &amp; tempore AD percurreretur, ac &longs;ecund&ugrave;&mtail; <lb/>imaginem ABCD; Itaque exi&longs;tente rectangulo DE, quod <lb/><arrow.to.target n="marg31"/><lb/>e&longs;set imago velocitatum illius motus &aelig;quabilis, foret idem <lb/><arrow.to.target n="marg32"/><lb/>&aelig;quale imagini ABCD &lpar;nam imagines ABCD, &amp; DG <lb/>homogene&aelig; &longs;unt&rpar; eodem modo imago rectangulum VL <lb/>&aelig;quale e&longs;set imagini EFHI. </s>
          <s id="s.000148">Cum ergo du&aelig; imagines re&shy;<lb/>ctangula DE, IL componantur ex rationibus temporum <lb/>AD ad EI, &amp; ex ea &aelig;quatricum AG ad EL; ex ij&longs;de&mtail; <lb/>prors&ugrave;s rationibus etiam imagines propo&longs;it&aelig; pr&aelig;dictis re&shy;<lb/>ctangulis &aelig;quales nectentur. </s>
          <s id="s.000149">Et ideo &longs;patia, qu&aelig; propo&shy;<lb/>&longs;itis imaginibus tran&longs;iguntur, qu&aelig;que ip&longs;is proportionalia <pb pagenum="13" xlink:href="022/01/019.jpg"/>&longs;unt, componentur ex rationibus temporum, &amp; ex rationi&shy;<lb/>bus &aelig;quatricum. </s>
        </p>
        <p type="margin">
          <s id="s.000150"><margin.target id="marg30"/><emph type="italics"/>Def.<emph.end type="italics"/> 6. <emph type="italics"/>Ax.<emph.end type="italics"/> 1.<!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000151"><margin.target id="marg31"/><emph type="italics"/>Cor.<emph.end type="italics"/> 3. <emph type="italics"/>Def.<emph.end type="italics"/> 3. <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000152"><margin.target id="marg32"/><emph type="italics"/>Pr.<emph.end type="italics"/> 2. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000153"><emph type="center"/><emph type="italics"/>Corollarium I.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000154"><emph type="italics"/>Hinc patet &longs;i line&aelig;, qu&aelig; in imagine velocitatum tempus <lb/>exhibet, aplicetur rectangulum &aelig;quale propo&longs;it&aelig; imagini ve&shy;<lb/>locitatum, fore vt latitudo eiu&longs;dem rectanguli, &longs;it velocitas <lb/>&aelig;quatrix propo&longs;it&aelig; imaginis.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000155"><emph type="center"/><emph type="italics"/>Corollarium II.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000156"><emph type="italics"/>Item constat, vbi tempora, vel &aelig;quatrices velocitates fue&shy;<lb/>rint &aelig;quales, rationem &longs;patiorum e&longs;&longs;e eandem, qu&aelig; &aelig;quatri&shy;<lb/>cum, vel qu&aelig; temporum.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000157"><emph type="center"/><emph type="italics"/>LEMMA.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000158"><emph type="italics"/>Si qu&aelig;libet ratio compo&longs;ita &longs;it ex quotcumque rationibus, <lb/>harum qu&aelig;libet nectetur ex propo&longs;ita, &amp; ex reliquis contra&shy;<lb/>ri&ograve; &longs;umptis rationibus. </s>
          <s id="s.000159">Sit A ad B compo&longs;ita ex rationibus E <lb/>&aelig;d F; G ad H; &amp; I ad K. <!-- KEEP S--></s>
          <s id="s.000160">Dico quamlibet ist arum puta G ad <lb/>K con&longs;tare ex rationibus A ad B, &amp; ex reliquis reciproc&egrave; &longs;um&shy;<lb/>ptis F ad E, &amp; I ad K. <!-- KEEP S--></s>
          <s id="s.000161">Vt E ad F, ita &longs;it A ad C, &amp; vt D ad B <lb/>&longs;ic I ad K; erit C ad D, vt G ad H; <expan abbr="ideoq;">ideoque</expan> C ad D, hoc e&longs;t G ad<emph.end type="italics"/><lb/><arrow.to.target n="table1"/><lb/><emph type="italics"/>H nectetur ex C ad A, &longs;eu F ad G, &amp; ex rationibus A ad B, <lb/>B ad D, &longs;iue K ad I. <!-- KEEP S--></s>
          <s id="s.000162">Quod &amp;c.<emph.end type="italics"/><!-- KEEP S--></s>
        </p>
        <pb pagenum="14" xlink:href="022/01/020.jpg"/>
        <table>
          <table.target id="table1"/>
          <row>
            <cell><emph type="italics"/>A<emph.end type="italics"/></cell>
            <cell><emph type="italics"/>E<emph.end type="italics"/></cell>
            <cell/>
            <cell/>
          </row>
          <row>
            <cell><emph type="italics"/>C<emph.end type="italics"/></cell>
            <cell><emph type="italics"/>F<emph.end type="italics"/></cell>
            <cell><emph type="italics"/>I.<emph.end type="italics"/></cell>
            <cell><emph type="italics"/>K<emph.end type="italics"/></cell>
          </row>
          <row>
            <cell><emph type="italics"/>D<emph.end type="italics"/></cell>
            <cell><emph type="italics"/>G<emph.end type="italics"/></cell>
            <cell/>
            <cell/>
          </row>
          <row>
            <cell><emph type="italics"/>B<emph.end type="italics"/></cell>
            <cell><emph type="italics"/>H<emph.end type="italics"/></cell>
            <cell/>
            <cell/>
          </row>
        </table>
        <p type="main">
          <s id="s.000163"><emph type="center"/>PROP. IV. THEOR. IV.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000164">TEmpora, quibus ab&longs;oluuntur duo motus componun&shy;<lb/>tur ex ratione &longs;patiorum, &amp; ex reciproca &aelig;quatri&shy;<lb/>cum. </s>
          <s id="s.000165">Cum enim &longs;patia <expan abbr="compon&atilde;tur">componantur</expan> ex ratione temporum, <lb/><arrow.to.target n="marg33"/><lb/>&amp; ex ea velocitatum &aelig;quatricum, &longs;equitur per pr&aelig;dictum <lb/>Lemma, qu&ograve;d tempora nectantur ex rationibus &longs;patiorum, <lb/>&amp; reciproca &aelig;quatricum. </s>
        </p>
        <p type="margin">
          <s id="s.000166"><margin.target id="marg33"/><emph type="italics"/>Pr.<emph.end type="italics"/> 3. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000167"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000168"><emph type="italics"/>Manife&longs;tum e&longs;t &longs;patia, vel &aelig;quatrices velocitates, &longs;i &longs;int <lb/>&aelig;quales, e&longs;&longs;e tempora in reliqua ratione reciproca &aelig;quatri&shy;<lb/>cum, vel &longs;patiorum non reciproca.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000169"><emph type="center"/>PROP. V. THEOR. V.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000170">&AElig;Quatrices velocitates componuntur ex rationibus <lb/>&longs;patiorum, &amp; reciproca temporum. </s>
        </p>
        <p type="main">
          <s id="s.000171">Cum &longs;patia componantur ex rationibus temporum, &amp; <lb/>velocitatum &aelig;quatricum, manife&longs;tum e&longs;t ex eodem Lem&shy;<lb/>mate, velocitates ip&longs;as necti ex rationibus &longs;patiorum, &amp; <lb/>reciproca temporum. </s>
        </p>
        <p type="main">
          <s id="s.000172"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000173"><emph type="italics"/>Deducitur, &aelig;quatrices velocitates e&longs;&longs;e vt tempora reciproc&egrave; <lb/>&longs;umpta, vel vt &longs;patia, &longs;i altera ratio fuerit &aelig;qualitatis.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000174"><emph type="center"/>D. <!-- KEEP S--></s>
          <s id="s.000175">E F. VII.<emph.end type="center"/><lb/><arrow.to.target n="marg34"/><!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000176"><margin.target id="marg34"/><emph type="italics"/>Tab.<emph.end type="italics"/> 2. <emph type="italics"/>Fig.<emph.end type="italics"/> 2.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000177">SI in gene&longs;ibus homogeneis AEC, GFK exi&longs;tente AB <lb/>ad BC &longs;icut GI ad IK, habeat AE ad BD eandem ra-<pb pagenum="15" xlink:href="022/01/021.jpg"/>tionem, ac GF ad IH, motus, qui fiunt iuxta illas gene&longs;es, <lb/>vocentur inter &longs;e &longs;imiles, &amp; ip&longs;&aelig; gene&longs;es dicentur &longs;imilium <lb/>motuum; quod ver&ograve; attinet ad rectas AE, BD, GF, IH apel&shy;<lb/>labimus applicatas ad homologa puncta A, B, G, I propor&shy;<lb/>tionales. </s>
        </p>
        <p type="main">
          <s id="s.000178"><emph type="center"/>PROP. VI. THEOR. VI.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000179">SI in imaginibus temporum homogeneis, applicat&aelig; v&shy;<lb/>nius fuerint ad homologa puncta, proportionales ap&shy;<lb/>plicatis alterius imaginis, motus, quorum &longs;unt ip&longs;&aelig; imagi&shy;<lb/>nes, &longs;imiles erunt. </s>
        </p>
        <p type="main">
          <s id="s.000180">Imagines temporum &longs;int &amp;MLABC, &amp;ONGIK, qu&aelig; </s>
        </p>
        <p type="main">
          <s id="s.000181"><arrow.to.target n="marg35"/><lb/>&longs;int homogene&aelig;, &amp; cum GI ad IK &longs;it vt AB ad BC, habeat <lb/>quoque AL ad BM eandem rationem, ac GN ad IO. Di&shy;<lb/>co, motus, quorum &longs;unt ill&aelig; imagines temporum inter &longs;e &longs;i&shy;<lb/>miles e&longs;&longs;e. </s>
        </p>
        <p type="margin">
          <s id="s.000182"><margin.target id="marg35"/><emph type="italics"/>Tab.<emph.end type="italics"/> 2. <emph type="italics"/>fig.<emph.end type="italics"/> 3.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000183">Sint apud ip&longs;as imagines eorundem motuum gene&longs;es, <lb/>&longs;cilicet EAC, FGK inter&longs;e homogene&aelig;. </s>
          <s id="s.000184">Exi&longs;tente AL ad <lb/>BM, vt GN ad IO, erit conuertendo BM ad AL vt IO ad <lb/>GN; &longs;ed vt BM ad AL ita ob gene&longs;im EA ad DB, &amp; vt IO <lb/><arrow.to.target n="marg36"/><lb/>ad GN, &longs;ic FG ad HI. ergo EA ad DB e&longs;t vt FG ad HI, erat <lb/>autem vt AB ad BC ita etiam GI ad IK, ergo motus &longs;unt <lb/><arrow.to.target n="marg37"/><lb/>&longs;imiles, &amp; ip&longs;&aelig; imagines &longs;imilium motuum. </s>
        </p>
        <p type="margin">
          <s id="s.000185"><margin.target id="marg36"/><emph type="italics"/>Def.<emph.end type="italics"/> 2. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000186"><margin.target id="marg37"/><emph type="italics"/>Def.<emph.end type="italics"/> 7. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000187"><emph type="center"/>PROP. VII. THEOR. VII.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000188">SI in imaginibus velocitatum vnius, applicate fuerint ex <lb/><arrow.to.target n="marg38"/><lb/>punctis homolog&egrave; &longs;umptis proportionales applicatis <lb/>alterius imaginis, motus iuxta ip&longs;as imagines erunt &longs;imi&shy;<lb/>les, ideoque ip&longs;&aelig; imagines &longs;imilium motuum. </s>
        </p>
        <p type="margin">
          <s id="s.000189"><margin.target id="marg38"/><emph type="italics"/>Tab.<emph.end type="italics"/> 2. <emph type="italics"/>fig.<emph.end type="italics"/> 4.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000190">Velocitatum imagines &longs;int ABCD, NPRT, &longs;itque AB <lb/>ad EF in eadem ratione, in qua NP ad TR; Dico exi&longs;tenti&shy;<lb/>bus etiam BF ad FC, vt PQ ad QR e&longs;&longs;e propo&longs;itas imagi&shy;<lb/>nes &longs;imilium motuum. </s>
          <s id="s.000191">Intelligantur eorundem motuum <pb pagenum="16" xlink:href="022/01/022.jpg"/>gene&longs;es GHKL, YZ 43. &amp; &longs;it pariter HI ad IK, vt &longs;egmen&shy;<lb/>tum ABFE ad EFCD. </s>
          <s id="s.000192">Sit &longs;imiliter Z <gap/> ad <gap/> 4 vt &longs;eg&shy;<lb/>mentum NPQV ad VQRT, ducti&longs;que applicatis IM, QV, <lb/>manife&longs;tum e&longs;t, vt velocitas AB &aelig;qualis e&longs;t velocitati GH, <lb/>&longs;ic EF &aelig;qualem fore ip&longs;i IM; nam quia &longs;patium <expan abbr="tran&longs;act&utilde;">tran&longs;actum</expan> <lb/>iuxta imaginem ABFE ad &longs;patium tran&longs;actum imagine <lb/><arrow.to.target n="marg39"/><lb/>EFCD e&longs;t vt illa ad hanc imaginem, nempe vt HI ad IK, <lb/>erit mobile in&longs;tanti F in puncto I, &amp; ideo inibi erit veloci&shy;<lb/>tas eadem, quam habet mobile in&longs;tanti F, &longs;cilicet &aelig;quales <lb/>erunt EF, IM. </s>
          <s id="s.000193">Eodem modo erunt &aelig;quales QV, <gap/> 2, &amp; <lb/>&longs;unt etiam &aelig;quales NP, YZ, ergo &longs;icut &longs;e habet AB ad EF, <lb/>ita erit GH ad MI, &amp; vt e&longs;t NP ad <expan abbr="Vq.">Vque</expan> ita erit YZ ad 2 <gap/><lb/>Pr&aelig;terea concipiatur figura OPRSXO &longs;imilis ip&longs;i ABCD, <lb/>&longs;cilicet &longs;it CB ad PR vt AB ad OP, vel &lpar;cum &longs;int BF ad <lb/>FC ita PQ ad QR, vt EF ad homologam XQ, erit &longs;eg&shy;<lb/>mentum ABFE ad &longs;ibi &longs;imile &longs;egmentum OPQX in dupli&shy;<lb/>cata ratione laterum homologorum EF ad XQ, &amp; item in <lb/><expan abbr="ead&etilde;">eadem</expan> duplicata ratione erunt inter&longs;e &longs;imilia <expan abbr="&longs;egm&etilde;ta">&longs;egmenta</expan> EFCD <lb/>ad XQRS, &longs;ed cum etiam OPQX &longs;egmentum ad NPQV, <lb/>&amp; XQRS ad &longs;egmentum VQRT &longs;int in eadem ratione <lb/>eiu&longs;dem QX ad QV, erit ex &aelig;quali &longs;egmentum ABFE ad <lb/>&longs;egmentum NPQV, vt &longs;egmentum EFCD ad VQRT, &amp; <lb/>permutando, &longs;egmentum ABFE ad &longs;egmentum EFCD ha&shy;<lb/>bebit eandem rationem, ac &longs;egmentum NPQV ad VQRT <lb/>&longs;cilicet erit HI ad IK vt Z <gap/> ad <gap/> 4, ob idque con&longs;tat ge&shy;<lb/>ne&longs;ium applicatas vnius proportionales e&longs;&longs;e applicatis al&shy;<lb/>terius, quare &longs;imiles motus erunt, qui fiunt iuxta imagines <lb/>velocitatum propo&longs;itas. </s>
        </p>
        <p type="margin">
          <s id="s.000194"><margin.target id="marg39"/><emph type="italics"/>Pr.<emph.end type="italics"/> 2. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000195"><emph type="center"/>PROP. VIII. THEOR. VIII.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000196">SPatia, qu&aelig; curruntur &longs;imilibus motibus &longs;unt in ratione <lb/>compo&longs;ita temporum, &amp; homologarum velocitatum, <lb/>inter quas &longs;unt extrem&aelig;, aut prim&aelig;. </s>
        </p>
        <pb pagenum="17" xlink:href="022/01/023.jpg"/>
        <p type="main">
          <s id="s.000197">Imagines velocitatum &longs;imilium motuum &longs;int BCDE, <lb/><arrow.to.target n="marg40"/><lb/>GMKI, &amp; iuxta eas percurrantur &longs;patia A, F. <!-- KEEP S--></s>
          <s id="s.000198">Dico i&longs;ta com&shy;<lb/>poni ex rationibus temporum BE ad GI, &amp; ex ea veloci&shy;<lb/>tatem extremarum ED ad IK. <!-- KEEP S--></s>
          <s id="s.000199">Fiat vt BE ad GI, ita BC <lb/>ad GH, intelligatur que GHLI figura &longs;imilis ip&longs;i BDE. Quo&shy;<lb/><arrow.to.target n="marg41"/><lb/>niam &longs;patium A ad F, hoc e&longs;t imago BCDE ad imaginem <lb/>GMKI componitur ex ratione imaginis BCDE ad figu&shy;<lb/>ram &longs;ibi &longs;imilem GHLI, &amp; ex ratione huius ad imaginem <lb/>GMKI: prior ratio e&longs;t duplicata homologorum lateru&mtail; <lb/>BE ad GI, &longs;eu e&longs;t compo&longs;ita ex BE ad GI, &amp; ex huic &longs;imi&shy;<lb/>li ratione ED ad IL, &amp; ratio altera, imaginis &longs;cilicet GHLI <lb/>ad imaginem GMKI e&longs;t, vt LI ad IK; ergo ex &aelig;quali ima&shy;<lb/>go BCDE ad imaginem GMKI, hoc e&longs;t &longs;patium A ad &longs;pa&shy;<lb/>tium F, componetur ex ratione temporum BE ad GI, &amp; ex <lb/>rationibus ED ad LI, &amp; IL ad IK, &longs;cilicet nectetur ex ra&shy;<lb/>tione BE ad GI, &amp; ED ad IK, qu&aelig; po&longs;trema cum &longs;it ratio <lb/>velocitatum extremarum ED ad IK; con&longs;tat, quod propo&shy;<lb/>&longs;uimus, &longs;patia &longs;imilium motuum componi ex ratione tem&shy;<lb/>porum, &amp; ex ratione homologarum velocitatum, hoc e&longs;t <lb/>extremarum. </s>
        </p>
        <p type="margin">
          <s id="s.000200"><margin.target id="marg40"/><emph type="italics"/>Tab.<emph.end type="italics"/> 2. <emph type="italics"/>Fig.<emph.end type="italics"/> 5</s>
        </p>
        <p type="margin">
          <s id="s.000201"><margin.target id="marg41"/><emph type="italics"/>Pr.<emph.end type="italics"/> 2. <emph type="italics"/>huiu.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000202"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000203"><emph type="italics"/>Si tempora fuerint &aelig;qualia, &longs;imilium motuum &longs;patia <expan abbr="er&utilde;t">erunt</expan> <lb/>vt extrem&aelig;, vel &longs;umm&aelig; velocitates, &amp; contra, &longs;i i&longs;t&aelig; &aelig;quales <lb/>&longs;int, erunt &longs;patia vt tempora.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000204"><emph type="center"/><emph type="italics"/>Corollarium II.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000205"><emph type="italics"/>Cum &longs;patia &longs;imilium motuum nectantur ex ratione tem&shy;<lb/>porum &amp; ex ea velocitatum &longs;ummarum, &longs;eu earum, qu&aelig; <expan abbr="s&utilde;t">sunt</expan> <lb/>ad in&longs;tantia &longs;imiliter &longs;umpta in rectis BE, GI, constat ex <lb/>lem: infra cor.<emph.end type="italics"/> 2. <emph type="italics"/>pr.<emph.end type="italics"/> 3. <emph type="italics"/>huius tempora componi ex rationi&shy;<lb/>bus &longs;patiorum &longs;imilium motuum, &amp; ex rec&igrave;proca dictarum <emph.end type="italics"/><pb pagenum="18" xlink:href="022/01/024.jpg"/><emph type="italics"/>velocitatum. </s>
          <s id="s.000206">Ex eadem ratione patet e&longs;&longs;e velocitates &longs;um&shy;<lb/>mas, vel homologas vti diximus in ratione compo&longs;ita dicto&shy;<lb/>rum &longs;patiorum, &amp; ip&longs;orum temporum.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000207"><emph type="center"/><emph type="italics"/>Corollarium III.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000208"><emph type="italics"/>Quare &longs;i alter&aelig; de dua<gap/>bus componentibus &aelig;qualis fuerit, <lb/>reliqua tant&ugrave;m computanda erit.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000209"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000210"><emph type="italics"/>Hinc emergit omnis fer&egrave; doctrina grauium cum <expan abbr="de&longs;cend&utilde;t">de&longs;cendunt</expan> <lb/>pror&longs;us libera, aut &longs;uper planis inclinatis ad horizonte&mtail;: <lb/>nec accidit veritates iam patefactas huc rur&longs;us lectoris taedio <lb/>afferre, &longs;ed libeat potius, rationem metiendarum imaginum, <lb/>quamuis longitudine immen&longs;arum, no&longs;tra methodo exponere.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000211"><emph type="center"/>DEF. VIII.<emph.end type="center"/><lb/><arrow.to.target n="marg42"/><!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000212"><margin.target id="marg42"/><emph type="italics"/>Tab.<emph.end type="italics"/> 2. <emph type="italics"/>Fig.<emph.end type="italics"/> 6.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000213">SInt inter binas parallelas AB, GH, et IK, PQ plan&aelig; fi&shy;<lb/>gur&aelig; ABHG, IKQP, &amp; in altera earum ducta altitudi&shy;<lb/>ne RV, &longs;int inter &longs;e ip&longs;&aelig; figur&aelig; talis natur&aelig;, vt cum &longs;it <lb/>GABH ad &longs;egmentum EABF factum per &aelig;quidi&longs;tantem <lb/>ip&longs;i GH &longs;icut VR ad RT, verificetur &longs;emper &lpar;ducta &aelig;qui&shy;<lb/>di&longs;tanti NTO ip&longs;i PQ&rpar; e&longs;&longs;e GH ad EF vt reciproc&egrave; NO ad <lb/>PQ tunc huiu&longs;modi figuras vocabimus inter &longs;e auuer&longs;as. </s>
        </p>
        <p type="main">
          <s id="s.000214"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000215"><emph type="italics"/>Sequitur ex vi nunc allat&aelig; deffin., lineam IK tunc e&longs;&longs;e in&shy;<lb/>finitam, cum AB fuerit punctum, &amp; ideo &longs;imul con&longs;tat figu&shy;<lb/>ram IPQK immen&longs;am e&longs;&longs;e longitudine vers&ugrave;s K aut I, aut <lb/>vtrinque, &longs;i nempe producerentur nunquam coitur&aelig; line&aelig; <lb/>QP, IK.<emph.end type="italics"/><!-- KEEP S--></s>
        </p>
        <pb pagenum="19" xlink:href="022/01/025.jpg"/>
        <p type="main">
          <s id="s.000216"><emph type="center"/>PROP. IX. THEOR. IX.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000217">REctangulum &longs;ub altitudine, &amp; ba&longs;i vnius auuer&longs;arum <lb/>ad ip&longs;am auuer&longs;am figuram, eandem habet <expan abbr="ration&etilde;">rationem</expan>, <lb/>ac altera auuer&longs;a figura ad rectangulum ex ba&longs;i in altitudi&shy;<arrow.to.target n="marg43"/><lb/>nem eiu&longs;dem huius figur&aelig;. </s>
        </p>
        <p type="margin">
          <s id="s.000219"><margin.target id="marg43"/><emph type="italics"/>Tab.<emph.end type="italics"/> <gap/>. <emph type="italics"/>fig.<emph.end type="italics"/> 7.</s>
        </p>
        <p type="main">
          <s id="s.000220">Sint auuer&longs;&aelig; figur&aelig; ACB, GFDEG. </s>
          <s id="s.000221">Dico rectangu&shy;<lb/>lum DF in DE ad figuram GFDEG, eandem habere ratio&shy;<lb/>nem ac figura ACBA ad rectangulum AB in BC. <!-- KEEP S--></s>
          <s id="s.000222">Sint pri&shy;<lb/>m&ugrave;m ABC, FDE anguli recti, &amp; ducta qualibet HI paral&shy;<lb/><arrow.to.target n="marg44"/><lb/>lela BC, &longs;it BAC ad HIA vt DF ad KF, erit ob naturam <lb/>auuer&longs;arum KL ad DE vt BC ad HI; itaque &longs;i ponatur e&longs;&longs;e <lb/>quidam motus ab F in D iuxta imaginem <expan abbr="velocitat&utilde;">velocitatum</expan> BAC, <lb/><arrow.to.target n="marg45"/><lb/>erit GFDEG imago temporis eiu&longs;dem motus; nam imago <lb/><arrow.to.target n="marg46"/><lb/>BAC ad imaginem HIA e&longs;t vt &longs;patium DF ad &longs;patium FK <lb/>&amp; velocitas BC ad <expan abbr="velocitat&etilde;">velocitatem</expan> HI vt reciproc&egrave; KL ad DE. <lb/><!-- KEEP S--></s>
          <s id="s.000223">Sit etiam alius motus, &longs;ed &aelig;quabilis, cuius imago velocita&shy;<lb/>tum &aelig;qualis &longs;it, &amp; homogenea ip&longs;i BAC, rectangulum <expan abbr="n&etilde;-pe">nen&shy;<lb/>pe</expan> AB in BM, &amp; ideo &longs;i fiat BM ad BC &longs;icut DE ad DN, <lb/>concipiaturque rectangulum FD in DN, erit hoc imago <lb/><arrow.to.target n="marg47"/><lb/>temporis dicti motus &aelig;quabilis, homogenea, &amp; &aelig;qualis <lb/>imagini GFDEG; nam <expan abbr="t&etilde;pora">tempora</expan>, &longs;cilicet imagines GFDEG, <lb/><arrow.to.target n="marg48"/><lb/>FD in DN rectangulum componuntur ex rationibus &longs;pa&shy;<lb/><arrow.to.target n="marg49"/><lb/>tiorum, hoc e&longs;t imaginum velocitatum inter&longs;e &aelig;qualium, <lb/>ABM, ACB, &amp; reciproca &aelig;quatricum pariter &aelig;qualium <lb/>BM, BM. </s>
          <s id="s.000224">Cum igitur rectangulum FD in DN &aelig;quale &longs;it <lb/><arrow.to.target n="marg50"/><lb/>imagini, &longs;eu figur&aelig; GFDEG, habebit eadem figur&atail; <lb/>GFDEG ad rectangulum FD in DE eandem rationem, <lb/>quam DN ad DE, hoc e&longs;t quam BC ad BM, &longs;eu quam re&shy;<lb/>ctangulum AB in BC ad rectangulum AB in BM, aut ad ei <lb/>&aelig;qualem figuram ABC; &amp; conuertendo, manife&longs;tum e&longs;t <lb/>quod propo&longs;uimus, nempe rectangulum FD in DE ad fi&shy;<lb/>guram GFDEG habere eandem <expan abbr="ration&etilde;">rationem</expan>, ac figura ACBA <pb pagenum="20" xlink:href="022/01/026.jpg"/>ad rectangulum AB in BC. quod erat demon&longs;trandum <lb/>primo loco. </s>
        </p>
        <p type="margin">
          <s id="s.000225"><margin.target id="marg44"/><emph type="italics"/>Def.<emph.end type="italics"/> 8. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000226"><margin.target id="marg45"/><emph type="italics"/>Def:<emph.end type="italics"/> 2. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000227"><margin.target id="marg46"/><emph type="italics"/>pr.<emph.end type="italics"/> 2. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000228"><margin.target id="marg47"/><emph type="italics"/>Def.<emph.end type="italics"/> 2. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000229"><margin.target id="marg48"/><emph type="italics"/>pr.<emph.end type="italics"/> 1. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000230"><margin.target id="marg49"/><emph type="italics"/>pr.<emph.end type="italics"/> 4. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000231"><margin.target id="marg50"/><emph type="italics"/>Cor. <!-- KEEP S--></s>
          <s id="s.000232">pr.<emph.end type="italics"/> 3. <emph type="italics"/>hu&shy;<lb/>ius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000233">2. Si ver&ograve; propo&longs;it&aelig; figur&aelig; &longs;int qu&aelig;cunque auuer&longs;&aelig; <lb/><arrow.to.target n="marg51"/><lb/>DAE, QPLMQ poterunt h&aelig; reuocari ad qua&longs;dam alias <lb/>FKG, RSZX, qu&aelig; &longs;int inter ea&longs;dem parallelas, queis com&shy;<lb/>prehenduntur propo&longs;it&aelig; figur&aelig;, ad eo vt exi&longs;tentibus re&shy;<lb/>ctis angulis KFG, RXZ &longs;int ip&longs;&aelig; bin&aelig; figur&aelig; ab ij&longs;dem pa&shy;<lb/>rallelis intercept&aelig;. </s>
          <s id="s.000234">inter &longs;e &aelig;qualiter analog&aelig; hoc e&longs;t du&shy;<lb/>ctis &aelig;quidi&longs;tantibus, vt vi&longs;um fuerit IHBC, VTNO, &longs;int <lb/>&longs;emper interiect&aelig; line&aelig; IH, BC, &amp; VT, NO &aelig;quales: hoc <lb/>modo non tant&ugrave;m liquet figuras FKG, DAE, nec no&ntail; <lb/>RSZX, PQML &aelig;quales inter &longs;e e&longs;&longs;e, ver&ugrave;m etiam FKG ad <lb/>IKH e&longs;&longs;e in eadem ratione, in qua QPLMQ ad QPNOQ, <lb/>quamobrem ex prima parte, rectangulum ZX in RM ad <lb/>figuram SRXZS, hoc e&longs;t rectangulum LM in altitudinem <lb/>figur&aelig; QPLMQ ad hanc ip&longs;am figuram habebit eandem <lb/>rationem, quam figura FKG ad rectangulum KF in FG, <lb/>vel quam figura DAE ad rectangulum DE in altitudinem <lb/>eiu&longs;dem huius figur&aelig; DAE; quo circa con&longs;tat omne pro&shy;<lb/>po&longs;itum. </s>
        </p>
        <p type="margin">
          <s id="s.000235"><margin.target id="marg51"/><emph type="italics"/>Tab.<emph.end type="italics"/> 2. <emph type="italics"/>Fig.<emph.end type="italics"/> 8.</s>
        </p>
        <p type="main">
          <s id="s.000236"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000237"><emph type="italics"/>Patet in prima parte repertum e&longs;&longs;e rectangulum FD i&ntail;<emph.end type="italics"/><lb/><arrow.to.target n="marg52"/><lb/><emph type="italics"/>DN &aelig;quale figur&aelig; GFDEG, lic&egrave;t h&aelig;c immen&longs;e longitudinis <lb/>&longs;it vers&ugrave;s G, &amp; ob id manife&longs;tum e&longs;t, qu&ograve;d quamuis aliqu&atail; <lb/>figura &longs;it &longs;in&egrave; fiue longa, non ideo &longs;emper magnitudinem ha&shy;<lb/>bet infinitam. </s>
          <s id="s.000238">Et &longs;imul illud con&longs;tat, vbi vna auuer&longs;arum, &longs;eu <lb/>vbi imago velocitatum, aut temporis &longs;it magnitudine termi&shy;<lb/>nata, etiam altera auuer&longs;arum, vel imaginum erit huiu&longs;&shy;<lb/>modi &amp;c.<emph.end type="italics"/><!-- KEEP S--></s>
        </p>
        <pb pagenum="21" xlink:href="022/01/027.jpg"/>
        <p type="margin">
          <s id="s.000239"><margin.target id="marg52"/><emph type="italics"/>Cor. <!-- KEEP S--></s>
          <s id="s.000240">pr.<emph.end type="italics"/> 18. <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000241"><emph type="center"/>PROP. X. THEOR. X.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000242">IN quouis parallelogrammo BD &longs;int deinceps diagona&shy;<lb/><arrow.to.target n="marg53"/><lb/>les AGC, AHC, AIC, ALC, ali&aelig;que numer&ograve; infinit&aelig;, <lb/>ita vt acta qu&aelig;libet recta EF parallela BA <expan abbr="&atilde;s">&longs;ecans</expan> ip&longs;as dia&shy;<lb/>gonales in punctis G, L, H, I, &longs;it &longs;emper DA ad AF, vt CD, <lb/>aut EF ad FG; quadratum ex DA ad quadratum AF vt <lb/>EF ad FH; cubus ex DA ad cubum ex AF vt EF ad FI; <lb/>quadroquadratum ex DA ad quadroquadratum ex AF <lb/>vt EF ad FL; &amp; &longs;ic continu&ograve; procedendo per infinitas ex <lb/>ordine pote&longs;tates: Stephanus de Angelis Author &longs;ubtilis, <lb/>ac celeberrimus, libro &longs;uo infin. parabolarum vocat trian&shy;<lb/>gulum rectilineum ABC parabolam primam, BAHC &longs;e&shy;<lb/>cundam; tertiam BAIC, quartam BALC, &amp; ita in infini&shy;<lb/>tum: His definitis docet ex Cauallerio parallelogrammum <lb/>BD ad quancunque dictarum parabolarum &longs;ibi in&longs;cripta&shy;<lb/>rum e&longs;&longs;e vt numerus, vel exponens parabol&aelig; vnitate au&shy;<lb/>ctus ad ip&longs;um exponentem, &longs;iue numerum parabol&etail;, qua&shy;<lb/>re ad primam habebit ip&longs;um parallelogrammum eandem <lb/>rationem, ac 2 ad 1; ad &longs;ecundam vt 3 ad 2; ad tertiam vt <lb/>4 ad 3, &amp; ita deinceps de reliquis; itaque per conuer&longs;io&shy;<lb/>nem rationis habebit ip&longs;um parallelogrammum ad exce&longs;&shy;<lb/>&longs;um illius &longs;upra quancunque parabolarum dictarum, &longs;cili&shy;<lb/>cet ad trilineum primum AGCD eandem rationem, quam <lb/>2 ad 1, ad &longs;ecundum quam 3 ad 1, &amp; &longs;ic deinceps quam <lb/>numerus trilinei vnitate auctus ad ip&longs;am vnitatem. </s>
          <s id="s.000244">Sed <lb/>e&longs;t etiam admonendum verticem dictarum parabolarum <lb/>e&longs;&longs;e punctum A, &amp; per con&longs;equens AB diametrum, &amp; BC <lb/>ordinatim aplicatam, &longs;eu ba&longs;im. </s>
        </p>
        <pb pagenum="22" xlink:href="022/01/028.jpg"/>
        <p type="margin">
          <s id="s.000245"><margin.target id="marg53"/><emph type="italics"/>Tab.<emph.end type="italics"/> 2. <emph type="italics"/>Fig.<emph.end type="italics"/> 9.</s>
        </p>
        <p type="main">
          <s id="s.000246"><emph type="center"/>PROP. XI. THEOR. XI.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000247">II&longs;dem adhuc manentibus, idem de Angelis mon&longs;trat eo&shy;<lb/>dem illo tractatu pr. <!-- REMOVE S-->3. &longs;i qu&aelig;cunque ex dictis parabo&shy;<lb/>lis &longs;ecta &longs;it qualibet recta parallela ba&longs;i BC, e&longs;&longs;e parabolam <lb/>ad re&longs;ectam portionem ver&longs;us verticem, vt pote&longs;tas ba&longs;is, <lb/>cuius exponens e&longs;t numerus parabol&aelig; vnitate auctus ad <lb/>&longs;imilem pote&longs;tatem ex ba&longs;i re&longs;ect&aelig; portionis; itaque i&ntail; <lb/>prima parabola e&longs;t vt quadratum ad quadratum, in &longs;ecun&shy;<lb/>da vt cubus ad cubum, &amp; &longs;ic de c&aelig;teris. </s>
          <s id="s.000248">Similiter &longs;i &longs;ece&shy;<lb/>tur quodlibet ex infinitis trilineis linea GF ba&longs;i CD paral&shy;<lb/>lela, erit trilineum ad &longs;uperius &longs;ui &longs;egmentum vt pote&longs;tas <lb/>ex DA, cuius exponens e&longs;t numerus trilinei vnitate auctus <lb/>ad &longs;imilem pote&longs;tatem ex AF. quare trilineum primu&mtail; <lb/>CAD ad GAF erit vt quadratum ex DA ad quadratum <lb/>ex FA, &longs;ecundum CHAD ad &longs;egmentum HAF vt cubus <lb/>ad cubum, &amp; ita in c&aelig;teris eodem ordine. </s>
        </p>
        <p type="main">
          <s id="s.000249"><emph type="center"/>PROP. XII. THEOR. XII.<emph.end type="center"/><lb/><arrow.to.target n="marg54"/><!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000250"><margin.target id="marg54"/><emph type="italics"/>Tab.<emph.end type="italics"/> 3. <emph type="italics"/>fig.<emph.end type="italics"/> 1.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000251">SIt mod&ograve; ACD angulus rectus, &amp; linea FE talis natur&aelig;, <lb/>vt deductis ad libitum rectis AF, BE parallelis ip&longs;i <lb/>CD, pote&longs;tas ex CA ad &longs;imilem pote&longs;tatem ex CB &longs;it reci&shy;<lb/>proc&egrave; vt alia qu&aelig;dam pote&longs;tas ex BE ad &longs;imilem huic po&shy;<lb/>te&longs;tatem ex AF; patet rectas CA, CD nondum iungi cum <lb/>EF, quamuis in immen&longs;um vn&agrave; producerentur. </s>
          <s id="s.000252">Ab hoc <lb/>proprietate VValli&longs;ius &amp; Fermatius &longs;ubtili&longs;&longs;imi authores <lb/>vocauerunt curuam FE nouam hyperbolam, &amp; eius a&longs;&shy;<lb/>&longs;ymptotos AC, CD. </s>
          <s id="s.000253">Omnes huiu&longs;modi hyperbol&aelig;, qu&aelig; <lb/>infinit&aelig; numero &longs;unt, terminantur ad vnam partem ma&shy;<lb/>gnitudine, cum hyperbola <expan abbr="c&otilde;munis">communis</expan>, &longs;eu Apolloniaca &longs;it in <lb/>vtranque partem magnitudine infinita. </s>
          <s id="s.000254">Quod ergo exi&shy;<lb/>mium e&longs;t, o&longs;tenderunt ip&longs;i authores rectangulum FA i&ntail; <pb pagenum="23" xlink:href="022/01/029.jpg"/>AC ad &longs;patium hyperbolicum qu&agrave; finitum e&longs;t, lic&egrave;t &longs;in&egrave; <lb/>fine longum, eandem habere rationem, quam differentia <lb/>exponentium pote&longs;tatum hyperbol&aelig; ad exponentem po&shy;<lb/>te&longs;tatis minoris. </s>
          <s id="s.000255">Quare &longs;i in hyperbola &longs;it vt cubus CB <lb/>ad cubum CA ita quadratum AF ad quadratum BE, erit <lb/>pr&aelig;dictum rectangulum CA in AF dimidium Spatij &longs;in&egrave; <lb/>fine producti A &amp; FA; at &longs;i quadratum CB ad quadratum <lb/>CA &longs;it vt recta AF ad rectam BE, rectangulum ip&longs;um CA <lb/>in AF &aelig;quale erit &longs;patio A &amp; FA, qu&ograve;d &longs;i pote&longs;tas CA vel <lb/>CB non fuerit altior pote&longs;tate ex BE, vel AF, tunc ip&longs;um <lb/>illud &longs;patium, infinitum quoque erit magnitudine, etenim <lb/>nullus exce&longs;&longs;us exponentis pr&aelig;dict&aelig; pote&longs;tatis ex CA &longs;u&shy;<lb/>pra exponentem pote&longs;tatis BE, habet ad numerum expo&shy;<lb/>nentis pote&longs;tatis BE rationem infinitam. </s>
        </p>
        <p type="main">
          <s id="s.000256"><emph type="center"/>DEMONSTRATIO.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000257">SVpradictum propo&longs;itum habetur in commercio epi&shy;<lb/>&longs;tolico Ioannis Valli&longs;ij Epi&longs;tola quarta, quem libellum <lb/>vn&agrave; cum alijs docti&longs;&longs;imis &longs;uis operibus Vincentius Viuia&shy;<lb/>nus ingens &aelig;ui no&longs;tri Geometra, antequam &longs;umma cu&mtail; <lb/>humanitate mi&longs;i&longs;&longs;et, eidem ip&longs;i quadraturam vnius ex di&shy;<lb/>ctis hyperbolis ex no&longs;tris principijs deductam, ac excogi&shy;<lb/>tatam, indicauimus. </s>
          <s id="s.000258">Cum ver&ograve; po&longs;tea nobis eueni&longs;&longs;et <lb/>vniuer&longs;aliorem ad alias hyperbolas &lpar;&longs;emper communi ex&shy;<lb/>cepta&rpar; accomodatam reperij&longs;&longs;e, huc debemus afferre, pri&shy;<lb/>m&ugrave;m vt quendam fructum &longs;cienti&aelig; huius; deinde cum di&shy;<lb/>ctorum authorum ip&longs;am propo&longs;itionis demon&longs;trationem <lb/>non habuerimus, &amp; demum quia ip&longs;arum hyperbolarum <lb/>men&longs;ura, ac quadratura in aquarum rationibus erunt po&shy;<lb/>ti&longs;&longs;imum ex v&longs;u. </s>
          <s id="s.000259">Sit igitur BC vna ex infinitis hyperbolis, <arrow.to.target n="marg55"/><lb/>quarum a&longs;&longs;ymptoti AE, EL; Sint etiam qu&aelig;cunque apli&shy;<lb/>cat&aelig; AB, DC a&longs;symptoto EL &aelig;quidi&longs;tantes, &amp; habeat <lb/>DE ad EA eandem rationem v. <!-- REMOVE S-->g. <!-- KEEP S--></s>
          <s id="s.000261">quam cubus ex AB ad <pb pagenum="24" xlink:href="022/01/030.jpg"/><arrow.to.target n="marg56"/><lb/>cubum DC. </s>
          <s id="s.000262">Patet &longs;i proponeretur illi auuer&longs;a figur&atail; <lb/><arrow.to.target n="marg57"/><lb/>FGK, e&longs;&longs;etque AE ad DE vt figura GFK ad figuram IHK <lb/>e&longs;&longs;e etiam FG ad IH vt DC ad AB, e&longs;t autem cubus ex <lb/>DC ad cubum ex AB vt AE ad ED; ergo etiam figur&atail; <lb/>FGK ad IHK &lpar;&longs;unt enim FG, IH parallel&etail;&rpar; habebit ean&shy;<lb/>dem rationem, ac cubus ex FG ad cubum ex IH: Itaqu&etail; <lb/>GFK erit comunis parabola, hoc e&longs;t quadratica, &longs;eu <expan abbr="&longs;ec&utilde;-">&longs;ecun&shy;<lb/></expan><arrow.to.target n="marg58"/><lb/>da in &longs;erie infinitarum parabolarum, &amp; ob id eadem GFK <lb/><arrow.to.target n="marg59"/><lb/>parabola ad rectangulum GF in FK erit vt 2 ad 3, in qua <lb/>ratione &longs;e habebit quoque rectangulum BA in AE ad &longs;pa&shy;<lb/>tium infinit&egrave; longum &amp; BM, et erit vt 2 ad 1; &longs;cilicet vt ex&shy;<lb/>ce&longs;&longs;us exponentis maioris pote&longs;tatis, qu&aelig; cubica e&longs;t, &longs;uper <lb/>numerum exponentis, qui hoc ca&longs;u e&longs;t tant&ugrave;m vnitas ra&shy;<lb/>dicis, e&longs;t ad hunc ip&longs;um exponentem, &longs;eu vnitatem line&aelig; <lb/>indicantem, quod concordat cum propo&longs;ita dictoru&mtail; <lb/>authorum. </s>
        </p>
        <p type="margin">
          <s id="s.000263"><margin.target id="marg55"/><emph type="italics"/>Tab.<emph.end type="italics"/> 3. <emph type="italics"/>Fig.<emph.end type="italics"/> 3.<!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000264"><margin.target id="marg56"/><emph type="italics"/>Tab.<emph.end type="italics"/> 3. <emph type="italics"/>fig<emph.end type="italics"/> 2.<!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000265"><margin.target id="marg57"/><emph type="italics"/>Def.<emph.end type="italics"/> 8. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000266"><margin.target id="marg58"/><emph type="italics"/>Pr.<emph.end type="italics"/> 10. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000267"><margin.target id="marg59"/><emph type="italics"/>Pr.<emph.end type="italics"/> 9. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000268"><emph type="center"/><emph type="italics"/>Exemplum aliud.<emph.end type="italics"/><emph.end type="center"/><lb/><arrow.to.target n="marg60"/></s>
        </p>
        <p type="margin">
          <s id="s.000269"><margin.target id="marg60"/><emph type="italics"/>In eadem fi&shy;<lb/>gur&aelig;.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000270">SIt etiam cubus ex DE ad cubum ex AE, &longs;icut quadra&shy;<lb/>to quadratum AB ad quadroquadratum DC, &amp; rur&shy;<lb/>&longs;us propo&longs;ita GKF auer&longs;a huius hyperbol&aelig;: patet &longs;i &longs;it AE <lb/>ad DE vt figura GFK ad figuram IKH, e&longs;&longs;e etiam FG ad </s>
        </p>
        <p type="main">
          <s id="s.000271"><arrow.to.target n="marg61"/><lb/>IH vt DC ad AB; cumque &longs;it cubus ex AE ad cubum ex <lb/>DE &longs;icut quadroquadratum ex DC ad <expan abbr="quadroquadrat&utilde;">quadroquadratum</expan> <lb/>ex AB, erit etiam quadroquadratum ex FG ad quadro&shy;<lb/>quadratum ex IH, vt cubus ex AE ad cubum ex DE; &longs;i <lb/>igitur intelligatur qu&aelig;dam ratio, qu&aelig; &longs;it &longs;ubduodecupla <lb/>tam rationis quadroquadratorum qu&agrave;m huic &longs;imilis cu&shy;<lb/>borum pr&aelig;dictorum, erit porr&ograve; FG ad IH triplicata, &amp; <lb/>AE ad ED quadruplicata eiu&longs;dem dict&aelig; &longs;ubduodecupl&aelig;; <lb/>quamobrem etiam ratio figur&aelig; GFK ad <expan abbr="figur&atilde;">figuram</expan> IHK, qu&aelig; <lb/>e&longs;&longs;e debet vt AE ad ED, erit quadruplicata eiu&longs;dem &longs;ub&shy;<lb/>duodecupl&aelig;: &amp; ide&ograve; &longs;i ponamus IK ad KI in ratione <pb pagenum="25" xlink:href="022/01/031.jpg"/>eiu&longs;dem &longs;ubduodecupl&aelig;, erit figura GFK illius natur&aelig;, vt <lb/><arrow.to.target n="marg62"/><lb/>&longs;it &longs;emper cubus ex FK ad cubum ex KI &longs;icut GF ad IH, &amp; <lb/>hoc modo eadem illa figura erit trilineum tertium, &longs;eu cu&shy;<lb/>bicum, ex quo ergo &longs;equitur, GFK ad HIK &longs;it in eadem ra&shy;<lb/>tione, in qua quadroquadratum ex FK ad quadroqua&shy;<lb/>dratum ex KI, hoc e&longs;t &longs;it vt AE ad ED; &longs;equiturque etiam <lb/><arrow.to.target n="marg63"/><lb/>ob hoc figuram GFK &longs;ubquadruplam e&longs;le circum&longs;cripti <lb/>rectanguli GF in FK; e&longs;t autem vt trilineum GFK ad <expan abbr="rect&atilde;-">rectan&shy;<lb/></expan><arrow.to.target n="marg64"/><lb/>gulum GF in FK circum&longs;criptum, &longs;ic rectangulum ABME <lb/>ad auuer&longs;am eidem trilineo figuram AB &amp; EA, ergo re&shy;<lb/>ctangulum ABME &longs;ubquadruplum erit eiu&longs;dem figur&aelig; <lb/>AB &amp; EA longitudinis infinit&aelig;, quare ip&longs;um rectangulum <lb/>erit &longs;ubtriplum portionis &amp; BM &amp; longitudinis pariter im&shy;<lb/>men&longs;&aelig;. </s>
          <s id="s.000272">Cum ita &longs;it, con&longs;tat exemplo hoc quoque, <expan abbr="eand&etilde;">eandem</expan> <lb/>illam rationem e&longs;&longs;e exce&longs;&longs;um maioris exponentis &longs;upr&atail; <lb/>minorem exponentem ad hoc ip&longs;um, dictarum <expan abbr="pote&longs;tat&utilde;">pote&longs;tatum</expan> <lb/>hyperbol&aelig;. </s>
        </p>
        <p type="margin">
          <s id="s.000273"><margin.target id="marg61"/><emph type="italics"/>Def.<emph.end type="italics"/> 8. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000274"><margin.target id="marg62"/><emph type="italics"/>Pr.<emph.end type="italics"/> 10. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000275"><margin.target id="marg63"/><emph type="italics"/>Pr.<emph.end type="italics"/> 10. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000276"><margin.target id="marg64"/><emph type="italics"/>Pr.<emph.end type="italics"/> 9. <emph type="italics"/>huius<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000277"><emph type="center"/>PROP. XIII. THEOR. XIII.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000278">SVperior demon&longs;tratio effecta fui&longs;&longs;et ampli&longs;&longs;ima, &longs;i pr&etail;&shy;<lb/>ponere volui&longs;&longs;emus <expan abbr="quadratur&atilde;">quadraturam</expan> vt datam omnis ge&shy;<lb/>neris parabolarum, &amp; trilineorum, ver&ugrave;m cum i&longs;ta pars <expan abbr="n&otilde;">non</expan> <lb/>&longs;it plen&egrave; tradita, vt videre e&longs;t quinto libro infinitarum pa&shy;<lb/>rabolarum eiu&longs;dem de Angelis, &longs;atius ideo duximus qua&shy;<lb/>draturam hyperbolarum &agrave; VVali&longs;io, &amp; Fermatio acuti&longs;&longs;i&shy;<lb/>mis illis viris propo&longs;itam omnino veram admittere, vt ind&egrave; <lb/>eam parabolarum &amp; trilineorum vniuer&longs;alem, quam adhuc <lb/>ab alijs non habemus, facillim&egrave;, compendios&egrave;que depro&shy;<lb/>meremus. </s>
          <s id="s.000279">Hanc igitur ita proponimus vt &longs;ubinde o&longs;ten&shy;<lb/>damus. </s>
        </p>
        <p type="main">
          <s id="s.000280">Si &longs;imiles pote&longs;tates applicatarum fuerint in eadem ra&shy;<lb/>tione, ac &longs;unt inter&longs;e pote&longs;tates qu&aelig;dam ali&aelig;, &amp; eiu&longs;dem <lb/>gradus diametrorum ab ip&longs;is applicatis ab&longs;ci&longs;&longs;arum v&longs;que <pb pagenum="26" xlink:href="022/01/032.jpg"/>ad verticem parabolarum, vel trilineorum; erit rectangu&shy;<lb/>lum ad parabolam &longs;ibi in&longs;criptam vt aggregatum <expan abbr="expon&etilde;-tium">exponen&shy;<lb/>tium</expan> vtriu&longs;que pote&longs;tatis ad exponentem altioris ip&longs;arum <lb/>pote&longs;tatum parabol&aelig;; &amp; ad trilineum vt aggregatum ex&shy;<lb/>ponentium pote&longs;tatum trilinei ad exponentem inferioris <lb/>pote&longs;tatis eiu&longs;demmet trilinei. </s>
          <s id="s.000281">Sic enim in expo&longs;ita figu&shy;<lb/>ra pr&aelig;dicta, &longs;i e&longs;&longs;et quadratum ex FG ad quadratum ex <lb/>IH, &longs;icut cubus ex FK ad cubum ex IH, e&longs;&longs;et rectangulum <lb/>GF in FK ad figuram GFK &lpar;qu&aelig; tunc foret trilineum, vt <lb/>5 ad 2; nam vbi pote&longs;tas ab&longs;ci&longs;&longs;arum maior e&longs;t illa applica. <lb/></s>
          <s id="s.000282">tarum e&longs;t &longs;emper GF trilineum. </s>
          <s id="s.000283">Simili modo, &longs;i &longs;it vt qua&shy;<lb/>dratum ex FK ad quadratum ex KI ita cubocubus ex FG <lb/>ad cubocubum ex IH; hoc e&longs;t &longs;i &longs;it cubus ex FG ad <expan abbr="cub&utilde;">cubum</expan> <lb/>ex IH, vt linea FK ad KI &lpar;tolluntur enim vtrinque ex &longs;imi&shy;<lb/>libus &longs;imiles rationes&rpar; erit &longs;igura GFK parabola, ad quam <lb/>&longs;ibi circum&longs;criptum rectangulum eandem habebit <expan abbr="ration&etilde;">rationem</expan>, <lb/>quam 4 ad 3, &amp; &longs;ic dicendum erit de omnibus alijs para&shy;<lb/>bolis atque trilineis. </s>
        </p>
        <p type="main">
          <s id="s.000284"><emph type="center"/>DEMONSTRATIO.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000285">VEr&ugrave;m vt propo&longs;itum o&longs;tendamus, e&longs;to qu&aelig;libet ex <lb/>parabolis GFK, nimirum quadratocubus ex FG ad <lb/>quadratocubum ex IH habeat eandem rationem, qua&mtail; <lb/>cubus ex FK ad cubum ex IK. Demon&longs;tro, rectangulum <lb/>GF in FK habere eandem rationem ad parabolam GFK, <lb/>quam aggregatum exponentium 8 ad maiorem exponen&shy;<lb/>tem 5. Prim&ugrave;m, quam rationem habet rectangulum GF in <lb/>FK ad parabolam GFK, eandem habebit rectangulum HI <lb/>in IK ad parabolam HIK &lpar;hoc enim demon&longs;trabimus in&shy;<lb/>fr&agrave;&rpar; permutandoque, erit rectangulum GF in FK ad re&shy;<lb/>ctangulum HI in IK, vt parabola GFK ad parabolam HIK; <lb/>componuntur ver&ograve; illa rectangula ex rationibus GF ad <lb/>IH, &amp; FK ad IK, ergo etiam parabola ad parabolam com-<pb pagenum="27" xlink:href="022/01/033.jpg"/>ponetur ex ij&longs;dem rationibus; &amp; quoniam ductis inuicem <lb/>exponentibus po&longs;&longs;unt con&longs;iderari quindecim rationes in&shy;<lb/>ter &longs;e &longs;imiles, ex quibus con&longs;tet tam ratio dictorum cubo&shy;<lb/>rum, qu&agrave;m huic &longs;imilis altera quadratocuborum, &amp; tunc <lb/>GF ad IH erit triplicata, et FK ad KI quintuplicata <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> <lb/>&longs;ubquindecupl&aelig; rationis, qu&aelig; &longs;it A ad B; ergo &longs;imul ad&shy;<lb/>ditis ij&longs;dem rationibus, quintuplicata &longs;cilicet, &amp; triplicata <lb/>exiliet ratio octuplicata ip&longs;ius A ad B; proptereaque pa&shy;<lb/>rabola GFK ad HIK, &longs;eu &longs;i con&longs;ideremus figuram &amp; BAEL <lb/>auuer&longs;am parabol&aelig; GFK, ita vt AE ad ED &longs;it vt para&shy;<lb/><arrow.to.target n="marg65"/><lb/>bola GFK ad <expan abbr="parabol&atilde;">parabolam</expan> HIK; AE ad ED erit pariter octu&shy;<lb/>plicata eiu&longs;dem A ad B; &amp; cum &longs;it ob naturam <expan abbr="auuer&longs;ar&utilde;">auuer&longs;arum</expan> <lb/>FG ad HI vt DC ad AB; erit DC ad AB triplicata <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> <lb/>rationis A ad B, qnare vt cubus AE ad cubum DE, it&atail; <lb/>quadratocubocubus DC ad quadratocubocubum ex <lb/>AB: rectangulum igitur ABME ad &longs;patium hyperbolicum <lb/>infin<gap/> &egrave; longum &amp; BM &amp; erit vt quinque ad tria, &amp; ad vni&shy;<lb/><arrow.to.target n="marg66"/><lb/>uer&longs;um &longs;patium &amp; BAE &amp; vt 5 ad 8, in qua nempe ratio&shy;<lb/>ne debet e&longs;&longs;e parabola GF<emph type="italics"/>K<emph.end type="italics"/> ad rectangulum GF in FK. <lb/><arrow.to.target n="marg67"/><lb/>Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000286"><margin.target id="marg65"/><emph type="italics"/>Def.<emph.end type="italics"/> 8. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000287"><margin.target id="marg66"/><emph type="italics"/>Pr.<emph.end type="italics"/> 12 <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000288"><margin.target id="marg67"/><emph type="italics"/>Pr.<emph.end type="italics"/> 9. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000289"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000290"><emph type="italics"/>Con&longs;tat &longs;i fuerit ratio A ad B e&ograve; &longs;ubmultiplicata rationis <lb/>applicatarum, quoties e&longs;t numerus exponentis pote&longs;tatis ab&shy;<lb/>&longs;ci&longs;&longs;arum eiu&longs;dem parabol&aelig;, e&longs;&longs;e ip&longs;am parabolam ad &longs;ui por&shy;<lb/>tionem in tam multiplicata ratione A ad B, ac e&longs;t numerus <lb/>aggregati exponentium ambarum pote&longs;tatum parabola. </s>
          <s id="s.000291">Nam <lb/>cum e&longs;&longs;et quadratocubus ex FG ad quadratocubum ex IH, &longs;i&shy;<lb/>cut cubus ex FK ad cubum ex IK, propo&longs;ita in&longs;uper e&longs;&longs;et A ad <lb/>B. &longs;ubquindecupla alterius dictarum &longs;imilium rationum ex <lb/>pote&longs;t atibus parabola, o&longs;ten&longs;um fuit rationem A ad B &longs;ubtri&shy;<lb/>plicatam ip&longs;ius GF ad IH, &amp; &longs;ubquintuplicatam alterius FK <lb/>ad KI, &amp; tandem o&longs;tendimus parabolam GFK ad portionem <emph.end type="italics"/><pb pagenum="28" xlink:href="022/01/034.jpg"/><emph type="italics"/>eius HIK e&szlig;e in octuplicata ratione eiu&longs;dem A ad B; quod <lb/>idem omnino diceretur &longs;i figura GFK trilineum e&longs;&longs;et. </s>
          <s id="s.000292">Ratio <lb/>autem A ad B dicetur impo&longs;terum logarithmica pote&longs;tatum <lb/>parabol&aelig;, &longs;eu trilinei, aut hyperbol&aelig;.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000293"><emph type="center"/>ASSVMPTVM.<emph.end type="center"/></s>
        </p>
        <p type="main">
          <s id="s.000294">REliquum e&longs;t vt o&longs;tendamus, parabolam GFK ad <lb/>portionem HIK e&longs;&longs;e vt rectangulum GF ad rectan&shy;<lb/>gulum HI in IK, &longs;cilicet e&longs;&longs;e in ratione compo&longs;ita ba&longs;ium, <lb/>&amp; altitudinum parabolarum, quod nempe &longs;ic o&longs;tendetur, <lb/>Sit vt &longs;upra FGK parabola, eiu&longs;que portio IHK; exi&longs;tenti&shy;<lb/>bus ver&ograve; applicatis FG, IH, fiat EG ad IE vt FK ad KI, &longs;it&shy;<lb/><arrow.to.target n="marg68"/><lb/>que IE ba&longs;is, et K vertex parabol&etail; IEK &longs;imilis ip&longs;i GFK pa&shy;<lb/>tet propter &longs;imilitudinem figurarum, e&longs;&longs;e parabolam GFK <lb/>ad parabolam IEK in eadem duplicata ratione FG ad IE, <lb/>in qua nempe e&longs;t rectangulum GF in FK ad &longs;ibi &longs;imile re&shy;<lb/>ctangulum EI in IK, ob idque rectangulum GF in FK ad <lb/>rectangulum EI in IK, cum &longs;int inter&longs;e vt parabola GFK ad <lb/>parabolam EIK, h&aelig;c ver&ograve; parabola ad ip&longs;am IHK habeat <lb/>eandem rationem, ac IE ad IH; &longs;eu ob eandem altitudinem <lb/>IK vt rectangulum EI in IK ad rectangulum HI in IK, erit <lb/>ex &aelig;quali parabola GFK ad parabolam HIK vt rectangu&shy;<lb/>lum GF in FK ad rectangulum HI in IK. <!-- KEEP S--></s>
          <s id="s.000295">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000296"><margin.target id="marg68"/><emph type="italics"/>Tab.<emph.end type="italics"/> 3. <emph type="italics"/>Fig.<emph.end type="italics"/> 2.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000297"><emph type="center"/>PROP. XIV. THEOR. XIV.<emph.end type="center"/><lb/><arrow.to.target n="marg69"/><!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000298"><margin.target id="marg69"/><emph type="italics"/>Tab.<emph.end type="italics"/> 2. <emph type="italics"/>fig.<emph.end type="italics"/> 3.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000299">IN quacunque hyperbola &lpar;excepta &longs;emper conica&rpar; cu&shy;<lb/>ius a&longs;&longs;ymptoti EA, EM, &longs;i &longs;it pote&longs;tas applicatarum DC <lb/>AB altior pote&longs;tate ab&longs;ci&longs;&longs;arum AE, ED &lpar;&longs;ic enim finit&atail; <lb/>erit magnitudine &longs;ecundum eam a&longs;&longs;ymptoton, qu&aelig; appli&shy;<lb/>catis parallela e&longs;t&rpar; &longs;patium ip&longs;um hyperbol&aelig; &amp; BAE &amp; <lb/>ad &longs;ui portionem &amp; CDE &amp; habebit eandem rationem, ac <lb/>rectangulum BAE ad rectangulum CDE, &longs;eu &lpar;a&longs;&longs;umpta <pb pagenum="29" xlink:href="022/01/035.jpg"/>ratione logarithmica A ad B pote&longs;tatum hyperbol&aelig;&rpar; <expan abbr="qu&atilde;">quam</expan> <lb/>pote&longs;tas ex A, cuius exponens e&longs;t differentia <expan abbr="exponenti&utilde;">exponentium</expan> <lb/>pote&longs;tatum hyperbol&aelig; ad &longs;imilem pote&longs;tatem ex B. <!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000300"><emph type="center"/>DEMONSTRATIO.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000301">QVam rationem habet rectangulum BAE ad &longs;patium <lb/>&amp; BAE &amp;, eandem habet rectangulum CDE ad </s>
        </p>
        <p type="main">
          <s id="s.000302"><arrow.to.target n="marg70"/><lb/>&longs;patium &amp; CDE, &amp; permutando erit rectangu&shy;<lb/>lum BAE ad CDE, &longs;icut &longs;patium &amp; BAE &amp; ad &longs;patiu&mtail; <lb/>&amp; CDE &amp;; &longs;i igitur in eadem propo&longs;ita hyperbola &longs;it po&shy;<lb/>te&longs;tas applicatarum DC, AB quintuplicata ip&longs;ius A ad B, <lb/>&amp; AE ad ED &longs;eptuplicata &longs;it eiu&longs;dem; erit &longs;eptuplicat&atail; <lb/>applicatarum in eadem ratione, ac quintuplicata ab&longs;ci&longs;&longs;a&shy;<lb/>rum; &longs;cilicet quadratoquadratocubus ex DC ad &longs;imilem <lb/>pote&longs;tatem ex AB erit vt quadratocubus ex AE ad qua&shy;<lb/>dratocubum ex DE, eritque &longs;ic maior pote&longs;tas applicata&shy;<lb/>rum, atque adeo componetur rectangulum EAB ad EDC <lb/>ex &longs;eptuplicata ip&longs;ius A ad B, qualis e&longs;t AE ad ED, &amp; &longs;ub&shy;<lb/>quintuplicata eiu&longs;dem A ad B, qu&aelig; e&longs;t AB ad DC; nimi&shy;<lb/>r&ugrave;m erit rectangulum EAB ad EDC in duplicata tantum <lb/>ratione ip&longs;ius A ad B: quare &longs;patium &amp; BAE &amp; ad id <lb/>&amp; CDE &amp;, qu&aelig; &longs;unt inter &longs;e, vt ip&longs;a rectangula, erit vt po&shy;<lb/>te&longs;tas ex A, cuius exponens e&longs;t differentia exponentium &amp; <lb/>S pote&longs;tatum hyperbol&aelig; ad &longs;imilem pote&longs;tatem ex B. <lb/><!-- KEEP S--></s>
          <s id="s.000303">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000304"><margin.target id="marg70"/><emph type="italics"/>Pr.<emph.end type="italics"/> 12. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000305"><emph type="center"/>PROP. XV. THEOR. XV.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000306">SI ab exponente pote&longs;tatis applicatarum hyperbol&etail; de&shy;<lb/>trahatur exponens minoris pote&longs;tatis ab&longs;ci&longs;&longs;arum, po&shy;<lb/>te&longs;tas reliqui exponetis erit applicatarum auuer&longs;&aelig; figur&aelig;, <lb/>in ab&longs;ci&longs;&longs;is ver&ograve; ade&longs;t vtrobique eadem pote&longs;tas. </s>
          <s id="s.000307">Itaque <lb/>cum in &longs;uperiori hyperbola re&longs;idui exponentis pote&longs;tas <pb pagenum="30" xlink:href="022/01/036.jpg"/>quadratum e&longs;&longs;et, porr&ograve; in eius auuer&longs;a e&longs;&longs;et pote&longs;tas appli&shy;<lb/>catarum quadratica, &amp; ab&longs;ci&longs;&longs;arum quadratocubica. </s>
        </p>
        <p type="main">
          <s id="s.000308"><emph type="center"/>DEMONSTRATIO.<emph.end type="center"/><lb/><arrow.to.target n="marg71"/><!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000309"><margin.target id="marg71"/><emph type="italics"/>Tab.<emph.end type="italics"/> 3. <emph type="italics"/>Fig.<emph.end type="italics"/> 3.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000310">ESto rur&longs;us hyperbola &amp; BAE &amp;, et &longs;icut dictum e&longs;t <lb/>AE ad ED &longs;it in &longs;eptuplicata ratione logarithmic&aelig; <lb/>rationis A ad B, at DC ad AB in quintuplicata, videlicet <lb/>quadratocubus ex AE ad quadratocubum ex DE eandem <lb/>habeat rationem, ac quadratoquadratocubus ex DC ad <lb/>&longs;imilem pote&longs;tatem ex AB; Dico in auuer&longs;a figura pote&longs;ta&shy;<lb/>tem aplicatarum e&longs;&longs;e quadratum, cuius <expan abbr="expon&etilde;s">exponens</expan> 2 e&longs;t dif&shy;<lb/>ferentia exponentium pote&longs;tatum hyperbol&aelig;; pote&longs;tatem <lb/>ver&ograve; ab&longs;ci&longs;&longs;arum eandem e&longs;&longs;e, ab&longs;ci&longs;&longs;arum eiu&longs;dem hyper&shy;<lb/>bol&aelig;. </s>
          <s id="s.000311">Sit vt &longs;upra FK ad KI vt hyperbola &amp; BAE &amp; ad <lb/>&amp; CDE &amp;, hoc e&longs;t, &longs;it vt pote&longs;tas ex A, cuius exponens </s>
        </p>
        <p type="main">
          <s id="s.000312"><arrow.to.target n="marg72"/><lb/>e&longs;t differentia exponentium pote&longs;tatum hyperbol&aelig; ad &longs;i&shy;<lb/>milem pote&longs;tatem ex B, &amp; ideo FK ad KI erit duplicata ip&shy;<lb/>&longs;ius A ad B, &longs;ed DC ad AB eiu&longs;dem illius logarithmic&aelig; <lb/>quintuplicata; e&longs;tque in hac eadem ratione etiam GF ad <lb/>IH; ergo cum duplicata huius &longs;it &longs;imilis quintuplicat&aelig; KF <lb/>ad KI &lpar;nam vtraque ratio continet decies A ad B&rpar; pater, <lb/>quadratum ex FG ad quadratum ex IH e&longs;&longs;e eam pote&longs;ta&shy;<lb/>tem, quam propo&longs;uimus euenire in applicatis auuer&longs;&aelig;, cum <lb/>ali&agrave;s in ab&longs;ci&longs;&longs;is &longs;it vtrobique pote&longs;tas eadem, nempe qua&shy;<lb/>dratocubi. </s>
          <s id="s.000313">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000314"><margin.target id="marg72"/><emph type="italics"/>Pr.<emph.end type="italics"/> 14. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000315"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000316"><emph type="italics"/>Patet ex noto trilineo, vel parabola FGK e&longs;&longs;e in auuer&longs;a, <lb/>&longs;cilicet in hyperbola &amp; BAE &amp; &lpar;qu&aelig; tunc e&longs;t &longs;emper magnitu&shy;<lb/>dine finita iuxta a&longs;symptoton EM &amp;&rpar; pote&longs;tatem <expan abbr="applicatar&utilde;">applicatarum</expan>, <lb/>qua pro exponente habet &longs;ummam exponentium pote&longs;tatum <lb/>parabol&aelig;, aut trilinei; nam cum e&szlig;et in trilineo pracedenti<emph.end type="italics"/><pb pagenum="31" xlink:href="022/01/037.jpg"/><emph type="italics"/>quadratum ex FG ad quadratum ex IH vt quadratocubus <lb/>ex FK ad quadratocubum ex IK, fuit equidem in hyperbol&atail; <lb/>quadratoquadratocubus ex DC<gap/> quadratoquadratocubum <lb/>ex AB &longs;icut quadratocubus ex AE ad &longs;imilem pote&longs;tatem ex <lb/>DE, &longs;cilicet inuariata pote&longs;tate ab&longs;er&longs;arum in ambabus au&shy;<lb/>uer&longs;is. </s>
          <s id="s.000317">Quare ex pote&longs;tatibus notis vnius auuer&longs;arum fa&shy;<lb/>cil&egrave; inote&longs;cent pote&longs;tates alterius, atque etiam illius magnitu&shy;<lb/>do. </s>
          <s id="s.000318">Nunc redeamus ad motus, nouamque adhuc methodum, <lb/>quam hoc loco re&longs;eruauimus, afferamus.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000319"><emph type="center"/>DEF. IX.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000320">SIt qu&aelig;dam Gene&longs;is ACBH, cuius imago temporis <lb/>&amp; DCB &amp;; item &longs;it FCBK gene&longs;is alterius motus ab <lb/><arrow.to.target n="marg73"/><lb/>eodem C in B; &amp; act&agrave; rect&agrave; OIGE ip&longs;i AFCD parallel&atail;, <lb/>ponantur CD, GE loco minimorum temporum, ita vt <expan abbr="t&etilde;-pore">ten&shy;<lb/>pore</expan> CD, dum mobile ex C affectum velocitate CA, <lb/>currat minimum &longs;patiolum indicatum per C, cui e&longs;t &aelig;qua&shy;<lb/>le &longs;patiolum aliud indicatum per G, quodque tran&longs;igitur <lb/>tempore GE velocitate GD &lpar;nam vt e&longs;&longs;ent illa &longs;patia i&ntail; <lb/>C, G &aelig;qualia, effectum fuit vt velocitas AC ad GD ean&shy;<lb/>dem reciproc&egrave; rationem haberet, ac tempus GE ad CD, <lb/>id quod patet ex natura gene&longs;is ACBH, &amp; imaginis &amp; <lb/>DCB &amp;&rpar; et hic rur&longs;us notatu digni&longs;&longs;imum e&longs;t nulli errori <lb/>obnoxium e&longs;&longs;e, qu&ograve;d &aelig;quabiles in illis minimis &longs;patiolis <lb/>intellexerimus motus, quamuis potius deberet videri, in <lb/>ij&longs;dem interuallis reperiri innumeras, ac in&aelig;quales veloci&shy;<lb/>tates, queis nempe efficerentur motus in&aelig;quabiles, qu&ograve;d <lb/>gene&longs;es in&aelig;quabiles &longs;int. </s>
          <s id="s.000321">Cur i&longs;ta &longs;e ita habeant, hic non <lb/>e&longs;t nobis di&longs;putandum, ego enim puto, non ex indiui&longs;ibili <lb/>velocitates alijs &longs;uccedere, &longs;ed reuera minutulum tempo&shy;<lb/>ris con&longs;iderari debere antequam motus diuer&longs;imod&egrave; pro&shy;<lb/>cedat, nempe ac &longs;i velocitas, qu&aelig; &longs;uccedere debet priori, <lb/>non ita &longs;it in promptu, aut non ita &longs;tatim mobile afficiat ad <pb pagenum="32" xlink:href="022/01/038.jpg"/>motum &longs;ibi proportionatum. </s>
          <s id="s.000322">Sed linquamus h&aelig;c alijs di&longs;&shy;<lb/>putanda: &longs;atis nobis &longs;it, methodum no&longs;tram, quoad <expan abbr="no&longs;tr&utilde;">no&longs;trum</expan> <lb/>e&longs;t, demon&longs;trare. </s>
          <s id="s.000323">Ijs igitur vt &longs;upra propo&longs;itis, concipia&shy;<lb/>tur adhuc tempore CD velocitate FC <expan abbr="&longs;pati&utilde;">&longs;patium</expan> exigi quod&shy;<lb/>dam, item aliud tempore EG, velocitateque GI, &amp; &longs;ic per <lb/>omnes qua&longs;cunque applicatas: qu&aelig;ritur, quod &longs;patiu&mtail; <lb/>vltim&ograve; exactum e&longs;&longs;et, hoc e&longs;t quam rationem id haberet ad <lb/>illud alterum &longs;patium, quod eodem tempore tran&longs;igitur <lb/>iuxta gene&longs;im HACB, cuius imago temporis CD &amp; B. <lb/><!-- KEEP S--></s>
          <s id="s.000324">I&longs;ti duo motus in exemplo e&longs;&longs;ent, &longs;i in quodam plano mo&shy;<lb/>ueretur formica, dum ip&longs;um planum vna eius extremitate <lb/>immobili circumduceretur, Sic formica difficili&ugrave;s <expan abbr="a&longs;c&etilde;de-ret">a&longs;cende&shy;<lb/>ret</expan> prout ip&longs;um planum magis ad horizontem erigeretur. <lb/></s>
          <s id="s.000325">Iam motus extremitatis plani circumact&aelig; habet gene&longs;im <lb/>ACBH, cuius temporis imago &amp; DCB &amp;, et altera gene&longs;is <lb/>FCBK tribueretur motui formic&aelig;, nam vt <expan abbr="dict&utilde;">dictum</expan> e&longs;t varius <lb/>motus formic&aelig; pendet ex latione plani, ide&ograve; velocitates <lb/>eiu&longs;dem &lpar;nam in plano immobili ponimus &aelig;quabiliter fer&shy;<lb/>ri&rpar; durant ij&longs;dem temporibus, quibus velocitates pr&aelig;cipu&aelig; <lb/>gene&longs;is ACBH. <!-- KEEP S--></s>
          <s id="s.000326">Sit denique LMSR imago velocitatum <lb/>iuxta gene&longs;im ACBH, cuius temporis imago CD &amp; B; pa&shy;<lb/>tet &longs;i &longs;it MP ad PS &longs;icut imago temporis CDEG ad ima&shy;<lb/>ginem &amp; BGE &amp;, fore LM ad PQ vt AC ad OG, &amp; con&shy;<lb/>cepta etiam figura MNOTS inter parallelas LMN, RST <lb/>ita vt &longs;it &longs;emper MN ad PO &longs;icut FC ad GI, nec non LM <lb/>ad MN vt AC ad FC. &lpar;&longs;unt enim initio motuum in C, aut <lb/>in&longs;tanti M, velocitates gene&longs;ium AC, CF, &longs;cilicet LM, MN; <lb/>&amp; in G, hoc e&longs;t in&longs;tanti P &longs;unt velocitates OC, GI; nimi&shy;<lb/>rum QP, PO&rpar; vocetur proinde gene&longs;is FCBK &longs;puria, ac <lb/>ad&longs;tricta imagini temporis &amp; DCB &amp;, cuius imago veloci&shy;<lb/>tatum MNTS pariter &longs;puria, homogenea tamen ip&longs;i legiti&shy;<lb/>m&aelig; LMSR. <!-- KEEP S--></s>
        </p>
        <pb pagenum="33" xlink:href="022/01/039.jpg"/>
        <p type="margin">
          <s id="s.000327"><margin.target id="marg73"/><emph type="italics"/>Tab.<emph.end type="italics"/> 3. <emph type="italics"/>fig.<emph.end type="italics"/> 4.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000328"><emph type="center"/>PROP. XVI. THEOR. XVI.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000329">SI &longs;int duo motus iuxta gene&longs;es legitimam, &amp; &longs;puriam, <lb/>erunt mobilium exacta &longs;patia, vt imagines inter&longs;e <lb/>homogene&aelig; velocitatum, legitima ad &longs;puriam. </s>
        </p>
        <p type="main">
          <s id="s.000330">E&longs;to gene&longs;is legitima ACBH, cuius imago temporis <lb/><arrow.to.target n="marg74"/><lb/>&amp; DCA &amp;, &amp; imago velocitatum MLRS. </s>
          <s id="s.000331">Sit etiam gene&shy;<lb/>&longs;is altera illi homogenea, &longs;ed &longs;puria, &amp; ad&longs;tricta imagini <lb/>temporis &amp; DCB &amp;, cuius imago velocitatum &longs;puria, prio&shy;<lb/>rique legitim&aelig; homogenea NMST. Dico, &longs;patia iuxta has <lb/>imagines tran&longs;acta e&longs;&longs;e vt ip&longs;&aelig; imagines legitima LMSR <lb/>ad &longs;puriam NMST. </s>
          <s id="s.000332">Cum temporis momenta M, P in&shy;<lb/>telligantur ex minimis temporibus, qu&aelig; proponi po&longs;&longs;unt, <lb/>inter&longs;e &aelig;qualibus, &amp; quibus &aelig;quabiliter perdurant ve&shy;<lb/>locitates, quas mobile &longs;ortitur in aduentu &longs;uo in punctis <lb/>C, G, erit vt velocitas FC ad velocitatem GI &longs;ic inter&longs;e <lb/><arrow.to.target n="marg75"/><lb/>&longs;patia, qu&aelig; i&longs;tis velocitatibus, temporibu&longs;que illis &aelig;qua&shy;<lb/>libus percurrerentur, in qua ratione e&longs;t etiam NM ad OP. <lb/></s>
          <s id="s.000333">Deinde momento M peragerentur &longs;patia proportionalia <lb/>velocitatibus FC, AC, &longs;eu rectis NM, ML, momento <lb/>autem P &longs;patia proportionalia velocitatibus GI, GD, <lb/>in qua ratione e&longs;t etiam OP ad PQ, &amp; &longs;ic deinceps <lb/>procedendo per &longs;ingula temporis MR momenta, adeo <lb/>vt, cum &longs;patium velocitate FC exactum ad id veloci&shy;<lb/>tate CA, &longs;it vt NM ad ML, &longs;patium velocitate IG ad id <lb/>exactum velocitate GD &longs;it vt OP ad PQ, &amp; &longs;int pr&aelig;terea <lb/>prim&aelig; inter&longs;e, hoc e&longs;t &longs;patia velocitatibus FC, GI tran&shy;<lb/>&longs;acta, proportionalia tertijs, &longs;patijs videlicet tran&longs;actis <lb/>velocitatibus ML, PQ ergo vt omnes prim&aelig; ad omnes <lb/>tertias quantitates, hoc e&longs;t omnia &longs;patia tran&longs;acta iuxta <lb/>gene&longs;im FCBK ad omnia &longs;patia iuxta gene&longs;im ACB, ita <lb/>erit &longs;umma &longs;ecundarum ad omnes quartas, &longs;cilicet i&longs;ta <lb/>erit imago NMST ad imaginem LMSR. <!-- KEEP S--></s>
          <s id="s.000334">Quod &amp; c. <!-- KEEP S--></s>
        </p>
        <pb pagenum="34" xlink:href="022/01/040.jpg"/>
        <p type="margin">
          <s id="s.000335"><margin.target id="marg74"/><emph type="italics"/>Tab.<emph.end type="italics"/> 3. <emph type="italics"/>Fig.<emph.end type="italics"/> 4.<!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000336"><margin.target id="marg75"/><emph type="italics"/>Pr.<emph.end type="italics"/> 3. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000337"><emph type="center"/>LIBER ALTER<emph.end type="center"/></s>
        </p>
        <p type="main">
          <s id="s.000338"><emph type="center"/>DE<emph.end type="center"/></s>
        </p>
        <p type="main">
          <s id="s.000339"><emph type="center"/>Motu Compo&longs;ito.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000340">MOtum appellamus compo&longs;itum, vbi dum fer&shy;<lb/>tur mobile, con&longs;ideratur habere plures i&ntail; <lb/>diuer&longs;as partes, vel <expan abbr="eti&atilde;">etiam</expan> in eandem partem <lb/>conatus, ex quibus oriatur tertia vis di&longs;tin&shy;<lb/>cta ab illis. </s>
          <s id="s.000341">Hunc librum, cum expleueri&shy;<lb/>mus, non pauca vn&agrave; cum priori, dicta erunt de motu, erit&shy;<lb/>que ea methodus, qua &longs;imul geometrica qu&aelig;dam, difficil&shy;<lb/>lima &longs;citu &longs;atis breuiter o&longs;tendemus. </s>
          <s id="s.000342">Nam vibrationes <lb/>pendulorum exigi temporibus; qu&aelig; &longs;int in &longs;ubduplicat&atail; <lb/>ratione longitudinum eorundem, plan&egrave; tandem con&longs;tabit <lb/>ali&agrave;s nobis di&longs;&longs;entientibus: aperiemus etiam, qua arte in&shy;<lb/>telligi queant anguli rectilinei curuilineis &aelig;quales; nec non <lb/>exponemus parabolas quibu&longs;dam &longs;piralibus &aelig;quales, vt <lb/>e&longs;t vulgata &longs;pirali Archimede&aelig;, c&ugrave;m videlicet ba&longs;is para&shy;<lb/>bol&aelig; radio circuli &longs;piralem continentis, &amp; dimidium huius <lb/>circumferenti&aelig; circuli altitudini eiu&longs;dem parabol&aelig;, &aelig;qua&shy;<lb/>les &longs;int. </s>
        </p>
        <p type="main">
          <s id="s.000343"><emph type="center"/>PROP. I. THEOR. I.<emph.end type="center"/><lb/><arrow.to.target n="marg76"/><!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000344"><margin.target id="marg76"/><emph type="italics"/>Tab.<emph.end type="italics"/> 4. <emph type="italics"/>Fig.<emph.end type="italics"/> 1.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000345">SI in eadem recta linea currantur &longs;patia temporibus <lb/>&aelig;qualibus, &amp; &longs;int motus &longs;implices, ac ad ea&longs;dem par&shy;<lb/>tes tendentes, eadem illa &longs;patia &longs;imul motu compo&longs;ito, ab <lb/>eodemque mobili duabus illis gene&longs;ibus affecto, vnicoque <lb/>ex dictis temporibus &aelig;qualibus, excurrentur. </s>
        </p>
        <pb pagenum="35" xlink:href="022/01/041.jpg"/>
        <p type="main">
          <s id="s.000346">Curratur LI iuxta imaginem velocitatum HAEF, et IO <lb/>iuxta aliam dict&aelig; homogeneam BAED. </s>
          <s id="s.000347">Dico LO &longs;um&shy;<lb/>mam dictorum &longs;patiorum LI, IO exactum iri vnico tem&shy;<lb/>pore AE, &longs;i nempe mobile feratur <expan abbr="&longs;ec&utilde;dum">&longs;ecundum</expan> vtranque ima&shy;<lb/>ginem. </s>
        </p>
        <p type="main">
          <s id="s.000348">Per quodlibet punctum, &longs;eu temporis momentum M <lb/>agatur recta GMC parallela HB, vel FD. <!-- KEEP S--></s>
          <s id="s.000349">Habebit mobi&shy;<lb/>le momento A, <expan abbr="d&utilde;">dum</expan> &longs;cilicet mouetur motu compo&longs;ito duas <lb/>&longs;imul velocitates AH, AB, ide&longs;t vnicam HB. </s>
          <s id="s.000350">Similiter mo&shy;<lb/>mento M habebit GC, &amp; momento E ip&longs;am FD. <!-- KEEP S--></s>
          <s id="s.000351">Itaque </s>
        </p>
        <p type="main">
          <s id="s.000352"><arrow.to.target n="marg77"/><lb/>erit HBDF imago velocitatum compo&longs;iti motus, qui fiet <lb/>tempore AE iuxta imaginem, qu&aelig; aggregatum e&longs;t <expan abbr="dictar&utilde;">dictarum</expan> <lb/>HAEF, ABDE. <!-- KEEP S--></s>
          <s id="s.000353">E&longs;t ver&ograve; LI ad IO vt imago HAEF ad <lb/>imaginem ABDE; ergo conuertendo, componendoqu&etail; <lb/>erit vt LI ad LO, &longs;ic imago HAEF ad imaginem HBDF; <lb/>propterea quemadmodum &longs;patium LI currebatur iuxt&atail; <lb/>imaginem HAEF, &longs;ic LO percurretur imagine HBDF &longs;olo, <lb/>eodemque tempore AE. <!-- KEEP S--></s>
          <s id="s.000354">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000355"><margin.target id="marg77"/><emph type="italics"/>Def.<emph.end type="italics"/> 3. <emph type="italics"/>prima.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000356"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000357"><emph type="italics"/>Hinc patet graue perpendiculariter, violenterque deiectum <lb/>minim&egrave; ad terram venturum aggregato virium, quarum vna <lb/>e&longs;t ab impellente impre&szlig;a, altera ver&ograve; &agrave; grauitate <expan abbr="depend&etilde;s">dependens</expan>. <lb/></s>
          <s id="s.000358">Nam ex impartita vt celerior fit ca&longs;us, quam vt graue in de&shy;<lb/>cur&longs;u &longs;uo po&longs;&longs;it ex acceleratione naturali eum gradum acqui&shy;<lb/>rere, quem cert&egrave; &longs;ponte &longs;ua tant&ugrave;m de&longs;cendens in fine eiu&longs;dem <lb/>altitudinis adeptum e&longs;&longs;et. </s>
          <s id="s.000359">Hoc ita verum e&longs;t, vt aliquando <lb/>minimum inter&longs;it, inter impetum ab ambabus cau&longs;is proue&shy;<lb/>nientem, &amp; eum, qui a &longs;ola oritur grauitate, quamobrem pa&shy;<lb/>rum is proficeret, qui conaretur maiorem impetum componere <lb/>in ca&longs;u grauis, illi nempe adiecta vi, mobile idem in decur&longs;u <lb/>impellente, vltra natiuam grauitatem, quod tamen fieri haud <lb/>dubi&egrave; po&longs;&longs;et, &longs;i ca&longs;us obliquus e&szlig;et.<emph.end type="italics"/></s>
        </p>
        <pb pagenum="36" xlink:href="022/01/042.jpg"/>
        <p type="main">
          <s id="s.000360"><emph type="italics"/>Illud quoque hac occa&longs;ione aperiendum e&longs;t, graue naturali&shy;<lb/>ter de&longs;cendens e&ograve; concitati&ugrave;s ferri, quoad potentia re&longs;i&longs;tentis <lb/>aeris &lpar;validior namque i&longs;ta fit, vbi mobilis ca&longs;us e&longs;t celerior&rpar; <lb/>vi grauitatis mobili inh&aelig;renti exaquatur, tunc enim cau&longs;&atail; <lb/>vlterioris accelerationis adempta e&longs;t, con&longs;umiturque in lucta&shy;<lb/>tione aeris contranitentis: quare tunc grane progrederetur <lb/>&aelig;quabili motu, id qu&ograve;d citi&ugrave;s euenire deberet &longs;i grane intr&atail; <lb/>aquam de&longs;cendat.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000361"><emph type="center"/>PROP. II. THEOR. II.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000362">SI in eadem recta duos motus &longs;ibi contrarios, &longs;implices, <lb/>ac eodem tempore peractos intelligamus, mobile di&shy;<lb/>ferentiam illorum &longs;patiorum, &longs;i vtroque motu e&longs;&longs;et affe&shy;<lb/>ctum, percurreret. <lb/><arrow.to.target n="marg78"/></s>
        </p>
        <p type="margin">
          <s id="s.000363"><margin.target id="marg78"/><emph type="italics"/>Tab.<emph.end type="italics"/> 4. <emph type="italics"/>Fig.<emph.end type="italics"/> 2.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000364">Curratur &agrave; puncto L &longs;patium LO imagine velocitatum <lb/>ABFG, &amp; codem tempore curratur etiam recta OM ex <lb/>puncto altero O, &longs;cilicet contrario motu, &amp; iuxta <expan abbr="imagin&etilde;">imaginem</expan> <lb/>AHIG pr&aelig;dict&etail; homogeneam. </s>
          <s id="s.000365">Dico mobile, <expan abbr="c&otilde;po&longs;ito">compo&longs;ito</expan> ex <lb/>vtri&longs;que motu, &amp; tempore ip&longs;o AG cur&longs;urum differentiam <lb/>LM dictorum &longs;patiorum LO, OM. </s>
        </p>
        <p type="main">
          <s id="s.000366">Prim&ugrave;m intra parallelas AB, GF non &longs;e &longs;ecent line&aelig; <arrow.to.target n="marg79"/><lb/>BF, HI, &amp; ducatur qu&aelig;libet DC &aelig;quidi&longs;tans AB, vel GF, <lb/>qu&aelig; fecet HI in E. <!-- KEEP S--></s>
          <s id="s.000368">Manife&longs;tum e&longs;t, mobile, compo&longs;ito <lb/>motu feratur habere duplicem velocitatem, vnam AB al&shy;<lb/>teram illi oppo&longs;itam AH, ob idque moueri ver&longs;us O &longs;ol&atail; <lb/>velocitate HB differentia dictarum inter&longs;e pugnantium <lb/>velocitatum: pariter momento D feretur mobile veloci&shy;<lb/>tate EC differentia duarum DE, DC, &amp; in&longs;tanti G habebit <lb/><arrow.to.target n="marg80"/><lb/>differentialem IF; ex quo &longs;equitur figuram BHEIFCB, dif&shy;<lb/>ferentiam imaginum ABFG, HAGI, aptatam tempori AC <lb/>imaginem e&longs;&longs;e velocitatum compo&longs;iti motus. </s>
          <s id="s.000369">Hoc po&shy;<lb/><arrow.to.target n="marg81"/><lb/>&longs;ito habebit LM ad LO eandem rationem, ac BHIF ad <lb/>ABFG; Propterea LM, qu&aelig; e&longs;t differentia &longs;patiorum LO, <pb pagenum="37" xlink:href="022/01/043.jpg"/>MO curretur iuxta imaginem BHIF, nempe compo&longs;ito <lb/>motu, &amp; tempore AG. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000370"><margin.target id="marg79"/><emph type="italics"/>Tab.<emph.end type="italics"/> 4. <emph type="italics"/>Fig.<emph.end type="italics"/> 2.<!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000371"><margin.target id="marg80"/><emph type="italics"/>Def.<emph.end type="italics"/> 3 <emph type="italics"/>prima.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000372"><margin.target id="marg81"/><emph type="italics"/>Pr.<emph.end type="italics"/> 2. <emph type="italics"/>prim&atail; <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000373">2. Se nunc &longs;ecent line&aelig; BF, HI in C. <!-- KEEP S--></s>
          <s id="s.000374">Ducatur CD pa&shy;<lb/><arrow.to.target n="marg82"/><lb/>rallela alteri &aelig;quidi&longs;tantium AB, GF. <!-- KEEP S--></s>
          <s id="s.000375">Con&longs;tat ex prima <lb/>parte, qu&ograve;d mobile compo&longs;ito motu, &amp; iuxta imaginem <lb/>HBC feretur ver&longs;us O tempore AD; &longs;it ergo &longs;patium, quod <lb/>curreretur illa imagine, PR, &amp; ob id LO ad PR eande&mtail; <lb/><arrow.to.target n="marg83"/><lb/>habebit rationem quam imago ABFG ad imagine&mtail; <lb/>HBC. </s>
        </p>
        <p type="margin">
          <s id="s.000376"><margin.target id="marg82"/><emph type="italics"/>Tab.<emph.end type="italics"/> 4. <emph type="italics"/>fig.<emph.end type="italics"/> 3.<!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000377"><margin.target id="marg83"/><emph type="italics"/>Pr.<emph.end type="italics"/> 2. <emph type="italics"/>prima<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000378">Similiter dum mobile mouetur tempore DG iuxta ima&shy;<lb/>gines DCIG, DCFG, feretur ver&egrave; &longs;ecund&ugrave;m imagine&mtail; <lb/><arrow.to.target n="marg84"/><lb/>FCI ver&longs;us L, quamobrem &longs;i &longs;patium, quod exigeretur <lb/>hac imagine &longs;it RQ, habebit i&longs;tud ad LO eandem rationem, <lb/><arrow.to.target n="marg85"/><lb/>quam imago CFI ad imaginem ABFG, &amp; ideo ex &aelig;quali <lb/>QR ad PR &longs;e habebit vt imago CFI ad imaginem HBC; &longs;i <lb/>igitur ponatur ABFG maior imagine AHIG, dempt&agrave; co&shy;<lb/>muniter AHCFG relinquetur HBC maior imagine CEI, &amp; <lb/>ideo etiam PR maior QR: curritur ver&ograve; PR vers&ugrave;s R tem&shy;<lb/>pore AD, &amp; RQ vers&ugrave;s P tempore DG, ergo toto tempo&shy;<lb/>re AG curretur PQ differentia &longs;patiorum PR, RQ Cum <lb/>ver&ograve; HBC ad CFI, &longs;it vt PR ad RQ, erit diuidendo vt ex&shy;<lb/>ce&longs;&longs;us imaginis HBC &longs;upra imaginem FCI ad imagine&mtail; <lb/>i&longs;tam, ita PQ ad QR, &amp; o&longs;ten&longs;um e&longs;t QR ad LO, &longs;icut ima&shy;<lb/>go FCI ad imaginem ABFG, ergo ex &aelig;quali exce&longs;&longs;us ima&shy;<lb/>ginis HBC &longs;upra imaginem AHIG habebit eandem ratio&shy;<lb/>nem ad imaginem AHIG, ac PQ ad LO, at e&longs;t in illa <expan abbr="ead&etilde;">eadem</expan> <lb/>ratione etiam LM ad LO &lpar;e&longs;t enim LO ad MO vt imago <lb/>ABFG ad imaginem AHIG&rpar; ergo PQ erit &aelig;qualis LM, <lb/>atque adeo mobile dum currit vtroque motu, hoc e&longs;t iux&shy;<lb/>ta &longs;imul duas imagines propo&longs;itas contrariorum motuum, <lb/>peraget &longs;patium LM vers&ugrave;s O &longs;ecund&ugrave;m imaginem, qu&aelig; <lb/>differentia e&longs;t propo&longs;itarum ABFG, AHIG, tempore AG. <lb/><!-- KEEP S--></s>
          <s id="s.000379">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <pb pagenum="38" xlink:href="022/01/044.jpg"/>
        <p type="margin">
          <s id="s.000380"><margin.target id="marg84"/><emph type="italics"/>Ex prim&atail; <lb/>parte.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000381"><margin.target id="marg85"/><emph type="italics"/>Pr.<emph.end type="italics"/> 2. <emph type="italics"/>prima.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000382"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000383"><emph type="italics"/>Deduc&igrave;tur, mobile nullum &longs;patium emen&longs;urum, vbi ima&shy;<lb/>gines &longs;implicium motuum fuerint aquales.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000384"><emph type="center"/>PROP. III. THEOR. III.<emph.end type="center"/><lb/><arrow.to.target n="marg86"/><!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000385"><margin.target id="marg86"/><emph type="italics"/>Tab.<emph.end type="italics"/> 4. <emph type="italics"/>Fig.<emph.end type="italics"/> 4.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000386">REperire eam velocitatem, eamque directionem, qu&aelig; <lb/>orirentur, &longs;i mobile pluribus eodem momento velo&shy;<lb/>citatibus, &longs;eu conatibus affectum e&longs;&longs;et. </s>
          <s id="s.000387">Opportet autem <lb/>non &longs;olum has velocitates, ver&ugrave;m etiam earum directio&shy;<lb/>nes manife&longs;tas e&longs;&longs;e. </s>
        </p>
        <p type="main">
          <s id="s.000388">Habeat mobile A, eodem momento conatum AB, quo <lb/>tendat in R; AC; quo in C; &amp; AD, quo in D. <!-- KEEP S--></s>
          <s id="s.000389">Qu&aelig;ritur ve&shy;<lb/>locitas, &amp; directio, quas mobile habiturum e&longs;&longs;et in multi&shy;<lb/>plici illa affectione &lpar;Nam actu vnam velocitatem, vnam&shy;<lb/>que tant&ugrave;m directionem &longs;ortiri debet&rpar; Ex duabus qui&shy;<lb/>bu&longs;que AD, AC intelligatur perfici parallelogrammum <lb/>ACED, &amp; ducta diametro AE fiat itidem aliud parallelo&shy;<lb/>grammum ABFE, cuius agatur diameter AF. </s>
          <s id="s.000390">Dico AF <lb/>e&longs;&longs;e qu&aelig;&longs;itam velocitatem, ac directionem, quibus mobile <lb/>ex illis pluribus conatibus motum &longs;uum in&longs;titueret. </s>
        </p>
        <p type="main">
          <s id="s.000391">Si mobili A currendum e&longs;&longs;et &aelig;quabili motu &longs;patium <lb/>AE, pertran&longs;iret eodem tempore tam rectam AD, qu&agrave;m </s>
        </p>
        <p type="main">
          <s id="s.000392"><arrow.to.target n="marg87"/><lb/>ip&longs;am AC; nam cum fertur ab A in E ver&egrave; de&longs;cendit ab A <lb/>in C, &amp; ab A in D motu pariter &aelig;quabili; ergo AD ad <lb/>AC, erit vt velocitas, qua curritur per AD ad velocitatem, <lb/>qua curritur per AC. <!-- KEEP S--></s>
          <s id="s.000393">Itaque &longs;i mobile dum e&longs;t in A in&shy;<lb/>telligatur affectum velocitatibus AD, AC habentibus di&shy;<lb/>rectiones ip&longs;as rectas AD, AC, perinde e&longs;&longs;et, ac &longs;i &longs;ola fo&shy;<lb/>ret mobili velocitas vn&acirc; cum directione AE. <!-- KEEP S--></s>
          <s id="s.000394">Eadem ra&shy;<lb/>tione AF velocitas habens directionem AF, &aelig;quipollebit <lb/>duabus velocitatibus AB, AE iuxta directiones rectas ea&longs;-<pb pagenum="39" xlink:href="022/01/045.jpg"/>dem ABAE; hoc &aelig;quiualebit tribus AB, AC, AD. <!-- KEEP S--></s>
          <s id="s.000395">Mo&shy;<lb/>bile igitur ex affectione trium illorum conatuum, vt &longs;up&shy;<lb/>po&longs;itum fuit, nitetur &longs;ecund&ugrave;m AF velocitate ip&longs;a AF <lb/>Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000396"><margin.target id="marg87"/><emph type="italics"/>Gal. </s>
          <s id="s.000397">pr. <!-- REMOVE S-->de mo&shy;<lb/>tu aquab.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000398"><emph type="center"/>DEF. I.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000399">ACcelerationem alicuius motus, tunc intelligimus, <expan abbr="c&utilde;">cum</expan> <lb/>velocitates, qu&aelig; &longs;ubinde mobili adueniunt, non de&shy;<lb/>lentur, &longs;ed pror&longs;us integr&aelig;, atque indelebiles mobili in ip&longs;o <lb/>motu per&longs;euerant. </s>
          <s id="s.000400">Ex quo &longs;equitur motum &longs;implicem di&shy;<lb/>ci, cum pr&aelig;terit&aelig; velocitates protinus euane&longs;cunt, ill&aelig;&shy;<lb/>que tantum con&longs;iderantur, qu&aelig; mobili &longs;ubinde oriun&shy;<lb/>tur. </s>
        </p>
        <p type="main">
          <s id="s.000401"><emph type="center"/>PROP. IV. PROB. II.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000402">IMaginem accelerationis cuiu&longs;cunque &longs;implicis motus <lb/>exhibere. </s>
        </p>
        <p type="main">
          <s id="s.000403">Imago velocitatum &longs;implicis motus e&longs;to rectangulum <lb/><arrow.to.target n="marg88"/><lb/>AFDC: &longs;ic motus e&longs;t &aelig;quabilis, vt acceleretur debent in&shy;<lb/><arrow.to.target n="marg89"/><lb/>&longs;tanti C vigere omnes velocitates in imagine AFDC <expan abbr="c&otilde;-prehen&longs;&aelig;">con&shy;<lb/>prehen&longs;&aelig;</expan>, &amp; item ducta quacunque BE parallela AF, vel <lb/><arrow.to.target n="marg90"/><lb/>CD, erit mobile momento B affectum omnibus antece&shy;<lb/>dentibus velocitatibus, comprehen&longs;is nempe ab imaginis <lb/>portione AFEB; quare &longs;i ponamus HLG imaginem e&longs;&longs;&etail; <lb/>accelerationis, itaut nempe tempus GL &aelig;quale &longs;it tempo&shy;<lb/>ri AC; item KL &aelig;quale tempori AB, erit vt figura CAFD <lb/>ad figuram BAFE, &longs;ic velocitas, qua mobile fertur <expan abbr="mom&etilde;-to">momen&shy;<lb/>to</expan> G ad velocitatem, quam habet in&longs;tanti K; &amp; ideo quia <lb/>ponitur imago &longs;implicis motus rectangulum AFDC, erit <lb/>rectangulum CF ad BF, hoc e&longs;t recta CA ad AB imm&ograve; <lb/>LG ad LK, vt GH ad KI; quamobrem GLH imago velo&shy;<lb/><arrow.to.target n="marg91"/><lb/>citatum huiu&longs;modi motus, erit triangulum. </s>
          <s id="s.000404">Quod &longs;i ima-<pb pagenum="40" xlink:href="022/01/046.jpg"/>go &longs;implicis motus fui&longs;&longs;et triangulum, imago velocitatum <lb/>accelerationis foret trilineum &longs;ecundum, &amp; ita pro&shy;<lb/>portionaliter de infinitis numero accelerationibus. </s>
        </p>
        <p type="margin">
          <s id="s.000405"><margin.target id="marg88"/><emph type="italics"/>Tab.<emph.end type="italics"/> 4. <emph type="italics"/>fig.<emph.end type="italics"/> <gap/>.</s>
        </p>
        <p type="margin">
          <s id="s.000406"><margin.target id="marg89"/><emph type="italics"/>Cor. <!-- KEEP S--></s>
          <s id="s.000407">def.<emph.end type="italics"/> 3. <emph type="italics"/>pri&shy;<lb/>mi.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000408"><margin.target id="marg90"/><emph type="italics"/>Def.<emph.end type="italics"/> 1. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000409"><margin.target id="marg91"/><emph type="italics"/>Def.<emph.end type="italics"/> 3 <emph type="italics"/>primi.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000410"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000411"><emph type="italics"/>Hinc obiter habemus, quo pacto imago velocitatum corpo&shy;<lb/>rum naturaliter de&longs;cendentium triangulum &longs;it. </s>
          <s id="s.000412">Nam quo&shy;<lb/>libet momento &longs;ui ca&longs;us habet graue idem in&longs;e principiu&mtail; <lb/>motus, &longs;eu grauitas, ex qua concipitur imago &longs;implicis motus <lb/>&longs;i nempe priores gradus velocitatis &longs;ubinde deperirent, at <lb/>quia in eius de&longs;cen&longs;u pror&longs;us per&longs;euerant &lpar;id enim &longs;upponi&shy;<lb/>tur ab&longs;trahendo ab aere&rpar; inde motus concitatur, &amp; fit vti di&shy;<lb/>ximus imago accelerationis triangulum.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000413"><emph type="center"/>AXIOMA<emph.end type="center"/></s>
        </p>
        <p type="main">
          <s id="s.000414">QV&aelig;libet linea, vt fluxus puncti concipi po&shy;<lb/>te&longs;t. </s>
        </p>
        <p type="main">
          <s id="s.000415"><emph type="center"/>AX. II.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000416">VT propo&longs;ita linea ex fluxu puncti exar&egrave;tur, du&ograve; tan&shy;<lb/>t&ugrave;m nece&longs;&longs;aria &longs;unt, &longs;cilicet motus, &amp; puncti di&shy;<lb/>rectio. </s>
        </p>
        <p type="main">
          <s id="s.000417"><emph type="center"/>PROP. V. THEOR. III.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000418">REcta, qu&aelig; pri&ugrave;s de&longs;cripta e&longs;t, pote&longs;t alijs &agrave; primis <lb/>velocitatibus, rur&longs;us exarari. </s>
        </p>
        <p type="main">
          <s id="s.000419">Nam punctum pote&longs;t fluere &longs;ecundum quamcunque <lb/>rectam, quocunque motu, ergo illam pote&longs;t etiam quibu&longs;&shy;<lb/>cunque velocitatibus affectum rur&longs;us exarare. </s>
        </p>
        <pb pagenum="41" xlink:href="022/01/047.jpg"/>
        <p type="main">
          <s id="s.000420"><emph type="center"/>PROP. VI. THEOR. IV.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000421">VT eadem recta ex fluxu puncti renouetur, opportet in <lb/>quocunque illius puncto &longs;eruari pri&longs;tinas directio<lb/>nes, </s>
        </p>
        <p type="main">
          <s id="s.000422">Cum, vti diximus, ad de&longs;criptionem line&aelig; duo tant&ugrave;m <lb/><arrow.to.target n="marg92"/><lb/>exigantur, nempe motus, &amp; puncti directio; motus ver&ograve; po&shy;<lb/>te&longs;t e&longs;&longs;e quilibet, &longs;equitur ergo directionem, alteram de <lb/>duobus, &longs;eruari debere. </s>
        </p>
        <p type="margin">
          <s id="s.000423"><margin.target id="marg92"/><emph type="italics"/>Ax.<emph.end type="italics"/> 2. <emph type="italics"/>buius. <lb/></s>
          <s id="s.000424">pr.<emph.end type="italics"/> 5. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000425"><emph type="center"/>DEF. II.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000426">LIneam dicimus curuam, in qua &longs;umptis duobus ad&shy;<lb/>libitum punctis, recta, qu&aelig; ip&longs;a puncta coniunge&shy;<lb/>ret, nullam cum propo&longs;ita linea partem &longs;it habitura com&shy;<lb/>munem. </s>
        </p>
        <p type="main">
          <s id="s.000427"><emph type="center"/>PROP. VII. THEOR. V.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000428">DIrectiones puncti de&longs;cribentis lineam, iuxta rectas <lb/>lineas concipi debent. </s>
        </p>
        <p type="main">
          <s id="s.000429">Dum punctum fluere intelligimus, ine&longs;t in eo &longs;ingulis <lb/>momentis certus, ac pr&aelig;fixus gradus velocitatis, quo tan&shy;<lb/>t&ugrave;m attento, rect&agrave;, <expan abbr="&aelig;quabiliq;">&aelig;quabilique</expan> motu in certam partem con&shy;<lb/>tenderet; at huiu&longs;modi iter, aliud non e&longs;t, qu&agrave;m directio <lb/>puncti, qua eius temporis momento profici&longs;citur; ergo iux&shy;<lb/>ta rectas lineas, directiones omnes con&longs;iderari opportet. </s>
        </p>
        <p type="main">
          <s id="s.000430"><emph type="center"/>PROP. VIII. THEOR. VI.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000431">TAngens, &amp; directio motus in quouis curu&aelig; puncto <lb/>e&longs;t vna, <expan abbr="atq;">atque</expan> eadem recta. </s>
        </p>
        <p type="main">
          <s id="s.000432">Nam in de&longs;criptione <expan abbr="cuiu&longs;cunq;">cuiu&longs;cunque</expan> rect&aelig; procedit pun&shy;<lb/><arrow.to.target n="marg93"/><pb pagenum="42" xlink:href="022/01/048.jpg"/>ctum iuxta tendentias rectas, obliquatur tamen ob &longs;ub&longs;e&shy;<lb/>quentes, ali&ograve; tendentes ni&longs;us, &amp; ob id di&longs;trahitur punctum <lb/>ip&longs;um &agrave; priori tendentia, idem accidit ex alia parte &longs;i re&shy;<lb/>flaxi&longs;&longs;et idem punctum, nempe hinc inde vnicam rectam <lb/>eandemque, continuantibus oppo&longs;itis ad idem punctum <lb/>directionibus, ergo directio, &amp; tangens vna, &amp; eadem e&longs;t <lb/>recta. </s>
        </p>
        <p type="margin">
          <s id="s.000433"><margin.target id="marg93"/><emph type="italics"/>Pr.<emph.end type="italics"/> 7. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000434"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000435"><emph type="italics"/>Hinc &longs;equitur, vnicam lineam dicendam e&longs;&longs;e, cum &agrave; quo&shy;<lb/>cunque illius puncto vnica tant&ugrave;m ex vtraque parte egre&shy;<lb/>ditur tangens.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000436"><emph type="center"/>DEF. III.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000437">QV&ograve;d &longs;i ex aliquo puncto du&aelig; tangentes hinc inde <lb/>egredientes angulum efficiant; tunc propo&longs;itam li&shy;<lb/>neam inflexam dicemus, &amp; punctum, in quo &longs;unt <lb/>contactus, inflexionis appellabitur. </s>
        </p>
        <p type="main">
          <s id="s.000438"><emph type="center"/><emph type="italics"/>Corollarium I.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000439"><emph type="italics"/>Ab hi&longs;ce deffinitionibus, &amp; priori coroll. </s>
          <s id="s.000440">manat artificium <lb/>componendi duas curuas, vel curuam &amp; rectam, adeout vni&shy;<lb/>cam lineam efforment, nullumque angulum; nempe cum &longs;ic <lb/>inuicem iungamus, vt tangentes ad punctum connexus, vnam <lb/>tant&ugrave;m rectam efficiant.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000441"><emph type="center"/><emph type="italics"/>Corollarium II.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000442"><emph type="italics"/>Sed &amp; illud patet, quibus angulis inflectantur line&aelig; inui&shy;<lb/>cem compo&longs;it&aelig;, &longs;i ad punctum inflexionis angulum tangen&shy;<lb/>tium ob&longs;eruauerimus, &longs;unt enim inter&longs;e &aelig;quales, lic&egrave;t diuer&shy;<lb/>&longs;a &longs;peciei, cum vnus &longs;it curuilineus, &amp; rectilineus alter.<emph.end type="italics"/></s>
        </p>
        <pb pagenum="43" xlink:href="022/01/049.jpg"/>
        <p type="main">
          <s id="s.000443"><emph type="center"/>PROP. IX. THEOR. VII.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000444">TAngens, &longs;eu directio motus in quocunque curu&aelig; <lb/>puncto e&longs;t illa recta, qu&aelig; vtrinque &longs;tatim cadens <lb/>extra curu&aelig; conuexum ad eandem, qu&agrave;m fieri pote&longs;t ex <lb/>vtraque parte accedit. </s>
        </p>
        <p type="main">
          <s id="s.000445">Nam alia qu&aelig;que recta tran&longs;iens per punctum conta&shy;<lb/>ctus ad &longs;ectionem magis accedere nequit, quin ip&longs;am illinc <lb/>&longs;ecet, ob id extra conuexum eius non cadet, ab altera ve&shy;<lb/>r&ograve; parte magis &agrave; propo&longs;ita curua &longs;eparabitur, quamobrem <lb/>nulla alia recta, qu&agrave;m tangens poterit &longs;imul extra curuam <lb/>e&longs;&longs;e, &amp; qu&agrave;m fieri pote&longs;t ad ip&longs;am accedere. </s>
        </p>
        <p type="main">
          <s id="s.000446"><emph type="center"/>DEF. IV.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000447">LIne&aelig; AC, AD occurrant &longs;ibi in A, quod punctum in&shy;<lb/><arrow.to.target n="marg94"/><lb/>telligatur transferri ab A in C vn&agrave; cum linea AD <lb/>&longs;emper &longs;ibi parallela, quo tempore punctum A currat ip&shy;<lb/>&longs;am latam lineam ex A in D. <!-- KEEP S--></s>
          <s id="s.000448">Manife&longs;tum e&longs;t idip&longs;um <lb/>punctum A de&longs;cripturum e&longs;&longs;e motu compo&longs;ito lineam <lb/>quandam AB diagonalem &longs;uperficiei parallelogramm&aelig; <lb/>ABCD. <!-- KEEP S--></s>
          <s id="s.000449">Vocamus ergo diagonalem illam &longs;emitam com&shy;<lb/>po&longs;iti motus, &amp; AC, AD latera illius. </s>
        </p>
        <p type="margin">
          <s id="s.000450"><margin.target id="marg94"/><emph type="italics"/>Tab.<emph.end type="italics"/> 4. <emph type="italics"/>fig.<emph.end type="italics"/> 6.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000451"><emph type="center"/><emph type="italics"/>Corollarium I.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000452"><emph type="italics"/>Manife&longs;tum e&longs;t mobile dum currit AB tran&longs;ire etiam AC, <lb/>AD, lic&egrave;t curu&aelig; &longs;int, nam ver&egrave; transfertur illo tempore, tam <lb/>ad lineam CB quam ad DB.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000453"><emph type="center"/><emph type="italics"/>Corollarium II.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000454"><emph type="italics"/>Pr&aelig;terea &longs;i ducerentur, aut&longs;int AC, CB, DA, DB, AB<emph.end type="italics"/><pb pagenum="44" xlink:href="022/01/050.jpg"/><emph type="italics"/>rect&aelig; line&aelig;, efficeretur ex ijs parallelogrammum ACBD, cu&shy;<lb/>ius diameter AB; quamobrem ex datis punctis C, A, D repe&shy;<lb/>riretur &longs;tatim punctum B, &longs;cilicet extremum &longs;emit&aelig; compo&shy;<lb/>&longs;iti motus, cuius latera ip&longs;&aelig; curu&aelig;, aut rect&aelig; AC, AD<emph.end type="italics"/> &mdash;. </s>
        </p>
        <p type="main">
          <s id="s.000455"><emph type="center"/>PROP. X. PROB. III.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000456">EX datis <expan abbr="quotcunq;">quotcunque</expan> lateribus compo&longs;iti motus, huius <lb/><arrow.to.target n="marg95"/><lb/>&longs;emit&aelig; terminum exhibere. </s>
        </p>
        <p type="margin">
          <s id="s.000457"><margin.target id="marg95"/><emph type="italics"/>Tab.<emph.end type="italics"/> 4. <emph type="italics"/>Fig.<emph.end type="italics"/> 7.</s>
        </p>
        <p type="main">
          <s id="s.000458">Si latera compo&longs;iti motus e&longs;&longs;ent duo tant&ugrave;m AB, AC. <lb/><!-- KEEP S--></s>
          <s id="s.000459">Facto parallelogrammo vt dictum e&longs;t, inueniretur pun&shy;<lb/>ctum E extremum motus: &amp; <expan abbr="qu&aelig;cunq;">qu&aelig;cunque</expan> &longs;it &longs;emita, &longs;eu mo&shy;<lb/>tus, pote&longs;t idem E &longs;upponi tanquam extremum alterius la&shy;<lb/>teris, adeoque, &longs;i motus con&longs;tet ex tribus lateribus AC, <lb/>AB, AD, perinde &longs;it ac &longs;i foret duorum laterum AE, AD; <lb/>nam AC, AD valent &longs;imul ac &longs;olum AE; cum ita &longs;it, facto <lb/>etiam parallelogrammo EADF ex datis punctis E, A, D, <lb/>habebitur F extremum &longs;emit&aelig;, cuius &longs;unt tria latera CA, <lb/>AD, AB &mdash; </s>
        </p>
        <p type="main">
          <s id="s.000460"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000461"><emph type="italics"/>Deducitur artificium de&longs;cribend&aelig; &longs;emit&aelig; AE, vel AF, &longs;i <lb/>nempe a&longs;&longs;umptis partibus AG, AH, AI in dictis lateribus, <lb/>qu&aelig; quidem &longs;ciantur percurri temporibus &aelig;qualibus, &longs;i per <lb/>ip&longs;as &longs;ingulas mobile punctum ferretur eo modo, quo in com&shy;<lb/>po&longs;ito motu nititur per ea&longs;dem directiones; reperietur in&shy;<lb/>quam punctum K in &longs;emita AE, atque L in &longs;emita AF: qua&shy;<lb/>re hoc modo &longs;umptis alijs, atque alijs partibus in ip&longs;is lateri&shy;<lb/>bus, reperientur alia, atque alia puncta ad ip&longs;am &longs;emita&mtail; <lb/>pertinentia, quorum tandem beneficio, facile erit qua&longs;itam <lb/>ferm&egrave; &longs;emitam exarare.<emph.end type="italics"/></s>
        </p>
        <pb pagenum="45" xlink:href="022/01/051.jpg"/>
        <p type="main">
          <s id="s.000462"><emph type="center"/>PROP. XI. PROB. IV.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000463">EX datis imaginibus velocitatum, iuxta quas &longs;implici <lb/><arrow.to.target n="marg96"/><lb/>motu currantur latera compo&longs;iti motus; datis item <lb/>tangentibus ad qu&aelig;cunque puncta ip&longs;orum laterum, repe&shy;<lb/>rire &longs;emitam compo&longs;iti motus, nec non directiones, <expan abbr="veloci-tate&longs;q;">veloci&shy;<lb/>tate&longs;que</expan> puncti de&longs;cribentis ip&longs;am &longs;emitam. </s>
        </p>
        <p type="margin">
          <s id="s.000464"><margin.target id="marg96"/><emph type="italics"/>Tab.<emph.end type="italics"/> 4. <emph type="italics"/>fig.<emph.end type="italics"/> 8.</s>
        </p>
        <p type="main">
          <s id="s.000465">Opportet tamen latera ip&longs;a, <expan abbr="itemq;">itemque</expan> imagines pr&aelig;dictas, <lb/>in imperatas &longs;ecari po&longs;&longs;e rationes, quamquam nos non la&shy;<lb/>teat, in lateribus curuis hoc effici non po&longs;&longs;e, pr&aelig;terqua&mtail; <lb/>aliquatenus in periph&aelig;rijs circulorum. </s>
        </p>
        <p type="main">
          <s id="s.000466">Sint AB, AF latera compo&longs;iti motus, qu&aelig; quidem &longs;eor&shy;<lb/>&longs;im currantur eodem tempore QM, &longs;cilicet AB iuxta ima&shy;<lb/>ginem MNPQ, et AF iuxta imaginem alteram ei homoge&shy;<lb/>neam TMQR. </s>
          <s id="s.000467">Ponatur AB circuli arcus, quem tangat re&shy;<lb/>cta BC &aelig;qualis QB, at AF lineam&apos;, qu&aelig; parabola &longs;it, con&shy;<lb/>tingat recta FG &aelig;qualis RQ Reperiemus illic&ograve; punctum <lb/><arrow.to.target n="marg97"/><lb/>H extremum &longs;emit&aelig; compo&longs;iti motus; &longs;unt enim data pun&shy;<lb/>cta A, F, B. <!-- KEEP S--></s>
          <s id="s.000468">Cum igitur mobile venerit in H. Dico, eo <lb/>temporis momento velocitatem, ac directionem HL, qu&aelig; <lb/>recta diameter e&longs;t parallelogrammi, cuius duo latera &longs;unt <lb/>dict&aelig; line&aelig; HI, HK; Iam vti diximus punctum H e&longs;t ex&shy;<lb/>tremum compo&longs;iti motus, quare eo momento, quo pun&shy;<lb/>ctum mobile e&longs;t in H, habet inibi ea&longs;dem illas velocitates, <lb/>quas haberet in B, et F, dum &longs;eor&longs;im illa latera excurri&longs;&longs;et; <lb/>&longs;cilicet con&longs;ideratur ip&longs;um mobile habens &longs;imul velocita&shy;<lb/>tem HI &aelig;qualem, ac &aelig;quedirectam, &longs;eu &aelig;quidi&longs;tantem <lb/>ip&longs;i CB, cui e&longs;t &aelig;qualis alia QP; &amp; velocitatem HK &aelig;qua&shy;<lb/>lem, &longs;imiliterque directam, ip&longs;i GF &aelig;quali RQ Cum ita <lb/><arrow.to.target n="marg98"/><lb/>&longs;it erit HL velocitas, &amp; directio qu&aelig;&longs;ita momento <expan abbr="q.">que</expan> Eo&shy;<lb/>dem modo, &longs;i &longs;it, vel fiat vt imago PNMQ ad ONMV <lb/>&lpar;ducta &longs;cilicet applicata SVO&rpar; ita BA ad AX, et ONMV <lb/>ad imaginem VMTS, vt XA ad AI, percurrentur AX, AI <lb/><arrow.to.target n="marg99"/><pb pagenum="46" xlink:href="022/01/052.jpg"/>eodem tempore MV, eritque ob id in X velocitas, &amp; dire&shy;<lb/>ctio, tangens ip&longs;a ZX &aelig;qualis VO, &amp; in I velocitas, &amp; di&shy;<lb/>rectio, tangens 2 I &aelig;qualis VS; Itaque datis punctis X, I, A <lb/><arrow.to.target n="marg100"/><lb/>dabitur etiam Y extremum &longs;emit&aelig; compo&longs;iti motus, cuius <lb/>latera AX, AI, &amp; ideo mobile dum e&longs;t in Y momento V <lb/>affectum erit duplici velocitate, hoc e&longs;t Y 4 &aelig;quali ve&shy;<lb/>locitati ZX, &longs;eu VO, ac &aelig;quidi&longs;tante eidem ZX, et veloci&shy;<lb/>tate altera Y 3 &aelig;quali, &amp; &aelig;qu&egrave;directa ip&longs;i 2 I: quare ex <lb/>datis punctis 4, Y, 3 inuenietur punctum S quartus angu&shy;<lb/>lus parallelogrammi habentis diametrum YI, qu&aelig; quidem <lb/><arrow.to.target n="marg101"/><lb/>erit directio, &amp; velocitas mobilis currentis compo&longs;ito mo&shy;<lb/>tu in&longs;tanti V. <!-- KEEP S--></s>
          <s id="s.000469">Cumque alia quotcunque puncta eadem <lb/>methodo reperire queamus, per qu&aelig; duci po&longs;&longs;it linea fer&egrave; <lb/>qu&aelig;&longs;itam &longs;emitam repr&aelig;&longs;entans, <expan abbr="atq;">atque</expan> emulans, patet idcir&shy;<lb/>co, quod propo&longs;uimus. </s>
        </p>
        <p type="margin">
          <s id="s.000470"><margin.target id="marg97"/><emph type="italics"/>Pr,<emph.end type="italics"/> 10. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000471"><margin.target id="marg98"/><emph type="italics"/>Ex pr.<emph.end type="italics"/> 3. <emph type="italics"/>hu.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000472"><margin.target id="marg99"/><emph type="italics"/>Pr.<emph.end type="italics"/> 2. <emph type="italics"/>primi <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000473"><margin.target id="marg100"/><emph type="italics"/>Cor.<emph.end type="italics"/> 2. <emph type="italics"/>def.<emph.end type="italics"/> 3. <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000474"><margin.target id="marg101"/><emph type="italics"/>Pr.<emph.end type="italics"/> 3. <emph type="italics"/>huius<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000475"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000476"><emph type="italics"/>Cum ver&ograve; directiones &longs;int idem, ac tangentes, liquet HL<emph.end type="italics"/><lb/><arrow.to.target n="marg102"/><lb/><emph type="italics"/>VS tangentes e&longs;&longs;e compo&longs;iti motus.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000477"><margin.target id="marg102"/><emph type="italics"/>Pr.<emph.end type="italics"/> 8 <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000478"><emph type="center"/>PROP. XII. THEOR. VIII.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000479">CVm imagines velocitatum, iuxta quas curruntur du&etail; <lb/><arrow.to.target n="marg103"/><lb/>rect&aelig;, qu&aelig; &longs;int latera compo&longs;iti motus, &longs;unt paral-<lb/>lelogrammum, &amp; triangulum; tunc &longs;emita compo&longs;iti motus <lb/>erit communis parabola. </s>
        </p>
        <p type="margin">
          <s id="s.000481"><margin.target id="marg103"/><emph type="italics"/>Tab.<emph.end type="italics"/> 5. <emph type="italics"/>Fig.<emph.end type="italics"/> 1.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000482">Tempore HM curratur latus AC iuxta imaginem velo&shy;<lb/>citatum HILM rectangulum, &amp; latus AB iuxta imaginem <lb/><arrow.to.target n="marg104"/><lb/>triangulum HMN; erit CA ad AB, vt imago <expan abbr="parallelogr&atilde;-mum">parallelogram&shy;<lb/>mum</expan> HILM ad aliam imaginem triangulum NHM. </s>
          <s id="s.000483">Fiat <lb/><arrow.to.target n="marg105"/><lb/><expan abbr="parellogr&atilde;mum">parellogrammum</expan> ACDB erit in D extremum &longs;emit&aelig; com&shy;<lb/>po&longs;iti motus, qu&aelig; &longs;i ponatur AFC; Dico e&longs;&longs;e parabolam. <lb/></s>
          <s id="s.000484">Sumatur in ip&longs;a linea quoduis punctum F, ab ip&longs;o dedu-<pb pagenum="47" xlink:href="022/01/053.jpg"/>cta FE parallela AB, vti etiam FG parallela AC, erunt <lb/><arrow.to.target n="marg106"/><lb/>AE, AG latera compo&longs;iti motus, cuius &longs;emita AF: Con&shy;<lb/>cipiatur mod&ograve; P momentum, quo mobile ade&longs;t in F, &amp; <lb/>ducta OPK parallela alteri HI, vel NL, erit imago MHIL ad <lb/><arrow.to.target n="marg107"/><lb/><expan abbr="imagin&etilde;">imaginem</expan> PHIK, hoc e&longs;t MH ad HP, vt CA ad AE, &longs;eu vt BD <lb/>ad GF. <!-- KEEP S--></s>
          <s id="s.000485">Pariter erit imago NHM ad <expan abbr="imagin&etilde;">imaginem</expan> OHP, hoc e&longs;t <lb/>quadratum ex MH ad <expan abbr="quadrat&utilde;">quadratum</expan> ex PH; imm&ograve; id ex BO ad <lb/>illud ex GF, vt BA ad AG; quamobrem punctum F cadet <lb/>in curuam parabolicam communem, cuius diameter AB, <lb/>&amp; ba&longs;is, &longs;eu ordinatim applicata BD, &longs;cilicet AFD erit ip&longs;a <lb/>curua parabolica. </s>
          <s id="s.000486">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000487"><margin.target id="marg104"/><emph type="italics"/>pr.<emph.end type="italics"/> 2. <emph type="italics"/>primum <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000488"><margin.target id="marg105"/><emph type="italics"/>Pr.<emph.end type="italics"/> 3. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000489"><margin.target id="marg106"/><emph type="italics"/>Ex eadem.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000490"><margin.target id="marg107"/><emph type="italics"/>Pr.<emph.end type="italics"/> 2. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000491"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000492"><emph type="italics"/>Quoniam graue, quod iaculatur extr&aelig; perpendiculum, li&shy;<lb/>berum ab omni obice, ni&longs;i turbaretur eius motus &agrave; propri&atail; <lb/>grauitate pergerct moueri &aelig;quabiliter iuxta directionem, ve&shy;<lb/>locitatemque ei traditam; habet ver&ograve; coniunctam grauita&shy;<lb/>tem, qua, ni&longs;i ab impre&longs;&longs;o impetu flecteretur motus, de&longs;cen&shy;<lb/>deret iuxta perpendiculum motu naturaliter concitato, cuius <lb/>imago velocitatum, triangulum e&longs;t; Hinc propterea gran&etail; <lb/>vltra perpendiculum proiectum de&longs;cribit in cur&longs;u &longs;uo, motu <lb/>&longs;cilicet compo&longs;ite, parabolam vulgatam. </s>
          <s id="s.000493">Ver&ugrave;m enim ver&ograve; <lb/>de&longs;criptionem i&longs;t am nece&longs;&longs;e aliquo pacto e&longs;t ex duabus cau&longs;is <lb/>vitiari, hoc est ab aeris re&longs;i&longs;tentia, &amp; perpendiculis non in&shy;<lb/>ter&longs;e parallelis, quippe in idem, <expan abbr="vnumq;">vnumque</expan> punctum, vniuer&longs;i <lb/>centrum, conuergentibus.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000494"><emph type="center"/>PROP. XIII. THEOR. IX.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000495">SI ab a&longs;&longs;umpto hyperbol&aelig; puncto, recta axi primo pa&shy;<lb/><arrow.to.target n="marg108"/><lb/>rallela deducatur, qu&aelig; ad &longs;ecundam diametrum per&shy;<lb/>tingat; Quadrilineum comprehen&longs;um ab ip&longs;a curua hy&shy;<lb/>perbolica. </s>
          <s id="s.000496">&amp; dictis tribus rectis, erit imago velocitatis il-<pb pagenum="48" xlink:href="022/01/054.jpg"/>lius motus de&longs;cribentis curuam parabolicam, cuius ba&longs;is <lb/>ad axem eius habet eandem rationem, quam duplus axis <lb/>propo&longs;it&aelig; hyperbol&aelig; ad ductam illam <expan abbr="&aelig;quidi&longs;t&atilde;tem">&aelig;quidi&longs;tantem</expan> inter <lb/>eiu&longs;dem hyperbol&aelig; a&longs;&longs;ymptotos interiectam. </s>
        </p>
        <p type="margin">
          <s id="s.000497"><margin.target id="marg108"/><emph type="italics"/>Tab.<emph.end type="italics"/> 5. <emph type="italics"/>fig.<emph.end type="italics"/> 2.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000498">Hyperbol&aelig; IRS &longs;it centrum H, &longs;emiaxis HI, a&longs;&longs;ymptoti <lb/>HT, NH, et SN parallela HI; t&ugrave;m ducta HM &longs;ecunda dia&shy;<lb/>metro hyperbol&aelig;, intelligatur de&longs;criptio parabol&aelig; AFD; <lb/>itaut duplus axis hyperbol&aelig;, hoc e&longs;t quadruplum ip&longs;ius <lb/>HI ad NT eandem habeat rationem, quam DB ba&longs;is pa&shy;<lb/>rabol&aelig; ad BA axim eiu&longs;dem. </s>
          <s id="s.000499">Dico quadrilineum HISM <lb/>e&longs;&longs;e imaginem velocitatum, iuxta quam motu compo&longs;ito <lb/>de&longs;cribitur parabola AFD; &amp; cum &longs;it homogenea imagi&shy;<lb/><arrow.to.target n="marg109"/><lb/>nibus HILM, HTM, e&longs;&longs;e quoque rectangulum HDLM ad <lb/>imaginem ip&longs;am HISM vt recta CA ad curuam AFD. <lb/></s>
          <s id="s.000500">Fiat rectangulum ACDB, et HM &longs;it tempus, quo curritur <lb/><arrow.to.target n="marg110"/><lb/>vtrunque latus AB, AC, nempe axis AB motu grauium <lb/>iuxta imaginem triangulum HTM, alterum ver&ograve; latus AC <lb/><arrow.to.target n="marg111"/><lb/>&aelig;quabili motu iuxta imaginem rectangulum HILM, quod <lb/>quidem erit HILM; etenim AB ad &longs;patium AC e&longs;t vt ima&shy;<lb/>go triangulum HMT ad imaginem rectangulum HILM, <lb/>&longs;cilicet e&longs;t vt MT ad duplam HI, vel vt NT ad quadru&shy;<lb/>plam HI, quemadmodum po&longs;uimus. </s>
          <s id="s.000501">Iam mon&longs;trauimus <lb/>lineam, qu&aelig; curritur iuxta illas imagines motu compo&longs;ito <lb/>parabolam e&longs;&longs;e, cuius diameter AB, &amp; ba&longs;is BD; &amp; pro&shy;<lb/>pterea erit ip&longs;a AFD &lpar;nam vnica tantum parabola ex <lb/>datis AB, BD po&longs;itione, ac magnitudine, axi &longs;cilicet, ac <lb/>ba&longs;i dari pote&longs;t&rpar; Ducatur nunc &agrave; quolibet puncto F dict&aelig; <lb/>parabol&aelig; rect&aelig; FE, FG parallelogrammum con&longs;tituentes <lb/>AEFG; &amp; P &longs;it momentum, quo mobile punctum inueni&shy;<lb/><arrow.to.target n="marg112"/><lb/>tur in F. <!-- KEEP S--></s>
          <s id="s.000502">Habebit inibi ip&longs;o temporis momento P veloci&shy;<lb/>tatem PQ iuxta directionem GF, &longs;unt ver&ograve; i&longs;t&aelig; directiones <lb/>&longs;ibi ip&longs;is perpendiculares; ergo recta, qu&aelig; diameter e&longs;&longs;et <lb/>rectanguli AEFG, &amp; ob id potenti&acirc; &aelig;qualis duabus PK, <lb/><arrow.to.target n="marg113"/><lb/>PQ erit gradus velocitatis, quem mobile habet momen-<pb pagenum="49" xlink:href="022/01/055.jpg"/>to F motu compo&longs;ito currens; ver&ugrave;m quia quadratum ex <lb/>PR &etail;quatur rectangulo ORQ vn&agrave; cum quadrato ex PQ, &amp; <lb/><arrow.to.target n="marg114"/><lb/>e&longs;t ob hyperbolam rectangulum ORQ &aelig;quale quadrato <lb/>ex HI, vel PK; ergo PR quadratum &aelig;quale erit duobus &longs;i&shy;<lb/>mul quadratis PQ, PK; itaque PR erit gradus velocitatis <lb/>pr&aelig;dicti mobilis in F momento P, compo&longs;itoque motu <lb/>currentis iuxta curuam parabolicam. </s>
          <s id="s.000503">Pariter momento <lb/>M, cum mobile e&longs;&longs;et in D velocitas compo&longs;iti motus foret <lb/>MS pote&longs;tate &aelig;qualis duabus MT, ML, ac demum in A <lb/>initio motus velocitas e&longs;t HI: quare HISM erit imago ve&shy;<lb/>locitatis motus compo&longs;iti dum mobile punctum de&longs;crip&longs;e&shy;<lb/><arrow.to.target n="marg115"/><lb/>rit curuam parabolicam AFD, e&longs;tque illa imago imagini&shy;<lb/>bus diui&longs;orum, &longs;eu &longs;implicium, motuum homogenea; ergo <lb/>con&longs;tat ba&longs;im etiam BD ad parabolam AFD eandem ha&shy;<lb/>bere rationem, quam rectangulum HILM ad quadrili&shy;<lb/>neum HISM. </s>
          <s id="s.000504">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000505"><margin.target id="marg109"/><emph type="italics"/>Def.<emph.end type="italics"/> 7. <emph type="italics"/>primi <lb/>&amp; pr.<emph.end type="italics"/> 12. <emph type="italics"/>pri&shy;<lb/>mi huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000506"><margin.target id="marg110"/><emph type="italics"/>Cor. <!-- KEEP S--></s>
          <s id="s.000507">pr.<emph.end type="italics"/> 4. <emph type="italics"/>hu.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000508"><margin.target id="marg111"/><emph type="italics"/>Pr.<emph.end type="italics"/> 2. <emph type="italics"/>primi <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000509"><margin.target id="marg112"/><emph type="italics"/>Ex pr.<emph.end type="italics"/> 12. <emph type="italics"/>hu.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000510"><margin.target id="marg113"/><emph type="italics"/>Pr.<emph.end type="italics"/> 3. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000511"><margin.target id="marg114"/><emph type="italics"/>Pr.<emph.end type="italics"/> 11. <emph type="italics"/>l.<emph.end type="italics"/> 2. <emph type="italics"/>co&shy;<lb/>nic.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000512"><margin.target id="marg115"/><emph type="italics"/>Def.<emph.end type="italics"/> 3. <emph type="italics"/>prima <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000513"><emph type="center"/><emph type="italics"/>Corollarium. <!-- REMOVE S-->I.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000514"><emph type="italics"/>Patet, cum latera compo&longs;iti motus &longs;int duo, &amp; &longs;ibi ip&longs;is per&shy;<lb/>pendicularia, tunc gradum velocitatis e&igrave;u&longs;dem motus compo&shy;<lb/>&longs;iti &aelig;qualem e&longs;&longs;e potenti&acirc; duobus &longs;imul gradibus, quos habet <lb/>mobile eodem momento, ac &longs;i &longs;eor&longs;im intelligatur in ip&longs;is ferri <lb/>lateribus.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000515"><emph type="center"/><emph type="italics"/>Corollarium. <!-- REMOVE S-->II.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000516"><emph type="italics"/>Si ver&ograve; con&longs;iderentur imagines primi &longs;ecundique Ca&longs;us <lb/>inter&longs;e homogenea, erit vt quadrilineum HISM primi ad<emph.end type="italics"/><lb/><arrow.to.target n="marg116"/><lb/><emph type="italics"/>quadrilineum ij&longs;dem literis notatum &longs;ecundi ca&longs;us, vt cur&shy;<lb/>ua illa parabolica ad hanc &longs;ecundi ca&longs;us parabolam.<emph.end type="italics"/></s>
        </p>
        <pb pagenum="50" xlink:href="022/01/056.jpg"/>
        <p type="margin">
          <s id="s.000517"><margin.target id="marg116"/><emph type="italics"/>Pr<emph.end type="italics"/> 2. <emph type="italics"/>prim&atail; <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000518"><emph type="center"/><emph type="italics"/>Corollarium. <!-- REMOVE S-->III.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000519"><emph type="italics"/>Illud etiam con&longs;tat, e&longs;&longs;e in vtroque ca&longs;u vt quadrilineum <lb/>HIRP ad ip&longs;um PRSM, ita AF ad FD.<emph.end type="italics"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000520"><emph type="center"/>PROP. XIV. THEOR. X.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000521">PRopo&longs;itis Spirali Archimedea prim&aelig; circulationis <lb/><arrow.to.target n="marg117"/><lb/>ABD, et AGF <expan abbr="c&otilde;muni">communi</expan> parabola, &longs;it FG ba&longs;is huius <lb/>&aelig;qualis radio DA, et GA &longs;it dimidium circumferenti&etail; cir&shy;<lb/>culi AEG; erit parabola AGF axem habens GA &aelig;qualis <lb/>propo&longs;it&aelig; &longs;pirali. </s>
        </p>
        <p type="margin">
          <s id="s.000522"><margin.target id="marg117"/><emph type="italics"/>Tab.<emph.end type="italics"/> 5. <emph type="italics"/>fig.<emph.end type="italics"/> 3.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000523">Sit PNK communis hyperbola, cuius coniugati &longs;emia&shy;<lb/><arrow.to.target n="marg118"/><lb/>xes &longs;int IK, IH, &amp; a&longs;&longs;ymptotos IO. <!-- KEEP S--></s>
          <s id="s.000524">E&longs;to etiam axis hy&shy;<lb/>perbol&aelig; huius, dupla &longs;cilicet IK, ad HO illi &etail;quidi&longs;tantem <lb/>vt FG ad AG. <!-- KEEP S--></s>
          <s id="s.000525">Iam con&longs;tat quadrilineum IHPK fore ima&shy;<lb/>ginem velocitatum, iuxta quam curreretur parabola AGF <lb/>tempore IH: &longs;i modo o&longs;tendimus hoc ip&longs;um <expan abbr="quadriline&utilde;">quadrilineum</expan> <lb/>e&longs;&longs;e pariter homogeneam imaginem alterius compo&longs;iti <lb/>motus, quo videlicet de&longs;cribitur &longs;piralis propo&longs;ita ABD, <lb/><arrow.to.target n="marg119"/><lb/>palam erit, ip&longs;am parabolam eidem illi &longs;pirali &aelig;qualem fu&shy;<lb/>turam. </s>
          <s id="s.000526">Ducatur recta KL, qu&aelig; &aelig;quidi&longs;tet IH; item ex <lb/>quouis puncto Q <expan abbr="t&etilde;poris">temporis</expan> IH alia deducatur recta QRMN <lb/>parallela IK: erit parallelogrammum rectangulum HIKL <lb/>imago velocitatum, iuxta quam curritur FG, et HIO trian&shy;<lb/>gulum imago, qua curritur AG motu grauium de&longs;cenden&shy;<lb/>tium: Ver&ugrave;m quia eodem tempore IH, &longs;i mobile currat <lb/>&aelig;quabili motu DA &aelig;qualem FG, e&longs;t eius imago idem re&shy;<lb/>ctangulum IHKL, curriturque illo eodem tempore IH &lpar;&longs;pi&shy;<lb/>rali exigente&rpar; omnis circuli circunferentia AGEA &aelig;qua&shy;<lb/>bili etiam motu ab extremitate A radij AD circumducti in <lb/>de&longs;criptione &longs;piralis; ob idque factum e&longs;t, vt IK ad HO e&longs;&shy;<lb/>&longs;et vt DA ad circunferentiam ip&longs;am AGEA; nam hoc mo-<pb pagenum="51" xlink:href="022/01/057.jpg"/>do rectangulum IH in HO e&longs;t imago velocitatum eiu&longs;&shy;<lb/>dem motus per AGEA. </s>
          <s id="s.000527">Ducatur nunc ex quocun&shy;<lb/><arrow.to.target n="marg120"/><lb/>que momento Q linea QRMN ip&longs;i IK &aelig;quidi&longs;tans, &amp; au&shy;<lb/>&longs;picato motu ex centro D momento I, vt nempe oriatur <lb/>&longs;piralis, intelligatur momento Q ventum e&longs;&longs;e in B, quamo&shy;<lb/>brem duct&acirc; DBE, erit rectangulum, &longs;eu imago QIKR ad <lb/>imaginem rectangulum HIKL, ita DB ad DE, in qua ra&shy;<lb/>tione, cum propter &longs;piralem, &longs;it etiam circunferentia AGE <lb/>ad circunferentiam AGEA, erit rectangulum IQ in HO <lb/>imago velocitatis per AGE, e&longs;tque velocitas iuxta tangen&shy;<lb/>tem in E ad velocitatem iuxta tangentem circulum BC in <lb/>B vt ED ad DB, &longs;eu vt HO ad QM; ergo cum iuxta <expan abbr="tang&etilde;-tem">tangen&shy;<lb/>tem</expan> in A, hoc e&longs;t in E velocitas &longs;it HO, erit &longs;ecund&ugrave;m tan&shy;<lb/>gentem circulum BC in B, ip&longs;a QM velocitas; propterea&shy;<lb/>que imago triangulum HIO, qu&aelig; in parabol&aelig; de&longs;criptio&shy;<lb/>ne erat per AG, nunc erit per omnes tangentes circulos &longs;u&shy;<lb/>binde cre&longs;centes ex D in E: &longs;cilicet momento I, erit mobi&shy;<lb/>li puncto &longs;ecund&ugrave;m DA, velocitas IK; momento Q du&mtail; <lb/>ade&longs;t in B, erit &longs;ecund&ugrave;m BE velocitas QR, &amp; iuxta <expan abbr="tang&etilde;-tem">tangen&shy;<lb/>tem</expan> in B circuli BC velocitas QM; qu&aelig; amb&aelig;, hoc e&longs;t ve&shy;<lb/>locitates QR, QM cum &longs;int normaliter direct&aelig;, erit eidem <lb/><arrow.to.target n="marg121"/><lb/>mobili in B iuxta &longs;piralem velocitas QN potentia ip&longs;is am&shy;<lb/>babus &aelig;qualis. </s>
          <s id="s.000528">Similiterque momento H cum mobil&etail; <lb/>fuerit in A, erit velocitas iuxta &longs;piralem, ip&longs;a HP &aelig;qualis <lb/>potenti&acirc; duabus velocitatibus HL iuxta radium, et HO <lb/>iuxta tangentem; &amp; &longs;ic omnino liquet, ip&longs;um quadrilineum <lb/>HIKP e&longs;&longs;e imaginem velocitatum tam in de&longs;criptione pa&shy;<lb/>rabol&aelig; AGF, qu&agrave;m &longs;piralis Archimede&aelig; DBA, &amp; cum &longs;it <lb/>in ij&longs;dem de&longs;criptionibus homogenea &longs;ibi ip&longs;i, con&longs;tat ip&shy;<lb/><arrow.to.target n="marg122"/><lb/>&longs;as curuis &aelig;quales e&longs;&longs;e. </s>
          <s id="s.000529">Nam vt imago illa ad &longs;e ip&longs;am ita <lb/>parabola ad &longs;piralem pr&aelig;dictam. </s>
          <s id="s.000530">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <pb pagenum="52" xlink:href="022/01/058.jpg"/>
        <p type="margin">
          <s id="s.000531"><margin.target id="marg118"/><emph type="italics"/>Pr.<emph.end type="italics"/> 13. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000532"><margin.target id="marg119"/><emph type="italics"/>Pr. <gap/>. </s>
          <s id="s.000533">prima.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000534"><margin.target id="marg120"/><emph type="italics"/>Pr.<emph.end type="italics"/> 2. <emph type="italics"/>prima.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000535"><margin.target id="marg121"/><emph type="italics"/>Pr.<emph.end type="italics"/> 8. <emph type="italics"/>huius &amp; <lb/>Cor. <!-- KEEP S--></s>
          <s id="s.000536">pr.<emph.end type="italics"/> 13.</s>
        </p>
        <p type="margin">
          <s id="s.000537"><margin.target id="marg122"/><emph type="italics"/>Pr.<emph.end type="italics"/> 2. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000538"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000539"><emph type="italics"/>Hinc aparet, &longs;piralem DB ad &longs;piralem DBG eandem habe&shy;<lb/>re rationem, quam quadrilineum QIKN ad quadrilineum <lb/>HIKP; pariterque rectam DA ad eandem &longs;piralem DCB ha&shy;<lb/>bere ip&longs;am rationem, ac rectangulum HIKL ad dictum qua&shy;<lb/>drilineum HIKP. </s>
          <s id="s.000540">Eodem fer&egrave; modo exhiberi pi&szlig;et ratio &longs;pi&shy;<lb/>ralis ad &longs;piralem, lic&egrave;t plurium inter&longs;e circulationum, eritque <lb/>pror&longs;us ea, quam habet vnum ad alterum eiu&longs;dem illius na&shy;<lb/>tur&aelig;, quadrilineorum.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000541"><emph type="center"/>PROP. XV. THEOR. XI.<emph.end type="center"/><lb/><arrow.to.target n="marg123"/><!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000542"><margin.target id="marg123"/><emph type="italics"/>Tab.<emph.end type="italics"/> 5. <emph type="italics"/>Fig.<emph.end type="italics"/> 4.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000543">SPiralis orta ex motu naturaliter accelerato per <expan abbr="radi&utilde;">radium</expan> <lb/>circuli comprehendentis &longs;piralem ip&longs;am, &amp; ex motu <lb/>&aelig;quabili circa <expan abbr="circumferenti&atilde;">circumferentiam</expan> eiu&longs;dem circuli, &aelig;qualis e&longs;t <lb/>ei curu&aelig; parabolic&aelig; nat&aelig; ex motu compo&longs;ito, cuius vnum <lb/>latus curritur iuxta imaginem trianguli, nempe motu gra&shy;<lb/>uium, alterum ver&ograve; latus iuxta imaginem trilinei &longs;ecundi, <lb/>habebitque parabola ip&longs;a axim &aelig;qualem radio, &amp; ba&longs;i&mtail; <lb/>terti&aelig; parti circunferenti&aelig; eiu&longs;dem circuli &longs;piralem com&shy;<lb/>prehendentis. </s>
        </p>
        <p type="main">
          <s id="s.000544">E&longs;to &longs;piralis ACB, qu&aelig; &longs;ignatur ex motu <expan abbr="p&utilde;cti">puncti</expan> A &aelig;qua<lb/>biliter lati circa circumferentiam ADA, dum nempe <expan abbr="eod&etilde;">eodem</expan> <lb/>tempore IF, punctum B currit &agrave; quiete lineam BA motu <lb/>grauium de&longs;cendentium; &longs;it ver&ograve; imago velocitatum dicti <lb/>motus &aelig;quabilis per ADA rectangulum HGFI, &amp; alte&shy;<arrow.to.target n="marg124"/><lb/>rius motus imago, &lpar;qu&aelig; triangulum erit&rpar; e&longs;to FEIM. Pa&shy;<lb/><arrow.to.target n="marg125"/><lb/>tet, quia ip&longs;&aelig; imagines ponuntur homogene&aelig;, e&longs;&longs;e rectan&shy;<lb/>gulum HGFI ad triangulum IFM vt ADA circumferentia <lb/>ad radium BA, &amp; propterea IM ad IH erit vt BA ad dimi&shy;<lb/>dium circunferenti&aelig; AEDA. </s>
          <s id="s.000546">Sumatur quodlibet <expan abbr="mom&etilde;-tum">momen&shy;<lb/>tum</expan> K, &amp; ducatur ONKL &aelig;quidi&longs;tans HM, puteturque <pb pagenum="53" xlink:href="022/01/059.jpg"/>eodem illo momento mobile <expan abbr="v&etilde;tum">ventum</expan> e&longs;&longs;e in C &longs;piralis pro&shy;<lb/>po&longs;it&aelig; BCA: agatur per ip&longs;um punctum radius BCD, &amp; &longs;ic <lb/>illo momento extremitas A currendo circa periph&aelig;riam <lb/>reperietur in D, eritque circunferentia AED ad ip&longs;am <lb/>AEDA, vt imago rectangulum OGFK ad <expan abbr="imagin&etilde;">imaginem</expan> GHIF, <lb/>hoc e&longs;t erit vt KF ad FI; at BC ad BD erit vt imago trian&shy;<lb/>gulum KFL ad triangulum FIM, nempe vt quadratum KF <lb/>ad quadratum FI, e&longs;t autem vt BD ad BC ita velocitas <lb/>iuxta tangentem in D ad velocitatem iuxta tangentem in <lb/>C circulum, cuius radius BC; &longs;cilicet ita velocitas IH ad <lb/>velocitatem KN, quadrati nempe IF ad quadratum KF, &amp; <lb/>ob id velocitates, qu&aelig; &longs;unt iuxta tangentes circulos &longs;ubin&shy;<lb/>de <expan abbr="cre&longs;c&etilde;tes">cre&longs;centes</expan> ex centro B, <expan abbr="er&utilde;t">erunt</expan> expre&longs;&longs;&aelig; in trilineo HNFIH <lb/>&longs;ecundo, cuius &longs;cilicet indoles e&longs;t vt ab&longs;ci&longs;&longs;arum quadrata <lb/>&longs;int vt applicat&aelig;. </s>
          <s id="s.000547">His compo&longs;itis, intellecti&longs;que erit in B, <lb/>momento F, nulla velocitas, in C momento K du&aelig; velo&shy;<lb/>citates quarum vn&agrave; KI mobile iret iuxta CD, &longs;ed cum al&shy;<lb/>tera &longs;it KN iuxta tangentem circulum, cuius radius CB, ne&shy;<lb/><arrow.to.target n="marg126"/><lb/>ctitur vna ex duabus illis, quibus <expan abbr="ei&longs;d&etilde;">ei&longs;dem</expan> potentia e&longs;t &aelig;qua&shy;<lb/><arrow.to.target n="marg127"/><lb/>lis, &amp; qua idem mobile mouetur iuxta &longs;piralem illo mo&shy;<lb/>mento K. <!-- KEEP S--></s>
          <s id="s.000548">Similiter cum mobile e&longs;t in D, &longs;cilicet momento <lb/>I, habebit velocitatem potentia &aelig;qualem HI, qua dirigitur <lb/>iuxta tangentem, &amp; velocitati IM, qua &longs;ecund&ugrave;m radium, <lb/>Itaque imago velocitatum mobilis de&longs;cribentis &longs;piralem <lb/>propo&longs;itis motibus tempore IF, ea erit, cuius applicat&aelig; <lb/>&longs;unt vbique &aelig;quales potentia ijs applicatis, qu&aelig; ab <expan abbr="eod&etilde;">eodem</expan> <lb/>momento intelligi queunt in imaginibus &longs;implicibus, nem&shy;<lb/>pe partialium motuum, HNFI, IFM. </s>
          <s id="s.000549">Cum pr&aelig;terea OT <lb/>ponatur tertia pars e&longs;&longs;e circumferenti&aelig; AEDA, &amp; e&longs;t <expan abbr="eti&atilde;">etiam</expan> <lb/>trilineum HFI vtpote &longs;ecundum tertia pars <expan abbr="parallelogr&atilde;-mi">parallelogram<lb/>mi</expan> HGFI, erit triangulum IFM ad trilineum ip&longs;um HFI vt <lb/><arrow.to.target n="marg128"/><lb/>BA, vel ei &aelig;qualis QO ad OT; curritur ver&ograve; vt &longs;upponi&shy;<lb/>tur OQ tempore IF iuxta imaginem triangulum IFM, ergo <lb/><arrow.to.target n="marg129"/><lb/>eodem tempore iuxta trilineum HNF curretur alterum la-<pb pagenum="54" xlink:href="022/01/060.jpg"/>tus OT, &longs;iue ba&longs;is parabol&aelig; QI. </s>
          <s id="s.000550">Si itaque parabola ip&longs;a <lb/>putetur e&longs;&longs;e ORI, in qua punctum R e&longs;to vbi mobile ade&longs;t <lb/>momento K, deducantur ver&ograve; ab eodem illo puncto RS <lb/>parallela axi QO, et RP &aelig;quidi&longs;tans QI, vel OT, profect&ograve; <lb/>in O, momento F, &longs;icuti in &longs;pirali, nulla erit mobili veloci&shy;<lb/>tas, &longs;ed cum e&longs;t in R momento K habebit geminam veloci&shy;<lb/>tatem, KL &longs;ecund&ugrave;m SR, et KN iuxta PR perpendicularem <lb/>ip&longs;i SR, qu&aelig; du&aelig; velocitates itidem component vnicam <lb/>potentia &longs;imul illis &aelig;qualem, &amp; cum idem dicatur de qui&shy;<lb/>bu&longs;cunque alijs punctis parabol&aelig;, momentis temporis FI <lb/>re&longs;pondentibus, manife&longs;tum e&longs;t &longs;pirali BCA, &amp; parabol&aelig; <lb/>ORI vnicam, eandemque e&longs;&longs;e imaginem velocitatum, pro&shy;<lb/>pterquam qu&ograve;d ip&longs;&aelig; curu&aelig;, qu&ograve;d &longs;int vt imagines, erunt <lb/>inter&longs;e &aelig;quales. <lb/><arrow.to.target n="marg130"/></s>
        </p>
        <p type="margin">
          <s id="s.000551"><margin.target id="marg124"/><emph type="italics"/>Cor. <!-- KEEP S--></s>
          <s id="s.000552">pr.<emph.end type="italics"/> 4. <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000553"><margin.target id="marg125"/><emph type="italics"/>Pr.<emph.end type="italics"/> 2. <emph type="italics"/>prim&atail;<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000554"><margin.target id="marg126"/><emph type="italics"/>Pr.<emph.end type="italics"/> 8. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000555"><margin.target id="marg127"/><emph type="italics"/>Cor. <!-- KEEP S--></s>
          <s id="s.000556">prop.<emph.end type="italics"/> 13. <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000557"><margin.target id="marg128"/><emph type="italics"/>Pr.<emph.end type="italics"/> 10. <emph type="italics"/>primi <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000558"><margin.target id="marg129"/><emph type="italics"/>Pr.<emph.end type="italics"/> 2. <emph type="italics"/>primi <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000559"><margin.target id="marg130"/><emph type="italics"/>Pr.<emph.end type="italics"/> 2. <emph type="italics"/>prima.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000560"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000561"><emph type="italics"/>Exemplo traditarum curuarum, po&longs;&longs;unt innumer&aelig; &longs;pira&shy;<lb/>les &longs;uis parabolis &aelig;quales excogitari, nec ideo res min&ugrave;s de&shy;<lb/>mon&longs;trabitur, &longs;i loco rectarum, &longs;eu laterum OT, OP compo&longs;iti <lb/>motus, &longs;ub&longs;tituantur circuli, aut circulorum arcus, qui ad re&shy;<lb/>ctos angulos &longs;e &longs;ecent, &longs;cilicet <expan abbr="c&utilde;">cum</expan> tangentes ad punctum infle&shy;<lb/>xionis, &longs;eu occur&longs;us ip&longs;arum curuarum &longs;ibi ip&longs;is perpendicu&shy;la<lb/>res fuerint. </s>
          <s id="s.000562">Qu&ograve;d &longs;i ip&longs;a curua latera ad rectos angulos non <lb/>&longs;e &longs;ecent curu&aelig; nihilominus ab ip&longs;o compo&longs;ito motu na&longs;cen&shy;<lb/>tes poterunt exhiberi curuas parabolicas exequantes, quarum <lb/>itidem latera &longs;int rect&aelig; eundem angulum, quem pr&aelig;dict&aelig; <expan abbr="t&atilde;-gentes">tan&shy;<lb/>gentes</expan>, comprehendentes. </s>
          <s id="s.000563">Sed de his &longs;atis, nunc dicamus ea <lb/>tempora, quibus duorum pendulorum &longs;imiles vibrationes ab&shy;<lb/>&longs;oluuntur, hoc e&longs;t Galilei &longs;ententiam demon&longs;trabimus, quam <lb/>quondam haud ruditer decepti fal&longs;am credidimus.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000564"><emph type="italics"/>Vincentius Viuianus eximius no&longs;tri &aelig;ui Geometra vt tue&shy;<lb/>retur Galilei &longs;ententiam, cuius digni&longs;&longs;im&egrave; &longs;e fui&longs;&longs;e di&longs;cipu&shy;<lb/>lum profitetur, tradidit mihi per admodum Reuerendum, at-<emph.end type="italics"/><pb pagenum="55" xlink:href="022/01/061.jpg"/><emph type="italics"/>que culti&longs;&longs;imum Patrem Io&longs;eph Ferronum &egrave; Societate Ie&longs;u, de&shy;<lb/>mon&longs;trationem &longs;uam ver&egrave; pulcherrimam, ac di&longs;erti&longs;&longs;im&egrave; <lb/>exaratam, qua vna potui&longs;&longs;em de Galilei a&szlig;erto &longs;atisfactus <lb/>e&longs;&longs;e; eam demon&longs;trationem, ij&longs;dem pror&longs;us verbis, ac figuris, <lb/>quibus ad me peruenit hic duxi reponendam, ne gloria&mtail;, <lb/>quam Vir tantus meretur, ip&longs;i videremur no&longs;tra, quam inde <lb/>&longs;ubdemus, demon&longs;tratione, &longs;ubripere.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000565"><emph type="italics"/>Inquit ergo.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000566">TEmpora naturalium de cur&longs;uum &longs;ph&aelig;rarum grauium <lb/><arrow.to.target n="marg131"/><lb/>per &longs;imiles, &longs;imiliterque ad horizontem inclinatos <lb/>arcus curuarum linearum in planis, aut verticalibus, aut <lb/>ad horizontem &aelig;qualiter inclinatis de&longs;criptarum, &amp; qu&aelig; <lb/>tot&aelig; &longs;int ad ea&longs;dem partes cau&aelig;, inter&longs;e &longs;unt in &longs;ubdupli&shy;<lb/>cata ratione chordarum eorundem arcuum homolog&egrave; <lb/>&longs;umptarum. </s>
        </p>
        <p type="margin">
          <s id="s.000567"><margin.target id="marg131"/><emph type="italics"/>Tab.<emph.end type="italics"/> 6. <emph type="italics"/>fig.<emph.end type="italics"/> 1. 2 <lb/>3. 4.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000568">Ex puncto A ad curuam lineam BCD extra ip&longs;am i&ntail; <lb/>plano po&longs;itam, &amp; in totum ad ea&longs;dem partes cauam, qu&aelig;&shy;<lb/>cunque ea &longs;it &lpar;vel nimirum pars aliqua circumferenti&aelig; <lb/>circuli, vel alicuius ex infinitis ellip&longs;ibus, aut parabolis, aut <lb/>hyperbolis, aut &longs;piralibus, aut cycloidibus, vel concoidis, <lb/>vel ci&longs;oidis, &longs;eu alterius cuiu&longs;cumque ex notis, vel ignotis <lb/>curuis educantur omnes rect&aelig; AB, AC, AD &amp;c. <!-- KEEP S--></s>
          <s id="s.000569">qu&aelig; &agrave; <lb/>punctis E, F, C, vel intra, vel extra eas &longs;umptis proportio&shy;<lb/>nalibus &longs;ecentur, ita vt &longs;it AB ad AE, &longs;icut AC ad AF, &amp; <lb/>&longs;icut AD ad AG &amp;c. <!-- KEEP S--></s>
          <s id="s.000570">&amp; hoc &longs;emper. </s>
          <s id="s.000571">Sic enim dubio pro&shy;<lb/>cul apparet, prout facillimum e&longs;t o&longs;tendere, lineam EFG <lb/>tran&longs;euntem per &longs;ingula puncta E, F, G &longs;ic inuenta, cur&shy;<lb/>uam <expan abbr="quoq;">quoque</expan> e&longs;&longs;e, &amp; eiu&longs;dem penitus natur&aelig;, ac data BCD <lb/>eique &longs;imilem, &longs;imiliterque cum ip&longs;a po&longs;itam, atque in to&shy;<lb/>tum cauam ad ea&longs;dem partes, ad quas ponitur caua ip&longs;&atail; <lb/>BCD. </s>
          <s id="s.000572">Concipiatur mod&ograve; planum, in quo manent huiu&longs;&shy;<lb/>modi &longs;imilium curuarum &longs;imiles arcus BCD, EFG, vel e&longs;&longs;e <lb/>ad horizontem erectum, nemp&egrave; verticale, vel ad ip&longs;u&mtail; <pb pagenum="56" xlink:href="022/01/062.jpg"/>horizontem inclinatum iuxta curuitates ip&longs;orum arcuum <lb/>BCD, EFG inflexas e&longs;&longs;e &longs;uperficies eidem plano erectas, <lb/>ita tamen, vt &longs;uper has po&longs;itis grauibus &longs;ph&aelig;ris in A, E per <lb/>ip&longs;as &longs;ic inflexas &longs;uperficies e&aelig;dem &longs;ph&aelig;r&aelig; naturaliter <lb/>decurrere queant; id quod &longs;an&egrave; accidet, cum arcus BCD <lb/>totus fuerit infra horizontalem IL ex arcus &longs;ubli&shy;<lb/>miori puncto B ductam, fuerintque ab hac continuati re&shy;<lb/>ce&longs;&longs;us, ac totus ad vnam partem perpendiculi BH: nam &longs;ic <lb/>talis quoque erit alter arcus EFG illi BCD &longs;imilis, &longs;imili&shy;<lb/>terque po&longs;itus. </s>
          <s id="s.000573">His omnibus &longs;ic manentibus: Dico tem&shy;<lb/>pus decur&longs;us &longs;ph&aelig;r&aelig; grauis E per &longs;imilem, &longs;imiliterque po&shy;<lb/>&longs;itum arcum EFG, e&longs;&longs;e in &longs;ubduplicata ratione chordarum <lb/>BO, EG arcus ip&longs;os &longs;ubtendentium. </s>
          <s id="s.000574">Secto enim bifariam <lb/>angulo BAD per rectam AC arcum BD &longs;ecantem in C, <lb/>atque arcum EFG in F, iungantur chord&aelig; BC, CD, et EF, <lb/>FG, qu&aelig; ex huiu&longs;modi curuarum natura cadent tot&aelig; intra <lb/>ip&longs;os arcus, &longs;ed in prima, &amp; &longs;ecunda figura ad partes poli <lb/>A, in tertia ver&ograve;, &amp; quarta ad oppo&longs;itas. </s>
        </p>
        <p type="main">
          <s id="s.000575">Et quoniam, ex talium curuarum gene&longs;i, e&longs;t vt BA ad <lb/>AE, ita DA, ad AG, erit BD ip&longs;i EG parallela, hoc e&longs;t <lb/>vtraque ad horizontem &aelig;qualiter inclinata, atque in ra&shy;<lb/>tione BA ad AE. <!-- KEEP S--></s>
          <s id="s.000576">Similiter cum &longs;it, vt BA ad AE, ita CA <lb/>ad AF, etiam BC, EF inter&longs;e &aelig;quidi&longs;tabunt, &longs;eu ad hori&shy;<lb/>zontem &aelig;qualiter inclinabuntur, eruntque in ratione ea&shy;<lb/>dem, ac BA ad AE. <!-- KEEP S--></s>
          <s id="s.000577">Idemque o&longs;tenditur de chordis CD, <lb/>FG, quare ex magni Galilei &longs;ententia de motu naturaliter <lb/>accelerato indubitanter &longs;equitur tempus decur&longs;us &longs;ph&aelig;r&aelig; <lb/>grauis ex B in D per binas chordas BC, CD ad tempus <lb/>decur&longs;us per vnicam BD, e&longs;&longs;e vt tempus decur&longs;us grauis <lb/>&longs;ph&aelig;r&aelig; ex E in G per binas EF, FG ad tempus decur&longs;us <lb/>per vnicam EG: eadem itidem ratione demon&longs;tratur &lpar;an&shy;<lb/>gulis pariter BAC, CAD bifariam &longs;ectis per rectas, qu&aelig; <lb/>&longs;imiles arcus BC, EF, ac CD, FG duas in partes diuidant&rpar; <lb/>ex quatuor vtrinque arcuum horum cordis, illas inter&longs;e <pb pagenum="57" xlink:href="022/01/063.jpg"/>homologas, &longs;imile&longs;que arcus &longs;ubtendentes ad horizonte m <lb/>e&longs;&longs;e &aelig;qualiter inclinatas, ac alteram alteri in ratione ea&shy;<lb/>dem, in qua &longs;unt rect&aelig; AB, AE &amp;c: ac propterea ex ea&shy;<lb/>dem Galilei &longs;cientia con&longs;tabit vtique, tempus decur&longs;us ex <lb/>B in C &longs;ph&aelig;r&aelig; grauis B per quatuor chordas quatuor par&shy;<lb/>tes arcus BCD &longs;ubtendentes ad tempus decur&longs;us per vni&shy;<lb/>cam BD, e&longs;&longs;e vt tempus decur&longs;us &longs;ph&aelig;r&aelig; grauis E ex E in <lb/>G per quatuor illis homologas chordas quatuor partes <lb/>arcus EFG pariter &longs;ubtendentes ad tempus decur&longs;us per <lb/>vnicam chordam EG: &amp; hoc &longs;emper ita euenire demon&shy;<lb/>&longs;trabitur quantacunque, &amp; maxima fuerit in perpetua an&shy;<lb/>gulorum bi&longs;ectione &aelig;qu&egrave;multiplicitas in vtroque arcu <lb/>talium chordarum homolog&egrave; &longs;umptarum, ac inter&longs;e pro&shy;<lb/>portionalium, &aelig;qualiterque ad horizontem inclinatarum: <lb/>Propterquam qu&ograve;d &longs;emper decur&longs;us ex B in D per aggre&shy;<lb/>gatum chordarum omnium in arcu BCD ad tempus de&shy;<lb/>cur&longs;us per &longs;olam chordam BD e&longs;&longs;e vt tempus decur&longs;us ex <lb/>E in G per aggregatum totidem chordarum in arcu EFG <lb/>ad tempus decur&longs;us per vnicam chordam EG; adeo vt de&shy;<lb/>nique iure optimo educi po&longs;&longs;e videatur, tempus decur&longs;us <lb/>grauis ex B in D per aggregatum infinitarum chordarum <lb/>totum arcum BCD con&longs;tituentium, &longs;eu tempus per ip&longs;um <lb/>arcum BCD ad tempus decur&longs;us per &longs;olam cordam BD <lb/>e&longs;&longs;e vt tempus decur&longs;us grauis ex E in G per aggregatum <lb/>totidem infinitarum chordarum dictis homolog&egrave; propor&shy;<lb/>tionalium, &aelig;qualiterque &longs;ingul&aelig; &longs;ingulis ad horizonte&mtail; <lb/>inclinatarum, ac totum arcum EFG conformantium, &longs;iue <lb/>vt tempus per ip&longs;um arcum EFG per &longs;olam chordam EG. <lb/></s>
          <s id="s.000578">Quocirca permutando, tempus, decur&longs;us &longs;ph&aelig;r&aelig; grauis B <lb/>per arcum BCD ad tempus decur&longs;us &longs;ph&aelig;r&aelig; grauis E per <lb/>arcum &longs;imilem, &longs;imiliterque po&longs;itum EG erit vt tempus <lb/>decur&longs;us per chordam BD ad tempus decur&longs;us per chor&shy;<lb/>dam EG; &longs;ed ex eadem Galilaica &longs;cientia de motu, tempus <lb/>decur&longs;us per chordam BD ad tempus decur&longs;us per &aelig;qua-<pb pagenum="58" xlink:href="022/01/064.jpg"/>iter inclinatam EG e&longs;t in &longs;ubduplicata ratione ip&longs;aru&mtail; <lb/>chordarum BD, EG; ergo tempus quoque decur&longs;us ex B <lb/>per arcum BCD ad tempus decur&longs;us ex E per arcum EFG <lb/>e&longs;t in eadem &longs;ubduplicata ratione chord&etail; BD ad chordam <lb/>EG, quod o&longs;tendendum propo&longs;uimus. </s>
        </p>
        <p type="main">
          <s id="s.000579"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000580"><emph type="italics"/>Ex mod&ograve; osten&longs;is &longs;uper prima, ac &longs;ecunda figura, manife&shy;<lb/>&longs;tum fit celeberrimum illud magni Galilei pronuntiatum, <lb/>qu&ograve;d videlicet, ratio temporum &longs;imilium vibrationum pen&shy;<lb/>dulorum &longs;it &longs;ubduplicata rationis longitudinum filorum ho&shy;<lb/>molog&egrave; &longs;umptorum, non tantum verum e&longs;&longs;e de vibrationibus <lb/>pendulorum per arcus &longs;imiles, &longs;imiliterque po&longs;itos, &longs;umptos <lb/>ex circulorum quadrantibus ad perpendiculum v&longs;que termi&shy;<lb/>nantes, &longs;ed etiam de vibrationibus per arcus quo&longs;cumque &longs;i&shy;<lb/>miles quadrantum &agrave; perpendiculo &longs;eiunctos: dummodo ip&longs;i <lb/>&longs;imiles arcus &longs;int quoque &longs;imiliter po&longs;iti: quales nimir&ugrave;m ap&shy;<lb/>parent in figuris prima, ac &longs;ecunda arcus BCD, EFG, dum <lb/>grauia B, E ex filis, aut ha&longs;tulis AB, AE circa punctum A <lb/>conuertibilibus appen&longs;a concipiantur.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000581"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000582"><emph type="italics"/>Si curua BCD, EFG in prima, &amp; &longs;ecunda figura fuerint <lb/>&longs;imiles arcus ex circulis commune centrum A habentibus; ac <lb/>in verticali plano po&longs;itis, &amp; in prima figura recta AB, AE <lb/>fuerint fila aut ha&longs;tul&aelig; qu&aelig;dam circa clauum A conuertibi&shy;<lb/>les, in &longs;ecunda ver&ograve; recta AB, AE concipiantur, vt ha&longs;tul&aelig; <lb/>inflexibiles, volubile&longs;que circa imum punctum E, atque ex <lb/>huiu&longs;modi filorum, aut ha&longs;tularum terminis B, E pendeant <lb/>graues &longs;ph&aelig;r&aelig; B, E &lpar;cum eadem &longs;int tempora prout a&longs;&longs;umi&shy;<lb/>tur quoque ab ip&longs;o met Ceua&rpar; tempora inquam decur&longs;uum <lb/>liberorum granium B, E per arcus BCD, EFG, ac tempor&atail;<emph.end type="italics"/><pb pagenum="59" xlink:href="022/01/065.jpg"/><emph type="italics"/>de&longs;cen&longs;uum ip&longs;orum grauium per eo&longs;dem arcus &lpar;vel hac &agrave; <lb/>filis pendeant, vel ab hastulis &longs;ustineantur&rpar; erit quoque tem&shy;<lb/>pus de&longs;cen&longs;us, &longs;eu vibrationis penduli B per arcum BCD ad <lb/>tempus de&longs;cen&longs;us, &longs;eu vibrationis penduli E per arcum EFD <lb/>in &longs;ubduplicata ratione chord&aelig; BD ad chordam EG; &longs;ed h&aelig;c <lb/>ratio chordarum BD, EG eadem e&longs;t, ac ratio filorum, aut ha&shy;<lb/>&longs;tularum AB, AE; Ergo tempus vibrationis penduli AB per <lb/>arcum BCD ad tempus vibrationis penduli AE per arcum il&shy;<lb/>li &longs;imilem, &longs;imiliterque po&longs;itum EFG est quoque in &longs;ubdupli&shy;<lb/>cata ratione longitudinum, vel filorum, aut ha&longs;tularum, ex <lb/>quibus eadem grauia pendula &longs;imiles vibrationes ab&longs;oluunt <lb/>BCD, EFG.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000583"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000584"><emph type="italics"/>C&aelig;ter&ugrave;m non me latet con&longs;tructionem, ac demonstratio-<emph.end type="italics"/><lb/><arrow.to.target n="marg132"/><lb/><emph type="italics"/>nem &agrave; nobis &longs;uperi&ugrave;s allatam nonnullis euidentiorem forta&longs;&longs;e <lb/>eua&longs;uram, &longs;i ommi&longs;&longs;a illa continua bi&longs;ectione angulorum &longs;i&shy;<lb/>miles, &longs;imiliterque po&longs;itos arcus ab&longs;cindentium ex &longs;imilibus <lb/>curuis ibidem de&longs;criptis; atque ommi&longs;&longs;a pariter continua co&shy;<lb/>niunctione chordarum, vt ibi factum fuit, horum vice, vt in <lb/>quinta figura, ex punctis B, D bin&aelig; tangentes curuam BCD <lb/>ducantur BH, DH, qu&aelig; omnin&ograve; mutu&ograve; &longs;e &longs;ecabunt in puncto <lb/>H &lpar;ob conditiones in ip&longs;a Theorematis expo&longs;itione vltimo lo&shy;<lb/>co po&longs;itas&rpar; atque ex E, G ip&longs;is BH, DH agantur &aelig;quidistan&shy;<lb/>tes, qu&aelig; iunct&aelig;, AH &longs;imul occurrent in I, curuamque EFG <lb/>contingent pariter ad E, G &lpar;qu&aelig; omnia &longs;i opus fuerit, facil&egrave; <lb/>demon&longs;trabuntur&rpar; ac in&longs;uper, &longs;i &agrave; puncto C, in quo iunct&atail; <lb/>AH &longs;ecat arcum BCD, agatur tangens LM primas BH, DH <lb/>&longs;ecans in LM; Per F ver&ograve;, in quo AICH &longs;ecat arcum EFG <lb/>agatur NO parallela tangenti LM, qu&aelig; curuam pariter EFG <lb/>tanget ad F, ac tangentes EI, GI &longs;ecabit ad NO: &amp; &longs;i iunctis <lb/>in&longs;uper AL, AM, eadem, quam nunc explicauimus, continue&shy;<lb/>tur con&longs;tructio per alias, atque alias tangentes, ac parallelas<emph.end type="italics"/><pb pagenum="60" xlink:href="022/01/066.jpg"/><emph type="italics"/>&amp;c. </s>
          <s id="s.000585">&longs;ic enim vnicuique harum curuarum circum&longs;cribetur <lb/>rectilineum, prim&ograve; ex binis tangentibus, &longs;ecund&ograve; ex tribus, <lb/>terti&ograve; ex quinque, quart&ograve; ex &longs;eptem, &amp; &longs;ic vlteri&ugrave;s iuxta re&shy;<lb/>liquos impares numeros &longs;ucce&longs;&longs;iu&egrave; &longs;umptos; atque omnia pa&shy;<lb/>ria talium &aelig;quidi&longs;tantium tangentium eam &longs;emper inter &longs;e <lb/>rationem &longs;eruabunt, quam habent chorda BD, EG, &longs;en quam <lb/>habent rect&aelig; BA, EA, <expan abbr="eruntq;">eruntque</expan> inter&longs;e &aelig;qualiter inclinat&aelig;; <lb/><expan abbr="adeoq;">adeoque</expan> tempora decur&longs;uum grauium B, E tam per &longs;ummas <lb/>binarum tangentium BH, HD, EI, IG, qu&agrave;m per minores <lb/>&longs;ummas, ex quinque &longs;imul chordis vtrinque &longs;umptas, aut <lb/>qu&agrave;m per alias &longs;emper minores &longs;ummas huiu&longs;modi tangen&shy;<lb/>tium iuxta quantumuis maiorem numerum imparem &aelig;qu&egrave; <lb/>multipliciter &longs;umptarum, erunt perpetu&ograve; proportionalia tem&shy;<lb/>poribus decur&longs;uum per chordas BD, EG; &amp; hoc &longs;emper; etiam&shy;<lb/>&longs;i per huiu&longs;modi decrementa aggregatorum ex tangentibus <lb/>vtrinque &aelig;qu&egrave;multipliciter &longs;umptis, deueniatur ad vltimus, <lb/>ac breui&longs;&longs;imas ip&longs;is arcubus circum&longs;criptiones polygonorum <lb/>ex lateribus numero innumerabiliter aqu&egrave;multiplicibus, hoc <lb/>e&longs;t ad ip&longs;os &longs;imiles, &longs;imiliterque po&longs;itos arcus BCD, EFG, <lb/>quorum &longs;ingula homologorum laterum, &longs;eu punctorum paria, <lb/>vt B, &amp; E; C et F; D, et G &amp;c. <!-- KEEP S--></s>
          <s id="s.000586">haberi po&szlig;unt tanquam tot <lb/>paria parallelarum, ac proportionalium tangentium ip&longs;os &longs;i&shy;<lb/>miles, ac &longs;imiliter po&longs;itos arcus con&longs;tituentia. </s>
          <s id="s.000587">Quapropter <lb/>ratio <expan abbr="quoq;">quoque</expan> temporum decur&longs;uum per ip&longs;os arcus, &longs;imilis erit <lb/>rationi temporum decur&longs;uum per chordas; &longs;ed horum decur&shy;<lb/>&longs;uum ratio &longs;ubdupla e&longs;t rationis inter ip&longs;as chordas. </s>
          <s id="s.000588">Quare, <lb/>&amp; alia hac methodo con&longs;taret propo&longs;itum.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000589"><margin.target id="marg132"/><emph type="italics"/>Tab.<emph.end type="italics"/> 6 <emph type="italics"/>fig.<emph.end type="italics"/> 5.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000590"><emph type="italics"/>Hactenus graui&longs;&longs;imus Vir; &longs;upere&longs;t mod&ograve;, vt quemadmo&shy;<lb/>dum annuimus, veritatem eandem no&longs;tra quoque methodo, <lb/>confirmemus, vt ijs, quibus &longs;atis probat demon&longs;tratio allata, <lb/>&longs;it nostra, quam afferemus, in experimentum traditarum h&ugrave;c <lb/><expan abbr="v&longs;q;">v&longs;que</expan> rerum; &amp; quibus &longs;ec&ugrave;s acciderit ex aliqua dubitatione, <lb/>h&aelig;c per demon&longs;trationes no&longs;tras pror&longs;us, &longs;tatimq tollatur. <lb/></s>
          <s id="s.000591">Illud etiam admoneo, eam rem non tantum me o&longs;ten&longs;urum,<emph.end type="italics"/><pb pagenum="61" xlink:href="022/01/067.jpg"/><emph type="italics"/>vt pulcherrima, <expan abbr="vtilimaq;">vtilimaque</expan> veritas pluribus demon&longs;trationi&shy;<lb/>bus aperiatur; ver&ugrave;m potius vt ampli&longs;&longs;ima Methodus, qua tum <lb/>vtemur, aliorum motuum demon&longs;trandorum in exemplum <lb/>veniat.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000592"><emph type="center"/>PROP. XVI. THEOR. XII.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000593">IN eadem recta CD coeant du&aelig; plan&aelig;, <expan abbr="inter&longs;eq;">inter&longs;eque</expan> &longs;imiles, <lb/><arrow.to.target n="marg133"/><lb/>ac pror&longs;us &aelig;quales figur&aelig; ADCA, BDCB, &amp; quidem <lb/>ita, vt ab eodem puncto M &longs;i ducatur MH parallela CA, <lb/>et ML ip&longs;i CB, &longs;it &longs;emper MH &aelig;qualis ML, quemadmo&shy;<lb/>dum &aelig;quales &longs;unt inter&longs;e CA, CB. <!-- KEEP S--></s>
          <s id="s.000594">Dico &lpar;&longs;i concipiatur <lb/>&longs;olidum eius indolis, vt ductis rectis BA, LH cadant i&longs;t&aelig; <lb/>omnin&ograve; in &longs;olidi i&longs;tius &longs;uperficie; ip&longs;um ver&ograve; &longs;olidum, quod <lb/>&longs;it BADC, &longs;ecetur plano quolibet &aelig;quidi&longs;tante figur&aelig; <lb/>BCD&rpar; fore, vt &longs;ectio i&longs;ta KFEIK, &longs;it pror&longs;us &longs;imilis, &aelig;qua&shy;<lb/>li&longs;que alteri contermin&aelig; AEI; &longs;ed opportet, vt palam e&longs;t, <lb/>coeuntes ill&aelig; figur&aelig; non in eodem plano reperiantur. </s>
        </p>
        <p type="margin">
          <s id="s.000595"><margin.target id="marg133"/><emph type="italics"/>Tab.<emph.end type="italics"/> 6. <emph type="italics"/>fig.<emph.end type="italics"/> 7.</s>
        </p>
        <p type="main">
          <s id="s.000596">Cum duo plana inuicem parallela KIE, BCD &longs;ecent <lb/>alia duo inter&longs;e item parallela ACB, HML, erunt commu&shy;<lb/>nes &longs;ectiones, inter&longs;e omnes &aelig;quidi&longs;tantes rect&aelig; line&aelig; KI, <lb/>GF, ML, CB. <!-- KEEP S--></s>
          <s id="s.000597">Cum ver&ograve; ob naturam &longs;olidi, &longs;ectiones <lb/>BAC, IHM triangula &longs;int rectilinea, erit vt BC ad CA, <lb/>ita KI ad IA. <!-- KEEP S--></s>
          <s id="s.000598">Sunt autem priores inter&longs;e &aelig;quales, ergo &amp; <lb/>po&longs;trem&aelig; KI, AI inter&longs;e &aelig;quabuntur. </s>
          <s id="s.000599">Eademque ratione <lb/>&longs;unt &aelig;quales HG, GF: &amp; quoniam ob &longs;imilitudinem figu&shy;<lb/>rarum angulus BCD &aelig;quatur angulo ACD, &amp; angulus <lb/>BCD &aelig;qualis angulo KIE &lpar;nam etiam CD, IE &longs;unt rect&aelig; <lb/>&aelig;quidi&longs;tantes, cum nempe &longs;int communes &longs;ectiones plani <lb/>DCA &longs;ecantis duo &aelig;quidi&longs;tantia KIE, BCD&rpar; ergo cu&mtail; <lb/>angulus pariter ACD &aelig;quet angulum AIE, erunt anguli <lb/>KIE, AIE, et FGE, HGF &aelig;quales. </s>
          <s id="s.000600">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <pb pagenum="62" xlink:href="022/01/068.jpg"/>
        <p type="main">
          <s id="s.000601"><emph type="center"/>PROP. XVII. THEOR. XIII.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000602">II&longs;dem manentibus. </s>
          <s id="s.000603">Dico triangula ACB, LHM e&longs;&longs;&etail; <lb/>&longs;imilia. </s>
          <s id="s.000604">Sunt enim parallel&aelig; &amp;c. <!-- KEEP S--></s>
          <s id="s.000605">inter&longs;e tam rect&aelig; CB, <lb/>ML, qu&agrave;m CA, MH; ideo anguli ACB, HML inter&longs;e <lb/>&aelig;quabuntur, &amp; &longs;unt circa eos proportionalia latera, nem. <lb/></s>
          <s id="s.000606">pe BC ad CA, vt LM, MH; ergo con&longs;tat propo&longs;itum. </s>
        </p>
        <p type="main">
          <s id="s.000607"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000608"><emph type="center"/><emph type="italics"/>Simul con&longs;tat rectas AB, LH inter&longs;e &aelig;quidi&longs;tare.<emph.end type="italics"/><emph.end type="center"/></s>
        </p>
        <p type="main">
          <s id="s.000609"><emph type="center"/>PROP. XVIII. THEOR. XIV.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000610">II&longs;dem vt &longs;upra manentibus, ita tamen vt ACD &longs;it an&shy;<lb/>gulus rectus &lpar;&longs;ic enim DC perpendicularis erit duabus <lb/>AC, CB&rpar; Dico &longs;olidum huiu&longs;modi ad pri&longs;ma, cuius ba&longs;is <lb/>ABC, &amp; altitudo CD eandem habere rationem, quam &longs;o&shy;<lb/>lidum rotundum ortum ex rotatione figur&aelig; CAD circ&atail; <lb/>axem CD ad cylindrum genitum ex conuer&longs;ione rectan&shy;<lb/>guli AC in CD circa eundem axem. <lb/><arrow.to.target n="marg134"/></s>
        </p>
        <p type="margin">
          <s id="s.000611"><margin.target id="marg134"/><emph type="italics"/>Tab.<emph.end type="italics"/> 6. <emph type="italics"/>Fig.<emph.end type="italics"/> 8.</s>
        </p>
        <p type="main">
          <s id="s.000612">Compleatur ip&longs;um pri&longs;ma, &amp; &longs;it quidem AQDPBC, <lb/>quod &longs;ecetur vn&agrave; cum propo&longs;ito &longs;olido per quoduis pla&shy;<lb/>num ba&longs;i ACB &aelig;quidi&longs;tans: fiet in pri&longs;mate &longs;ectio trian&shy;<lb/>gulum OMN &longs;imile, &aelig;qualeque ip&longs;i ACB, &amp; in altero &longs;o&shy;<lb/>lido triangulum LHM eidem ACB &longs;imile. </s>
          <s id="s.000613">Triangulum <lb/>ACB pri&longs;matis ad <expan abbr="tri&atilde;gulum">triangulum</expan> idem &longs;olido propo&longs;ito com&shy;<lb/>mune, e&longs;t vt circulus radio CA de&longs;criptus ad circulum <lb/>eundem; Item triangulum NOM &longs;ectio pri&longs;matis e&longs;t ad <lb/>triangulum LHM &longs;ectionem propo&longs;iti &longs;olidi, vt circulus ex <lb/>radio MO de&longs;criptus ad circulum radio MH. </s>
          <s id="s.000614">Cum dein&shy;<lb/>de idem dicatur de alijs omnibus &longs;ectionibus pri&longs;matis, &amp; <pb pagenum="63" xlink:href="022/01/069.jpg"/>propo&longs;iti &longs;olidi erunt omnes &longs;imul prim&aelig;, qu&aelig; inter&longs;&etail; </s>
        </p>
        <p type="main">
          <s id="s.000615"><arrow.to.target n="marg135"/><lb/>&aelig;quales &longs;unt, ad omnes &longs;imul &longs;ecundas vt omnes terti&aelig;, <lb/>his partibus inter&longs;e &aelig;qualibus, ad omnes quartas; &longs;cilicet <lb/>erunt omnia triangula pri&longs;matis, &longs;eu ip&longs;um pri&longs;ma ad om&shy;<lb/>nia triangula propo&longs;iti &longs;olidi, &longs;eu ad ip&longs;um &longs;olidum, vt om&shy;<lb/>nes circuli eius cylindri, qui oritur ex conuer&longs;ione figur&aelig; <lb/>ADCA circa axem CD, hoc e&longs;t vt ip&longs;um &longs;olidum rotun&shy;<lb/>dum, &longs;eu cylindrus ad omnes &longs;imul circulos &longs;olidi rotundi <lb/>geniti ex rotatione figur&aelig; AHDCA circa axem <expan abbr="ips&utilde;">ipsum</expan> CD, <lb/>&longs;eu ad ip&longs;um propo&longs;itum &longs;olidum. </s>
          <s id="s.000616">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000617"><margin.target id="marg135"/><emph type="italics"/>lemm&aelig;<emph.end type="italics"/> 18. <emph type="italics"/>in <lb/>libro de dim. <lb/></s>
          <s id="s.000618">parab. </s>
          <s id="s.000619">Euang. <lb/><!-- REMOVE S-->T&etail;rricel.<emph.end type="italics"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000620"><emph type="center"/>PROP. XIX. THEOR. XV.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000621">ET rur&longs;us ip&longs;a manente figura patet, &longs;i ducantur HR, <lb/>LS parallel&aelig; MD, fore non &longs;olum figuram AHDPA, <lb/>&longs;imilem, ac &aelig;qualem BLDQB; ver&ugrave;m etiam APRHA ip&longs;i <lb/>BLSQB: Cum ita &longs;it, aio, eundem cylindrum ad &longs;oli&shy;<lb/>dum rotundum genitum, ex volutatione figur&aelig; APD cir&shy;<lb/>ca eundem axem CD eandem rationem habere, ac pri&longs;ma <lb/><expan abbr="pr&aelig;dict&utilde;">pr&aelig;dictum</expan>, cuius ba&longs;is ACB, altitudo AP ad &longs;olidum, quod <lb/>&longs;upere&longs;t ex ip&longs;o pri&longs;mate, dempto &longs;olido ACBLDHA. <!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000622">Nam ex pr&aelig;terita propo&longs;itione nouimus, dictum pri&longs;ma <lb/>ad &longs;olidum eius partem ACBLDHA e&longs;&longs;e vt cylindrus or&shy;<lb/>tus ex conuer&longs;ione rectanguli CP circa axem CD ad par&shy;<lb/>tem eius rotundum circa axem eundem CD conuer&longs;a fi&shy;<lb/>gura ADC, ergo per conuer&longs;ionem rationis, erit id quod <lb/>propo&longs;uimus. </s>
        </p>
        <p type="main">
          <s id="s.000623"><emph type="center"/>DEF. IV.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000624">QVodcunque ex dictis propo&longs;itis &longs;olidis vocetur ab <lb/>ea figura, iuxta quam intelligitur ortum. </s>
          <s id="s.000625">Scilicet <lb/>ACBLDHA dicatur &agrave; figura AHDCA, &amp; alte&shy;<lb/>rum, quod fuit re&longs;iduum pr&aelig;dictum dicatur &agrave; figura AH&shy;<lb/>DPA. <!-- KEEP S--></s>
        </p>
        <pb pagenum="64" xlink:href="022/01/070.jpg"/>
        <p type="main">
          <s id="s.000626"><emph type="center"/>PROP. XX. THEOR. XVI.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000627">SI &agrave; quibu&longs;cunque figuris fuerint duo &longs;olida, h&aelig;c inter&shy;<lb/><arrow.to.target n="marg136"/><lb/>&longs;e erunt vt &longs;olida alia genita ex conuer&longs;ione illarum <lb/>figurarum circa communem &longs;ectionem &longs;imilium, &aelig;qua&shy;<lb/>lium, ac inter&longs;e coeuntium figurarum. </s>
        </p>
        <p type="margin">
          <s id="s.000628"><margin.target id="marg136"/><emph type="italics"/>Tab.<emph.end type="italics"/> 6. <emph type="italics"/>Fig.<emph.end type="italics"/> 9.</s>
        </p>
        <p type="main">
          <s id="s.000629">Solidum &agrave; figura ABC &longs;it CAFDBC, &amp; quod e&longs;t &agrave; fi&shy;<lb/>gura GLH e&longs;to HGILH. </s>
          <s id="s.000630">Dico illud ad hoc &longs;olidum e&longs;&longs;e <lb/>vt rotundum natum ex conuer&longs;ione figur&aelig; ABC circ&atail; <lb/>axem CE ad rotundum ortum ex <expan abbr="c&otilde;uer&longs;ione">conuer&longs;ione</expan> figur&aelig; GLH <lb/>circa axem HL. <!-- KEEP S--></s>
          <s id="s.000631">Opportet tamen angulos ACF, GHI <lb/>&aelig;quales e&longs;&longs;e. </s>
          <s id="s.000632">Intelligantur pri&longs;mata triangularia, quorum <lb/>ba&longs;es ACF, GHI, &amp; altitudines CE, HL; hoc e&longs;t &longs;int ip&longs;a <lb/>&longs;olida pri&longs;matica AFCEBD, GIHLMK. </s>
          <s id="s.000633">Solidum &agrave; figu&shy;<lb/><arrow.to.target n="marg137"/><lb/>ra ABC ad pri&longs;ma AFCEBD habet eandem rationem, <lb/>quam &longs;olidum rotundum ortum ex conuer&longs;ione &longs;igur&aelig; <lb/>ABC circa axem CE ad cylindrum natum ex rotatione <lb/>ABEC circa eundem axem CE; hic ver&ograve; cylindrus ad cy&shy;<lb/>lindrum alium natum ex rotatione rectanguli GMLH cir&shy;<lb/>ca axem HL e&longs;t vt pri&longs;ma, cuius ba&longs;is ACF, altitudineque <lb/>CE ad alterum pri&longs;ma ba&longs;em habens GHI &longs;imilem ip&longs;i CF <lb/>&lpar;nam circa angulos &aelig;quales H, C &longs;unt latera etiam pro&shy;<lb/>portionalia, nempe &aelig;qualia&rpar; &amp; altitudinem HL. <!-- KEEP S--></s>
          <s id="s.000634">Solidum <lb/>pr&aelig;terea, hoc e&longs;t pri&longs;ma GKHM ad &longs;olidum, quod e&longs;t &agrave; <lb/><arrow.to.target n="marg138"/><lb/>plano GLH habet eandem rationem, ac cylindrus, qui fit <lb/>ex conuer&longs;ione rectanguli HM circa axem HL ad &longs;olidum <lb/>rotundum ortum ex circumactione figur&aelig; GLH circa ip&shy;<lb/>&longs;um axem HL, ergo ex &aelig;quali erit &longs;olidum &agrave; figura ABC <lb/>ad &longs;olidum &agrave; figura GLH, vt rotundum ex rotatione figu&shy;<lb/>r&aelig; ABC circa axem CE ad rotundum alterum ex conuer&shy;<lb/>&longs;ione alterius figur&aelig; GLH circa axem HL. <!-- KEEP S--></s>
          <s id="s.000635">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <pb pagenum="65" xlink:href="022/01/071.jpg"/>
        <p type="margin">
          <s id="s.000636"><margin.target id="marg137"/>18. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000637"><margin.target id="marg138"/><emph type="italics"/>Ex eadem.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000638"><emph type="center"/>PROP. XXI. THEOR. XVI.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000639">PRopo&longs;itis ij&longs;dem &longs;olidis, erunt inter &longs;e, vt momenta fi&shy;<lb/>gurarum a quibus &longs;unt, qu&aelig; tamen figur&aelig; &longs;u&longs;pen&longs;&aelig; <lb/>&longs;int ex longitudinibus deductis ab ip&longs;arum grauitatu&mtail; <lb/>centris v&longs;que ad coeuntium figurarum communes illas &longs;e&shy;<lb/>ctiones. </s>
        </p>
        <p type="main">
          <s id="s.000640">Figur&aelig;, &agrave; quibus &longs;unt &longs;olida, ponantur ABC, GLH, <expan abbr="c&etilde;-">cen&shy;<lb/></expan><arrow.to.target n="marg139"/><lb/>tra grauitatum illarum M, N; axes, &longs;iue communes &longs;ectio&shy;<lb/>nes coeuntium binarum inter&longs;e &longs;imilium, ac &aelig;qualium fi&shy;<lb/>gurarum &agrave; quibus dicuntur ip&longs;a &longs;olida; &amp; demum MO, NP <lb/>perpendiculares &longs;int ab ip&longs;is centris ad illas communes &longs;e&shy;<lb/>ctiones deduct&aelig; CE, HL. Dico, &longs;olidum &agrave; plana figur&atail; <lb/>ABC ad &longs;olidum a plana GHL eandem habere rationem, <lb/>ac momentum figur&aelig; ABC pendentis ex MO ad momen&shy;<lb/><arrow.to.target n="marg140"/><lb/>tum alterius figur&aelig; &longs;u&longs;pen&longs;&aelig; ex NP, &longs;unt enim h&aelig;c &longs;oli&shy;<lb/>da inter&longs;e, vt rotunda, quorum genetrices figur&aelig; ABC, <lb/>GLH circa axes CE, HL, huiu&longs;modi ver&ograve; &longs;olida &longs;unt vt <lb/><arrow.to.target n="marg141"/><lb/>momenta propo&longs;ita; ergo &longs;olidum &agrave; plana figura ABC ad <lb/>&longs;olidum &agrave; plana GLH, erit vt momentum figur&aelig; ABC <lb/>&longs;u&longs;pen&longs;&aelig; ex MO ad momentum GLH pendentis ex NP. <lb/><!-- KEEP S--></s>
          <s id="s.000641">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000642"><margin.target id="marg139"/><emph type="italics"/>Tab.<emph.end type="italics"/> 6. <emph type="italics"/>fig.<emph.end type="italics"/> 10.</s>
        </p>
        <p type="margin">
          <s id="s.000643"><margin.target id="marg140"/><emph type="italics"/>pr.<emph.end type="italics"/> 20. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000644"><margin.target id="marg141"/><emph type="italics"/>Ter. <!-- REMOVE S-->lem.<emph.end type="italics"/> 31. <lb/><emph type="italics"/>in libro </s>
          <s id="s.000645">di&shy;<lb/>men. parabol&aelig;. <emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000646"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000647"><emph type="italics"/>Cum ip&longs;a illa momenta nectantur ex rationibus figurarum <lb/><arrow.to.target n="marg142"/><lb/>ABC, GLH, &amp; ex longitudinibus, ex quibus pendent ip&longs;&aelig; fi&shy;<lb/>gura &lpar;nam habentur vt grauia&rpar; ex ij&longs;dem etiam rationibus <lb/>componentur &longs;olida, qua &longs;unt ab ip&longs;is figuris&mdash;<emph.end type="italics"/></s>
        </p>
        <pb pagenum="66" xlink:href="022/01/072.jpg"/>
        <p type="margin">
          <s id="s.000648"><margin.target id="marg142"/>Ex mechani&shy;<lb/>cis,</s>
        </p>
        <p type="main">
          <s id="s.000649"><emph type="center"/>PROP. XXII. THEOR. XVII.<emph.end type="center"/><lb/><arrow.to.target n="marg143"/><!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000650"><margin.target id="marg143"/><emph type="italics"/>Tab.<emph.end type="italics"/> 7. <emph type="italics"/>fig.<emph.end type="italics"/> 1.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000651">IMagines velocitatum, &longs;eu &longs;patia, qu&aelig; curruntur accele&shy;<lb/>ratis motibus, &longs;unt vt &longs;olida ab imaginibus &longs;implicium <lb/>motuum, ex quibus ip&longs;i gignuntur accelerati. </s>
        </p>
        <p type="main">
          <s id="s.000652">Sint imagines &longs;implicium motuum ABC, GLH, &amp; &longs;oli&shy;<lb/>da ab ip&longs;is imaginibus &lpar;angulis ACQ, GHD &longs;emper re&shy;<lb/>ctis, aut &longs;altem &aelig;qualibus&rpar; intelligantur ABCRQ, GLHD. <lb/>Dico, vt &longs;unt inter&longs;e i&longs;ta &longs;olida, &longs;ic e&longs;&longs;e homolog&egrave; &longs;patium <lb/>exactum tempore AC motu accelerato ex &longs;implici motu <lb/>imaginis ABC ad &longs;patium tran&longs;actum tempore GH motu <lb/>item accelerato ex &longs;implici imagine priori homogene&atail; <lb/>GLH: &longs;ecetur &longs;olidum ABCRQ plano &aelig;quidi&longs;tanti QCR, <!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000653"><arrow.to.target n="marg144"/><lb/>quod faciat in &longs;olido ip&longs;o &longs;ectionem TSVX: erit h&aelig;c figu&shy;<lb/>ra pror&longs;us &longs;imilis, ac &aelig;qualis contermin&aelig; ABVI; quare <lb/><arrow.to.target n="marg145"/><lb/>cum in accelerato motu velocitas, qu&aelig; habetur momen&shy;<lb/>to C ad velocitatem momento S &longs;it vt imago ABC &longs;im&shy;<lb/><arrow.to.target n="marg146"/><lb/>plex ad &longs;egmentum eius ABVS: erit etiam QCR &aelig;qualis <lb/>ABC ad &longs;ectionem &longs;olidi TSVX, qu&aelig; &aelig;quatur ABVS, vt <lb/>illa eadem velocitas momento C mobili inh&aelig;rens ad ve&shy;<lb/>locitatem momento S alterius accelerati motus. </s>
          <s id="s.000654">E&longs;t au&shy;<lb/>tem &longs;ectio TSVX ad libitum &longs;umpta; ergo &longs;olidum ABC&shy;<lb/><arrow.to.target n="marg147"/><lb/>QR pote&longs;t &longs;umi merito vt imago velocitatum accelerati <lb/><arrow.to.target n="marg148"/><lb/>motus, cuius &longs;implex imago ABC: &amp; eodem modo &longs;oli&shy;<lb/>dum alterum vicem geret imaginis velocitatum alterius <lb/>motus ex &longs;implici imagine GLH, itaque erit ob homoge&shy;<lb/>neitatem &longs;patium tran&longs;actum motu accelerato iuxta &longs;im&shy;<lb/>plicem imaginem ABC ad &longs;patium tran&longs;actum motu ac&shy;<lb/>celerato iuxta &longs;implicem imaginem GLH, <expan abbr="t&etilde;poribus">temporibus</expan> AC, <lb/>GH, vt &longs;olidum ABCQR ad ALHD, </s>
        </p>
        <pb pagenum="67" xlink:href="022/01/073.jpg"/>
        <p type="margin">
          <s id="s.000655"><margin.target id="marg144"/>16. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000656"><margin.target id="marg145"/>4. <emph type="italics"/>huius,<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000657"><margin.target id="marg146"/>16. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000658"><margin.target id="marg147"/><emph type="italics"/>Def.<emph.end type="italics"/> .3. <emph type="italics"/>primi <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000659"><margin.target id="marg148"/><emph type="italics"/>&amp; Def.<emph.end type="italics"/> 1. <emph type="italics"/>hu&shy;<lb/>ius vn&agrave; cum <lb/>pr.<emph.end type="italics"/> 4. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000660"><emph type="center"/>PROP. XXII. THEOR. XVIII.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000661">SInt nunc CE, HL communes &longs;ectiones imaginum &longs;im&shy;<lb/><arrow.to.target n="marg149"/><lb/>plicium ABC, GLH, &longs;i extenderentur cum &longs;ujs &aelig;qua&shy;<lb/>libus, ac &longs;imilibus coeuntibus figuris. </s>
          <s id="s.000662">E&longs;to pariter M cen&shy;<lb/>trum grauitatis imaginis ABC, et N grauitatis alterius ima&shy;<lb/>ginis GLH; actis dem&ugrave;m MO, NP perpendicularibus ad <lb/>ip&longs;as CE, HL. Dico, &longs;patium accelerati motus ab imagine <lb/>&longs;implici ABC ad <expan abbr="&longs;pati&utilde;">&longs;patium</expan> accelerati alterius motus ab ima&shy;<lb/>gine &longs;implici GLH componi ex ratione imaginis ABC ad <lb/>imaginem GLH, &amp; ex ea perpendicularis MO ad perpen&shy;<lb/>dicularem NP. <!-- KEEP S--></s>
          <s id="s.000663">Cum h&aelig;c ip&longs;a &longs;patia &longs;int o&longs;ten&longs;a, vt &longs;oli&shy;<lb/><arrow.to.target n="marg150"/><lb/>da &agrave; figuris ABC, GLH; h&aelig;c ver&ograve; &longs;unt vt momenta ip&longs;a&shy;<lb/><arrow.to.target n="marg151"/><lb/>rum figurarum &longs;u&longs;pen&longs;arum ex MO, NP. <!-- KEEP S--></s>
          <s id="s.000664">Ergo quemad&shy;<lb/>modum momenta i&longs;ta nectuntur ex rationibus figurarum <lb/><arrow.to.target n="marg152"/><lb/>tanquam magnitudinum ABC ad LGH, &amp; di&longs;tantiarum <lb/>MO ad NP, ita pariter ex his nectentur propo&longs;ita &longs;patia. </s>
        </p>
        <p type="margin">
          <s id="s.000665"><margin.target id="marg149"/><emph type="italics"/>Tab.<emph.end type="italics"/> 7. <emph type="italics"/>Fig.<emph.end type="italics"/> 2.<!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000666"><margin.target id="marg150"/>21. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000667"><margin.target id="marg151"/>20. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000668"><margin.target id="marg152"/><emph type="italics"/>Ex mechani&shy;<lb/>cis.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000669"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000670"><emph type="italics"/>Patet communes &longs;ectiones CE, HL e&longs;&longs;e &aelig;quidi&longs;tantes ap&shy;<lb/>plicatis AB, HL, qu&aelig; in imaginibus &longs;umuntur perpendicula&shy;<lb/>res rectis AC, GH. nam HL est recta, in quam coeunt figura <emph.end type="italics"/><lb/><arrow.to.target n="marg153"/><lb/><emph type="italics"/>plan&aelig; &longs;imiles, ac &aelig;quales.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000671"><margin.target id="marg153"/><emph type="italics"/>Pr<emph.end type="italics"/> 2. <emph type="italics"/>prim&atail; <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000672"><emph type="center"/>PROP. XXIV. THEOR. XIX.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000673">SI imagines &longs;implicium motuum fuerint &longs;imiles, &longs;imili&shy;<lb/>terque &longs;u&longs;pen&longs;&aelig;, imagines velocitatum accelerato&shy;<lb/>rum motuum erunt in triplicata ratione temporum &longs;impli&shy;<lb/>cium motuum, aut in triplicata homologarum, vel extre&shy;<lb/>marum velocitatum eorundem &longs;implicium motuum. </s>
        </p>
        <p type="main">
          <s id="s.000674">Cum centra grauitatum &longs;imilium imaginum, &longs;eu figu&shy;<lb/><arrow.to.target n="marg154"/><pb pagenum="68" xlink:href="022/01/074.jpg"/>rarum, &longs;int puncta in ij&longs;dem figuris &longs;imiliter po&longs;ita, ponun&shy;<lb/>tur ver&ograve; imagines &longs;imiliter &longs;u&longs;pen&longs;&aelig;, ergo &longs;equitur ip&longs;as <lb/>longitudines e&longs;&longs;e vt latera homologa dictarum imaginum, <lb/>&longs;cilicet vt tempus AC ad tempus FG, vel vt extrem&aelig; ve&shy;<lb/>locitates BC ad KE. <!-- KEEP S--></s>
          <s id="s.000675">Quamobrem imagines ip&longs;&aelig;, cum &longs;int <lb/>in duplicata ratione laterum homologorum, &longs;i huic dupli&shy;<lb/>cat&aelig; addatur alia ratio &longs;imilis rationi longitudinum, fiet <lb/>ratio imaginum velocitatum, &longs;eu &longs;patiorum acceleratorum <lb/>motuum ex &longs;implicibus illis deriuantium triplicata tempo&shy;<lb/>rum, vel extremarum velocitatum &longs;implicium motuum. </s>
        </p>
        <p type="margin">
          <s id="s.000676"><margin.target id="marg154"/><emph type="italics"/>Tab.<emph.end type="italics"/> 7. <emph type="italics"/>Fig.<emph.end type="italics"/> 3.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000677"><emph type="center"/>PROP. XXV. THEOR. XX.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000678">SI ver&ograve; &longs;implices motus extiterint &longs;imiles, <expan abbr="&aelig;qualibu&longs;q;">&aelig;qualibu&longs;que</expan> <lb/>temporibus ab&longs;oluantur, imagines acceleratorum <lb/><arrow.to.target n="marg155"/><lb/>motuum erunt in &longs;ola ratione amplitudinum imaginum <lb/>&longs;implicium. </s>
        </p>
        <p type="margin">
          <s id="s.000679"><margin.target id="marg155"/><emph type="italics"/>Tab.<emph.end type="italics"/> 7. <emph type="italics"/>fig.<emph.end type="italics"/> 4.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000680">Sint imagines &longs;imilium, ac &longs;implicium motuum BAC, <lb/>KFG, quarum grauitatis centra D, H, erunt ex hypothe&longs;i <lb/><arrow.to.target n="marg156"/><lb/>tempora AC, FG &aelig;qualia; &amp; ideo &longs;patia, &longs;cilicet imagines <lb/><arrow.to.target n="marg157"/><lb/>velocitatum BAC, KFG habebunt eandem rationem, <lb/>quam &longs;umm&aelig;, aut extrem&aelig; motuum &longs;implicium velocita&shy;<lb/>tes, &longs;cilicet, quam amplitudines imaginum, &longs;eu gene&longs;um: <lb/>&longs;unt ver&ograve; di&longs;tanti&aelig; DE, HI pariter &aelig;quales, quia AC, FG <lb/><arrow.to.target n="marg158"/><lb/>&aelig;quales &longs;unt; ergo cum &longs;patia acceleratorum motuum ne&shy;<lb/>ctantur ex imaginibus &longs;implicium motuum ABC, KFG, &amp; <lb/>ex di&longs;tantijs DE ad HI, liquet ip&longs;a &longs;patia e&longs;&longs;e in vnica, &longs;o&shy;<lb/>laque ratione amplitudinum BC, KG, aut amplitudinum <lb/>gene&longs;um. </s>
        </p>
        <p type="margin">
          <s id="s.000681"><margin.target id="marg156"/>8 <emph type="italics"/>primi huius<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000682"><margin.target id="marg157"/>2 <emph type="italics"/>primi huius<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000683"><margin.target id="marg158"/>23. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000684"><emph type="center"/>PROP. XXVI. THEOR. XX.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000685">AT &longs;i &longs;implicium, &longs;imiliumque motuum fuerint imagi&shy;<lb/>nes &aelig;qu&egrave; ampl&aelig;, imagines acceleratorum motuum, <pb pagenum="69" xlink:href="022/01/075.jpg"/>&longs;iue tempora erunt in duplicata ratione temporum i&longs;to&shy;<lb/>rum, vel illorum motuum. </s>
        </p>
        <p type="main">
          <s id="s.000686">Amplitudines imaginum &longs;implicium, velocitatumque <lb/><arrow.to.target n="marg159"/><lb/>BAC, KFG &longs;unto BC, KG, qu&aelig; &aelig;quales &longs;int. </s>
          <s id="s.000687">Dico &longs;pa&shy;<lb/>tia acceleratorum motuum ab illis &longs;implicibus imaginibus <lb/>fore in duplicata ratione temporum AC ad FG &lpar;qu&etail; &longs;em&shy;<lb/>per in acceleratis ponuntur eadem, ac in &longs;implicibus, nec <lb/>aliter e&longs;&longs;e po&longs;&longs;unt.&rpar; Vt FG ad GK, ita &longs;it AC ad CL, &amp; <lb/>intelligatur LAC imago alterius motus &longs;imilis motui, cuius <lb/>imago BAC, vel KFG. <!-- KEEP S--></s>
          <s id="s.000688">Facil&egrave; demon&longs;trabitur ip&longs;am fi&shy;<lb/><arrow.to.target n="marg160"/><lb/>guram LAC &longs;imilem e&longs;&longs;e ip&longs;i KFG, &amp; ad BAC eande&mtail; <lb/>habere rationem, quam LC ad BC. <!-- KEEP S--></s>
          <s id="s.000689">Cum ergo imago BAC <lb/>ad imaginem KFG componatur ex ratione imaginis BAC <lb/>ad LAC &lpar;qu&aelig; &longs;unt vt BC ad CL&rpar; &amp; ex ratione imagi&shy;<lb/>nis ALC ad imaginem KFG, qu&aelig; &longs;unt in ratione compo&shy;<lb/>&longs;ita LC ad KG, et AC ad FG: priores ver&ograve; du&aelig; rationes <lb/>componunt vnicam &aelig;qualitatis, ergo relinquitur, imagi&shy;<lb/>nem BAC ad imaginem KFG e&longs;&longs;e vt AC ad FG; &longs;patium <lb/>ver&ograve; accelerati motus ex &longs;implici imagine BAC ad accele&shy;<lb/>ratum ex &longs;implici KFG nectitur ex ratione imaginum &longs;im&shy;<lb/><arrow.to.target n="marg161"/><lb/>plicium ip&longs;arum, &amp; ex ea di&longs;tantiarum DE, HI &agrave; centris <lb/>grauitatum deductarum D, H, et &longs;unt h&aelig; rect&aelig; in eadem <lb/>ratione, ac altitudines AC, FG &lpar;nam in figuris, &longs;eu imagi&shy;<lb/>nibus &longs;imilium motuum BAC, LAC centra grauitatum <lb/>&longs;unt in eadem recta parallela ip&longs;i BC, &amp; in LAC, KFG <lb/>&longs;unt in punctis &longs;imiliter po&longs;itis, adeo ut, &longs;icut po&longs;itum e&longs;t, <lb/>ratio ip&longs;arum di&longs;tantiarum in ip&longs;is figuris LAC, KFG, &longs;eu <lb/>BAC, KEG eadem &longs;it, ac laterum homologorum LC ad <lb/>KG, vel AC ad FG&rpar; ergo &longs;patium accelerati motus ex &longs;im&shy;<lb/>plici imagine KFG, erit vt quadratum ex AC ad quadra&shy;<lb/>tum ex FG, nempe in duplicata ratione temporum &longs;impli&shy;<lb/>cium motuum. </s>
        </p>
        <pb pagenum="70" xlink:href="022/01/076.jpg"/>
        <p type="margin">
          <s id="s.000690"><margin.target id="marg159"/><emph type="italics"/>Tab.<emph.end type="italics"/> 7. <emph type="italics"/>fig.<emph.end type="italics"/> 5.<!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000691"><margin.target id="marg160"/><emph type="italics"/>Def.<emph.end type="italics"/> 7. <emph type="italics"/>primi <lb/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000692"><margin.target id="marg161"/>23. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000693"><emph type="center"/>PROP. XXVII. THEOR. XXI.<emph.end type="center"/><lb/><arrow.to.target n="marg162"/><!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000694"><margin.target id="marg162"/><emph type="italics"/>Tab.<emph.end type="italics"/> 7. <emph type="italics"/>fig.<emph.end type="italics"/> 6.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000695">DEm&ugrave;m &longs;i &longs;int imagines, qu&aelig;cunque velocitatum &longs;im&shy;<lb/>plicium, &longs;imiliumque motuum, imagines accelera&shy;<lb/>torum motuum, &longs;eu &longs;patia ijs motibus exacta componen&shy;<lb/>tur ex duplicata temporum ratione, &amp; ex ea amplitudi&shy;<lb/>num, vel applicatarum homologarum earundem imagi&shy;<lb/>num. </s>
        </p>
        <p type="main">
          <s id="s.000696">Imagines &longs;imilium, &longs;impliciumque motuum &longs;int BAC, <lb/>KFG. Dico, imagines acceleratorum motuum ab illis &longs;im&shy;<lb/>plicibus deriuantium habere rationem compo&longs;itam ex du&shy;<lb/>plicata temporum AC ad FG, &amp; amplitudinum imaginum <lb/>dictarum, vel gene&longs;um. </s>
          <s id="s.000697">Intelligatur alius &longs;imilis motus, <lb/>cuius velocitatum imago &longs;it DFG &aelig;qu&egrave;ampla, ac homo&shy;<lb/>genea ip&longs;i BCA; nimirum &longs;it DG &aelig;qualis BC. <!-- KEEP S--></s>
          <s id="s.000698">Quoniam <lb/>imago accelerati motus ex &longs;implici imagine BA ad imagi&shy;<lb/>nem accelerati ex &longs;implici imagine KFG componitur ex <lb/>ratione imaginis accelerati motus, cuius &longs;implex imago <lb/>BAC ad imaginem accelerati motus ex &longs;implici DFG, &amp; <lb/>ex imagine huius accelerati motus ad accelerati imaginem <lb/>&agrave; &longs;implici KFG; e&longs;t autem prior ratio imaginum, &longs;eu &longs;pa&shy;<lb/>tiorum acceleratis motibus percur&longs;orum ip&longs;a temporum </s>
        </p>
        <p type="main">
          <s id="s.000699"><arrow.to.target n="marg163"/><lb/>duplicata AC ad FG, &amp; altera dictarum imaginum, &longs;eu <lb/>&longs;patiorum item acceleratis motibus confectorum, &amp; quo&shy;<lb/><arrow.to.target n="marg164"/><lb/>rum &longs;implices imagines &longs;unt DFG, KFG, e&longs;t eadem, ac ra&shy;<lb/>tio amplitudinum DG, &longs;eu BC ad KG. </s>
          <s id="s.000700">Ergo cum i&longs;t&aelig; <lb/>amplitudines &longs;int e&aelig;dem, ac ill&aelig; gene&longs;um, con&longs;tat propo&shy;<lb/>&longs;itam rationem acceleratorum motuum ex &longs;implicibus <lb/>imaginibus BAC, KFG habere rationem compo&longs;itam ex <lb/>duplicata temporum AC ad FG, &amp; ex ea amplitudinum <lb/>imaginum &longs;implicium BC ad KG, &longs;eu amplitudinum gene&shy;<lb/>&longs;um. </s>
          <s id="s.000701">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <pb pagenum="71" xlink:href="022/01/077.jpg"/>
        <p type="margin">
          <s id="s.000702"><margin.target id="marg163"/><emph type="italics"/>Pr.<emph.end type="italics"/> 26 <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000703"><margin.target id="marg164"/>25. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000704"><emph type="center"/>PROP. XXVIII. THEOR. XXII.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000705">SI gene&longs;es &longs;imilium, &longs;impliciumque motuum fuerint <lb/>&aelig;qu&egrave;ampl&aelig;, imagines acceleratorum motuum erunt <lb/>in duplicata ratione temporum, vel altitudinum ip&longs;arum <lb/>gene&longs;um. </s>
        </p>
        <p type="main">
          <s id="s.000706">Gene&longs;es &longs;imilium, ac &longs;implicium motuum &longs;unto ABC, <lb/><arrow.to.target n="marg165"/><lb/>DEF, quarum amplitudines &aelig;quales &longs;int AC, DF. Dico, <lb/>imagines, &longs;iue &longs;patia acceleratorum motuum e&longs;&longs;e in dupli&shy;<lb/>cata ratione temporum, vel altitudinum BC ad EF. </s>
          <s id="s.000707">Cum <lb/>AC, DF &longs;int gradus velocitatum in extremitatibus &longs;impli&shy;<lb/>cium decur&longs;uum, etiam imagines velocitatum, iuxta ip&longs;as <lb/>gene&longs;es, qu&aelig; &longs;int inter&longs;e homogene&aelig;, erunt &aelig;qu&egrave;ampl&aelig;, <lb/>&amp; &longs;unt &longs;imilium motuum; ergo imagines acceleratorum <lb/><arrow.to.target n="marg166"/><lb/>motuum, iuxta &longs;implices illas gene&longs;es, aut imagines &aelig;qu&egrave;&shy;<lb/>amplas erunt in duplicata ratione temporum: &longs;unt autem <lb/>imagines velocitatum &aelig;qu&egrave;ampl&aelig;, &longs;imiliumque motuum, <lb/><arrow.to.target n="marg167"/><lb/>hoc e&longs;t &longs;patia BC ad EF vt ip&longs;a tempora; ergo &longs;patia acce&shy;<lb/>leratorum, propo&longs;itorumque motuum erunt in ratione du&shy;<lb/>plicata altitudinum BC, EF &longs;implicium gene&longs;um, ABC, <lb/>DEF. <!-- KEEP S--></s>
          <s id="s.000708">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000709"><margin.target id="marg165"/><emph type="italics"/>Tab.<emph.end type="italics"/> 7. <emph type="italics"/>Fig.<emph.end type="italics"/> 7.</s>
        </p>
        <p type="margin">
          <s id="s.000710"><margin.target id="marg166"/>26. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000711"><margin.target id="marg167"/>26. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000712"><emph type="center"/>PROP. XXIX. THEOR. XXIII.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000713">SI gene&longs;es &longs;imilium, &longs;impliciumque motuum fuerint <lb/>&aelig;qu&egrave;alt&aelig;, imagines, &longs;iue &longs;patia, acceleratorum mo&shy;<lb/>tuum erunt vt tempora, vel reciproc&egrave; vt amplitudines ge&shy;<lb/>ne&longs;um ip&longs;orum &longs;implicium motuum. </s>
        </p>
        <p type="main">
          <s id="s.000714">Gene&longs;es &longs;imilium, &longs;impliciumque motuum, ac inter&longs;e <lb/><arrow.to.target n="marg168"/><lb/>homogene&aelig; &longs;int BAC, DEF, qu&aelig; habeant altitudines <lb/>AC, EF &aelig;quales. </s>
          <s id="s.000715">Dico, imagines acceleratorum motuum <lb/>e&longs;&longs;e inter &longs;e, vt tempora dictorum &longs;implicium motuum, vel <lb/>reciproc&egrave; vt amplitudines ip&longs;arum gene&longs;um. </s>
          <s id="s.000716">Concipian-<pb pagenum="72" xlink:href="022/01/078.jpg"/>tur imagines velocitatum <expan abbr="&longs;implici&utilde;">&longs;implicium</expan> motuum, &longs;cilicet GHI <lb/>iuxta gene&longs;im BAC, et MKL iuxta <expan abbr="alter&atilde;">alteram</expan> gene&longs;im DEF, &amp; <lb/>quia, vtpot&egrave; homogene&etail;, &longs;unt inter &longs;e vt &longs;patia &etail;qualia AC <lb/>ad EF, <expan abbr="er&utilde;t">erunt</expan> ip&longs;&aelig; imagines &etail;quales inter &longs;e, <expan abbr="c&utilde;">cum</expan> ver&ograve; ob &longs;imili <lb/><expan abbr="tudin&etilde;">tudinem</expan> motuum e&aelig; ip&longs;&aelig; imagines nectantur ex rationibus <lb/>GI ad ML, &amp; ex ea, quam habet HI ad KL, &longs;equitur e&longs;&longs;e <lb/>GI ad ML, vt KL ad IH, &amp; demum quia acceleratorum <lb/>motuum &longs;patia &agrave; &longs;implicibus imaginibus GHI, MKL ne&shy;<lb/>ctuntur ex duplicata temporum HI ad KL, &amp; ex ea ampli&shy;<lb/><arrow.to.target n="marg169"/><lb/>tudinum GI ad ML, &longs;iue ex ea, quam habet KL ad HI, re&shy;<lb/>linquitur, &longs;patia acceleratis illis motibus confecta e&longs;&longs;e in <lb/>&longs;ola, <expan abbr="vnicaq;">vnicaque</expan> ratione temporum HI ad KL, vel in ei &etail;qua&shy;<lb/>li ratione, reciproca amplitudinum imaginum ML ad GI, <lb/>vel gene&longs;um DF ad BC. <!-- KEEP S--></s>
          <s id="s.000717">Quod &amp;c, </s>
        </p>
        <p type="margin">
          <s id="s.000718"><margin.target id="marg168"/><emph type="italics"/>Tab.<emph.end type="italics"/> 7. <emph type="italics"/>fig.<emph.end type="italics"/> 8.</s>
        </p>
        <p type="margin">
          <s id="s.000719"><margin.target id="marg169"/>27. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000720"><emph type="center"/>PROP. XXX. THEOR. XXIV.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000721">QV&aelig;cunque fuerint gene&longs;es &longs;imilium, &longs;impliciumque <lb/>motuum, dum inter&longs;e homogene&aelig;, &longs;patia accelera&shy;<lb/>tis motibus ex illis &longs;implicibus exacta nectentur <lb/>ex duplicata ratione altitudinum, &amp; reciproca amplitudi&shy;<lb/>num earundem &longs;implicium gene&longs;um, </s>
        </p>
        <p type="main">
          <s id="s.000722">Sint qu&aelig;cunque &longs;imilium motuum gene&longs;es BAC, KFG. <lb/><arrow.to.target n="marg170"/><lb/>Dico, &longs;patia acceleratorum motuum, ab ijs &longs;implicibus de&shy;<lb/>riuantium, componi ex duplicata ratione altitudinum AC <lb/>ad FG, &amp; ex ratione extremarum velocitatum, &longs;eu ampli&shy;<lb/>tudinum reciproc&egrave; &longs;umptarum ip&longs;arum gene&longs;um: e&longs;to alia <lb/>gene&longs;is DFG illis homogenea, &amp; motu pariter &longs;imilis cum <lb/>ij&longs;dem gene&longs;ibus. </s>
          <s id="s.000723">Eadem &longs;it amplitudine &aelig;qualis BAC, <lb/>&amp; altitudo eius &longs;it FG, &longs;patia acceleratorum motuum ex <lb/><arrow.to.target n="marg171"/><lb/>&longs;implicibus gene&longs;ibus &aelig;quales amplitudines habentibus, <lb/>&amp; &longs;imilium motuum BAC, DFG &longs;unt in duplicata ratione <lb/>rectarum, &longs;eu altitudinum AC ad FG, &amp; &longs;patia accelera&shy;<lb/><arrow.to.target n="marg172"/><pb pagenum="73" xlink:href="022/01/079.jpg"/>torum motuum ex &longs;implicibus gene&longs;ibus, qu&aelig; &longs;int in ea&shy;<lb/>dem altitudine DFG, KFG, &longs;unt in reciproca ratione am&shy;<lb/>plitudinum, &longs;eu primarum velocitatum KG ad DG, vel <lb/>BC; ex &aelig;quali igitur &longs;patia acceleratorum motuum ex <lb/>propo&longs;itis &longs;implicibus gene&longs;ibus BAC, KFG nectentur ex <lb/>ratione duplicata altitudinum AC ad FG, &amp; reciproca <lb/>amplitudinum KG ad BC earundem gene&longs;um BAC, <lb/>KFG. <!-- KEEP S--></s>
          <s id="s.000724">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000725"><margin.target id="marg170"/><emph type="italics"/>Tab.<emph.end type="italics"/> 7. <emph type="italics"/>fig.<emph.end type="italics"/> 6.<!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000726"><margin.target id="marg171"/>28. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000727"><margin.target id="marg172"/>29. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000728"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000729"><emph type="italics"/>At quia in &longs;patijs, qu&aelig; accelerato motu peraguntur; non <lb/>&longs;eruatur ratio altitudinum gene&longs;um &longs;implicium, ex quo ori&shy;<lb/>tur in hac methodo qu&aelig;dam percipiendi difficultas; ideo &longs;e&shy;<lb/>quenti problemate, alij&longs;que iam notis veritatibus, rem plan&egrave; <lb/>illu&longs;trabimus, ac &longs;imul doctrina v&longs;um trademus.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000730"><emph type="center"/>PROP. XXXI. PROB. VI.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000731">EX datis &longs;patijs accelerato motu confectis, cogniti&longs;&shy;<lb/>que primis, aut po&longs;tremis &longs;imilium, &longs;impliciumque <lb/>motuum velocitatibus, reperire tempora ip&longs;orum de&shy;<lb/>cur&longs;uum. </s>
        </p>
        <p type="main">
          <s id="s.000732">Spatia motibus acceleratis exacta &longs;unt C, D, &amp; velo&shy;<lb/><arrow.to.target n="marg173"/><lb/>tates, &longs;eu amplitudines gene&longs;um ponantur e&longs;&longs;e A, B, &longs;cili&shy;<lb/>cet A principio motus per C, &amp; B initio motus per D, qu&aelig;&shy;<lb/>ritur ratio temporum, quibus exiguntur propo&longs;ita &longs;patia. <lb/></s>
          <s id="s.000733">Vt A ad B, ita fiat C ad E, &amp; inter E, et D &longs;umatur F me&shy;<lb/>dia proportionalis. </s>
          <s id="s.000734">Dico ip&longs;a tempora e&longs;&longs;e vt E ad F. <lb/><!-- KEEP S--></s>
          <s id="s.000735">Componuntur &longs;patia acceleratis motibus exacta ex ratio&shy;<lb/><arrow.to.target n="marg174"/><lb/>ne quadratorum temporum, &amp; ex ea amplitudinum, &longs;eu <lb/>homologarum velocitatum in &longs;implicibus motibus, &longs;imili&shy;<lb/><arrow.to.target n="marg175"/><lb/>bu&longs;que &longs;umptarum; &amp; ideo temporum quadrata necten&shy;<lb/>tur ex ratione &longs;patiorum C ad D, &amp; ex reciproca ampli-<pb pagenum="74" xlink:href="022/01/080.jpg"/>tudinum E ad C; temporum igitur quadrata erunt vt E ad <lb/>D, ip&longs;a ver&ograve; tempora vt E ad F. <!-- KEEP S--></s>
          <s id="s.000736">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000737"><margin.target id="marg173"/><emph type="italics"/>Tab.<emph.end type="italics"/> 8. <emph type="italics"/>Fig.<emph.end type="italics"/> 1.<!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000738"><margin.target id="marg174"/>27. <emph type="italics"/>huiuij<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000739"><margin.target id="marg175"/><emph type="italics"/>lem. </s>
          <s id="s.000740">pr.<emph.end type="italics"/> 3. <emph type="italics"/>pri&shy;<lb/>mi huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000741"><emph type="center"/>PROP. XXXII. PROB. VII.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000742">EXdatis &longs;patijs accelerato motu tran&longs;actis, datis item <lb/>primis velocitatibus &longs;imilium, &longs;impliciumque mo&shy;<lb/>tuum, inuenire altitudines &longs;implicium gene&longs;um, ex quibus <lb/><arrow.to.target n="marg176"/><lb/>propo&longs;ita &longs;patia effecta &longs;unt. </s>
        </p>
        <p type="margin">
          <s id="s.000743"><margin.target id="marg176"/><emph type="italics"/>Tab.<emph.end type="italics"/> 7. <emph type="italics"/>fig.<emph.end type="italics"/> 1. <lb/>30. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000744">Spatia &longs;int E, D reliquis, vt &longs;upra, manentibus: quoniam <lb/>&longs;patia accelerato motu tran&longs;acta componuntur ex ratio&shy;<lb/>nibus amplitudinum gene&longs;um &longs;implicium, &longs;imiliumqu&etail; <lb/>motuum reciproc&egrave; &longs;umptarum B ad A, &longs;iue E ad C, &amp; ex <lb/>ea quadratorum altitudinum ip&longs;arum gene&longs;um; erit ratio <lb/>dictarum altitudinum duplicata C ad D; quare F, &longs;i &longs;it me&shy;<lb/>dia proportionalis, non inter E, &amp; D &lpar;vt antea po&longs;uimus&rpar; <lb/>&longs;ed inter C ad D; erit &longs;an&egrave; C ad F ratio altitudinum gene&shy;<lb/>&longs;um &longs;implicium, &longs;imiliumque motuum, quam quereba&shy;<lb/>mus. </s>
        </p>
        <p type="main">
          <s id="s.000745"><emph type="center"/><emph type="italics"/>Exemplum primum.<emph.end type="italics"/><emph.end type="center"/></s>
        </p>
        <p type="main">
          <s id="s.000746">SI idem graue naturaliter cadens percurrerit &agrave; quiete <lb/>duo &longs;patia; tempora erunt in ratione &longs;ubduplicat&atail; <lb/>eorundem &longs;patiorum. </s>
        </p>
        <p type="main">
          <s id="s.000747">Ex Cor. <!-- KEEP S--></s>
          <s id="s.000748">pr: 4. huius con&longs;tat rectangula e&longs;&longs;e gene&longs;es &longs;im&shy;<lb/>plicium motuum grauium naturaliter de&longs;cendentium, &amp; <lb/>ex def. <!-- REMOVE S-->7. primi liquet ea&longs;dem gene&longs;es e&longs;&longs;e motuum &longs;imi&shy;<lb/>lium. </s>
          <s id="s.000749">Cumque eiu&longs;dem mobilis naturaliter cadentis ve&shy;<lb/>locitas &agrave; quiete &longs;it vna, eademque; &longs;implices motus erunt <lb/>ij, vt gene&longs;um &longs;imilium, &longs;impliciumque motuum amplitu&shy;<lb/>dines &aelig;quales &longs;int, proptereaque, vt in figura pr&aelig;cedentis <lb/>propo&longs;itionis &aelig;quales erunt C, E, atque adeo &longs;patiu&mtail; <lb/>C, &longs;iue E ad D erit in duplicata ratione temporum E ad F. <!-- KEEP S--></s>
        </p>
        <pb pagenum="75" xlink:href="022/01/081.jpg"/>
        <p type="main">
          <s id="s.000750"><emph type="center"/><emph type="italics"/>Exemplum II.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000751"><emph type="center"/>PROP. XXXIV. THEOR. XXVII.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000752">TEmpora &longs;imilium vibrationum &longs;unt in &longs;ubduplicata <lb/>ratione arcuum exactorum, &longs;eu longitudinum pen&shy;<lb/>dulorum, quorum &longs;unt vibrationes. </s>
          <s id="s.000753">Sint grauia pendula <lb/>LA, LF, qu&aelig; ab eadem recta LF di&longs;cedentia currant &longs;u&longs;&shy;<lb/><arrow.to.target n="marg177"/><lb/>pen&longs;a ex L duos &longs;imiles arcus circulares FI, AC. <!-- KEEP S--></s>
          <s id="s.000754">Dico <lb/>tempora horum de&longs;cen&longs;uum e&longs;&longs;e in ratione &longs;ubduplicat&atail; <lb/>arcuum FI, AC, &longs;eu longitudinum filorum, aut ha&longs;tularum <lb/>FA, LA. </s>
          <s id="s.000755">Ducamus quamcumque rectam LBG, erit AB <lb/>ad BC, vt FG ad GI, &amp; cum pr&aelig;terea velocitates pendu&shy;<lb/>lorum a quiete in A, F &longs;int &aelig;quales, pariterque velocita&shy;<lb/>tes &aelig;quales a quiete in B, G; erit velocitas in A ad veloci&shy;<lb/>tatem in B, vt velocitas in F ad velocitatem in G, quare <lb/>con&longs;ideratis arcubus ABC, FGI, vt altitudines rect&etail;, &lpar;qu&aelig; <lb/>item forent in B, G proportionaliter &longs;ect&etail;&rpar; gene&longs;um &longs;imi&shy;<lb/><arrow.to.target n="marg178"/><lb/>lium &longs;impliciumque motuum, quarum amplitudines &aelig;qua <lb/>les &longs;unt, erunt &longs;patia in acceleratis decur&longs;ubus per FI, AC <lb/>in ratione duplicata temporum, &longs;cilicet ip&longs;i arcus, aut lon&shy;<lb/>gitudines LF, LA erunt in ratione duplicata temporu&mtail;. <lb/></s>
          <s id="s.000756">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000757"><margin.target id="marg177"/><emph type="italics"/>Tab.<emph.end type="italics"/> 8. <emph type="italics"/>fig.<emph.end type="italics"/> 2.<!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000758"><margin.target id="marg178"/><emph type="italics"/>Def.<emph.end type="italics"/> 7. <emph type="italics"/>primi.<gap/><emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000759">Idem demon&longs;tratum e&longs;&longs;et beneficio imaginum, qu&aelig; vt&shy;<lb/>pote eorundem illorum motuum &longs;implicium, forent etiam <lb/>&longs;imilium, &amp; &longs;unt amplitudines &aelig;quales, etenim e&aelig;de&mtail; <lb/>&longs;unt, ac gene&longs;um ergo rur&longs;us &longs;patia, hoc e&longs;t arcus ABC, <lb/>FGI, nempe longitudines filorum IF, AC erunt in ratione <lb/>duplicata temporum. </s>
          <s id="s.000760">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <pb pagenum="76" xlink:href="022/01/082.jpg"/>
        <p type="main">
          <s id="s.000761"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000762"><emph type="italics"/>Vides, qu&agrave;m breuiter rei di&longs;ficillim&aelig; demon&longs;trationem at&shy;<lb/>tulimus, nec dubium, quin illa extendi queat ad qua&longs;cum&shy;<lb/>que lineas decur&longs;uum, dummodo &longs;imiles, ac &longs;imiliter po&longs;itas in <lb/>ij&longs;dem, vel &aelig;qualibus ab horizonte planis elenatis, quemad&shy;<lb/>modum Dominus Viuianus pulcherrim&egrave; propo&longs;uit.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000763"><emph type="center"/><emph type="italics"/>Exemplum III.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000764"><emph type="center"/>PROP. XXXV. THEOR. XXVIII.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000765">TEmpora lationum &agrave; quiete per plana eandem eleua&shy;<lb/><arrow.to.target n="marg179"/><lb/>tionem habentia &longs;unt homolog&egrave; vt longitudines <lb/>planorum. </s>
        </p>
        <p type="margin">
          <s id="s.000766"><margin.target id="marg179"/><emph type="italics"/>Tab.<emph.end type="italics"/> 8. <emph type="italics"/>fig.<emph.end type="italics"/> 3.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000767">Sint plana AB, AC eandem eleuationem AD habentia. <lb/></s>
          <s id="s.000768">Dico tempus lationis per AC ad id per AB e&longs;&longs;e vt AC ad <lb/>AB. &lpar;h&aelig;c Torricellij propo&longs;itio, <expan abbr="expo&longs;itioq;">expo&longs;itioque</expan> e&longs;t, hancque <lb/>eandem veritatem ex no&longs;tris principijs demon&longs;trare <expan abbr="vis&utilde;">visum</expan> <lb/>e&longs;t, non vt de re illa dubitemus, imm&ograve; contr&agrave;, qu&ograve;d de e&atail; <lb/>plen&egrave; &longs;atisfacti &longs;imus, ex eo rur&longs;us demon&longs;trandam &longs;u&longs;ce&shy;<lb/>pimus, vt exinde methodus no&longs;tra, qu&agrave;m vera &longs;it, eluce&longs;&shy;<lb/>cat&rpar; Momentum de&longs;cen&longs;us inplano AC ad id de&longs;cen&longs;us &longs;u&shy;<lb/><arrow.to.target n="marg180"/><lb/>per plano AB e&longs;t vt AB ad AC; &longs;unt autem <expan abbr="de&longs;cendenti&utilde;">de&longs;cendentium</expan> <lb/>grauium, etiam &longs;uper planis inclinatis motus, quos &longs;impli&shy;<lb/>ces appellamus, inter &longs;e &longs;imiles, nempe quorum gene&longs;es <lb/><arrow.to.target n="marg181"/><lb/>&longs;unt rectangula; ergo habebimus &longs;implices gene&longs;es, vnam, <lb/>cuius altitudo AC amplitudoque AB; alteram, cuius am&shy;<lb/>plitudo AC, altitudo autem AB; itaque propo&longs;itis &longs;patijs <lb/>AC, AB, primi&longs;que velocitatibus AB, AC, &longs;i fiat AB ad AC <lb/>vt CA ad EA, erit EA ad AB duplicata <expan abbr="t&etilde;porum">temporum</expan>, &amp; ideo <lb/><arrow.to.target n="marg182"/><lb/>ratio temporum per AC, AB erit CA ad AB. <!-- KEEP S--></s>
          <s id="s.000769">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <pb pagenum="77" xlink:href="022/01/083.jpg"/>
        <p type="margin">
          <s id="s.000770"><margin.target id="marg180"/><emph type="italics"/>Tor. <!-- REMOVE S-->pr.<emph.end type="italics"/> 2. <emph type="italics"/>de <lb/>motu <expan abbr="graui&utilde;">grauium</expan>.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000771"><margin.target id="marg181"/><emph type="italics"/>Cor pr.<emph.end type="italics"/> 4. <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000772"><margin.target id="marg182"/>31. <emph type="italics"/>vel<emph.end type="italics"/> 27. <emph type="italics"/>hu.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000773"><emph type="center"/><emph type="italics"/>Exemplum IV.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000774"><emph type="center"/>PROP. XXXVI. THEOR. XXIX.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000775">II&longs;dem pror&longs;us manentibus demon&longs;trarunt Gallileus, ac <lb/>Torricellius, gradus velocitatum acqui&longs;itos in B, et C <lb/>eiu&longs;dem mobilis de&longs;cendentis &agrave; quiete in A pares e&longs;&longs;e; <lb/>idip&longs;um nos o&longs;tendemus. </s>
        </p>
        <p type="main">
          <s id="s.000776">Cum tempora &longs;int vt AC ad AB, &amp; velocitates &agrave; quie&shy;<lb/>te in ratione reciproca temporum, &longs;cilicet vt AB ad AC, <lb/><arrow.to.target n="marg183"/><lb/>&longs;int deinde velocitates e&aelig; vt amplitudines imaginum &longs;im&shy;<lb/>plicium, &longs;imiliumque illorum motuum &lpar;nam amplitudines <lb/>imaginum velocitatum &longs;unt pror&longs;us e&aelig;dem, ac ill&aelig; gene&shy;<lb/>&longs;um&rpar; erunt ip&longs;&aelig; imagines &longs;implicium motuum &aelig;quales; <lb/>nam tempora, qu&aelig; &longs;ummuntur vt altitudines imaginum <lb/>reciprocantur, vt dictum e&longs;t, amplitudinibus, &longs;eu primis &agrave; <lb/><arrow.to.target n="marg184"/><lb/>quiete velocitatibus, at in motibus acceleratis ip&longs;&aelig; inte&shy;<lb/>gr&aelig; imagines &longs;implicium motuum &longs;unt loco graduum ve&shy;<lb/>locitatum in extremo &longs;patiorum acqui&longs;itorum; ergo in B, et <lb/>C gradus velocitatum &aelig;quales erunt. </s>
        </p>
        <p type="margin">
          <s id="s.000777"><margin.target id="marg183"/><emph type="italics"/>Tab.<emph.end type="italics"/> 8. <emph type="italics"/>fig.<emph.end type="italics"/> 4. <lb/>33. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000778"><margin.target id="marg184"/>4. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000779"><emph type="center"/>PROP. XXXVII. THEOR. XXX.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000780">SI &aelig;qualia pondera, &longs;u&longs;pen&longs;a &longs;int ex filis, quorum par&shy;<lb/>tes inter&longs;e &aelig;quales, pr&aelig; tractione &aelig;qualiter elongen<lb/>ter tempora in reditu ip&longs;orum filorum, cum ab ip&longs;is graui&shy;<lb/>bus &longs;tatim liberantur, &aelig;qualia erunt. </s>
          <s id="s.000781">Hoc prim&ugrave;m <expan abbr="dem&otilde;-">demon&shy;<lb/></expan><arrow.to.target n="marg185"/><lb/>&longs;trabimus alia via, tum methodo no&longs;tra, vt de ea aliud <lb/>exemplum tradamus. </s>
          <s id="s.000782">Sint funiculi AB, DC, &amp; ex ijs <lb/>pendeant &aelig;qualia grauia B, C, adeo vt &longs;umptis hinc ind&egrave; <lb/>partibus &aelig;qualibus eorundem funiculorum, con&longs;tet ip&longs;as <lb/>&aelig;qualiter ab ip&longs;is grauibus trahi, atque produci. </s>
          <s id="s.000783">Dico, &longs;i <lb/>elongationes &longs;int HB, GC, &amp; omnibus &longs;ic &longs;tantibus pon-<pb pagenum="78" xlink:href="022/01/084.jpg"/>dera &longs;ubmoueantur ex B, et C funiculis c&aelig;&longs;is, fore vt e&aelig;&shy;<lb/>dem extremitates re&longs;tituantur in H, et G &aelig;qualibus tem&shy;<lb/>poribus. </s>
          <s id="s.000784">Sit AE &aelig;qualis DC, erit porr&ograve; elongatio facta <lb/>per idem graue B, qu&aelig; &longs;it EF, &aelig;qualis GC; propterea li&shy;<lb/>beratis funiculis ad B, et C, eodem tempore re&longs;tituetur C <lb/>in G, ac E in F, quo tempore etiam B in H re&longs;titutum fue&shy;<lb/>rit; nam vno puncto in primum &longs;uum locum redito, etiam <lb/>alia &longs;ingula in &longs;uum locum perueni&longs;&longs;e, opportebit. </s>
        </p>
        <p type="margin">
          <s id="s.000785"><margin.target id="marg185"/><emph type="italics"/>Tab.<emph.end type="italics"/> 8. <emph type="italics"/>fig.<emph.end type="italics"/> 5.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000786"><emph type="center"/><emph type="italics"/>Exemplum.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000787">HAc occa&longs;ione de funiculis erit non iniucunda di&longs;er&shy;<lb/>tatio, remque &longs;ic adhuc intactam promouebimus, <lb/>&longs;imulque demon&longs;trabimus. </s>
        </p>
        <p type="main">
          <s id="s.000788">Idip&longs;um propo&longs;itum no&longs;tris principijs &longs;ic demon&longs;tra&shy;<lb/>mus. </s>
        </p>
        <p type="main">
          <s id="s.000789">Sint <expan abbr="ead&etilde;">eadem</expan>, qu&aelig; &longs;upra, &longs;cilicet conceptis in filo AB quot&shy;<lb/>libet partibus inter&longs;e &aelig;qualibus, <expan abbr="l&otilde;gitudin&etilde;que">longitudinenque</expan> totam im&shy;<lb/>plentibus, h&aelig; &longs;ingul&aelig; &aelig;qualiter &agrave; pondere B trahentur, <lb/>eritque BH &longs;umma omnium dictarum partium elongatio&shy;<lb/>num, &amp; eodem pacto EF erit &longs;umma elongationum <expan abbr="parti&utilde;">partium</expan> <lb/>omnium in AE contentarum, ab eodemque pondere effe&shy;<lb/>ctarum; propterea vt AB ad BH, ita erit AE ad EF; quamo <lb/>brem velocitas etiam puncti B &longs;ublato pondere B erit ad <lb/>velocit atem puncti E ob eandem detractionem, vt BH ad <lb/>EF, vel BA ad EA &lpar;nam quot &longs;unt partes concept&etail; i&ntail; <lb/>vtraque fili longitudine, totidem &longs;unt etiam impetus inter <lb/>&longs;e &aelig;quales&rpar; idem o&longs;tenderemus &longs;i loco ponderis B, minus <lb/>quodcumque &longs;u&longs;penderemus, vt &longs;cilicet puncta B, et E ad <lb/>quemuis locum &longs;uperius remanerent, librarenturque cum <lb/>re&longs;i&longs;tentijs <expan abbr="parti&utilde;">partium</expan> e&ograve; elongatarum, ergo tran&longs;itus ex B in H, <lb/><arrow.to.target n="marg186"/><lb/>&amp; puncti E in F &longs;ubducto pondere B erunt motus &longs;imilium <lb/>&longs;impliciumque; &longs;ed motus ex C in G exempto pondere C <lb/>e&longs;t pror&longs;us idem, ac motus E in F, ergo motus &longs;imiles, ac <pb pagenum="79" xlink:href="022/01/085.jpg"/>&longs;implices ex B in H, &amp; ex C in G, ex quibus fiunt accele&shy;<lb/>rati, gene&longs;es habebunt, quarum prim&aelig; velocitates, &longs;eu am&shy;<lb/>plitudines proportionales &longs;unt altitudinibus earundem, <lb/>&longs;patijs nimirum CG, BH accelerato motu exigendis; qua&shy;<lb/>mobrem componentur ex ratione ip&longs;arum velocitatum, <lb/>&longs;eu amplitudinum CG ad BH, &amp; ex ea quadratorum tem&shy;<lb/>porum, qu&aelig; proinde &aelig;qualitatis erit; itaque etiam huius <lb/>&longs;ubduplicata; hoc e&longs;t tempora in tran&longs;itibus accelarato <lb/>motu exactis, erunt paria. </s>
        </p>
        <p type="margin">
          <s id="s.000790"><margin.target id="marg186"/><emph type="italics"/>pr.<emph.end type="italics"/> 4. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000791"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000792"><emph type="italics"/>Hinc patet, vbi &aelig;qu&egrave; cra&longs;&longs;is filis eiu&longs;demque materiei vel <lb/>cedenti&aelig; &longs;u&longs;pen&longs;a &longs;int &aelig;qualia pondera, tunc primas velocita&shy;<lb/>tes, &longs;ubductis ponderibus, fore in eadem ratione <expan abbr="elongation&utilde;">elongationum</expan>, <lb/>vel longitudinum filorum.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000793"><emph type="center"/>PROP. XXXVIII. THEOR. XXXI.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000794">SI extremitatibus funiculorum ex vna parte <expan abbr="firmator&utilde;">firmatorum</expan>, <lb/>ac eandem cra&longs;&longs;itiem habentium, nec non eiu&longs;dem <lb/>c&aelig;denti&aelig; exi&longs;tentium, fuerint &longs;u&longs;pen&longs;a &aelig;qualia pondera, <lb/>qu&aelig; inde ij&longs;dem longitudinibus &longs;eruatis, quomodo opor&shy;<lb/>tet tollantur, erunt &longs;patia recur&longs;uum, temporibus &longs;impli&shy;<lb/>cium motuum exacta in ratione longitudinum pendulo&shy;<lb/>rum. </s>
        </p>
        <p type="main">
          <s id="s.000795">Sit funiculus AC &aelig;qu&egrave; cra&longs;&longs;us ac BD, &amp; &longs;u&longs;pen&longs;is <lb/>hinc inde ponderibus &aelig;qualibus, elongatio primi funiculi <lb/>&longs;it CE, &amp; alterius &longs;it DF. <!-- KEEP S--></s>
          <s id="s.000796">Dico &longs;patia temporibus &longs;impli&shy;<lb/>cium imaginum, ab extremitatibus &longs;olutis exacta, fore i&ntail; <lb/>ratione longitudinum ip&longs;orum funiculorum. </s>
        </p>
        <p type="main">
          <s id="s.000797">Iam con&longs;tat CE ad DF e&longs;&longs;e, vt AC ad BD, in qua ratione <lb/>&longs;unt etiam velocitates &agrave; quiete, dum pondera &longs;ubduceren&shy;<lb/>tur ex E, et F, vel ex alijs punctis quibu&longs;cunque &longs;i &aelig;qualia <pb pagenum="80" xlink:href="022/01/086.jpg"/>pondera &longs;u&longs;pen&longs;a fui&longs;&longs;ent maioris, vel minoris ponderis, <lb/>&longs;ic enim concipiuntur gene&longs;es &longs;imilium, &longs;impliciumque <lb/>motuum, quarum altitudines &aelig;quantur elongationibus <lb/>funiculorum; propterea &longs;patia recur&longs;uum temporibus &longs;im&shy;<lb/>plicium motuum exacta, nectentur ex rationibus duplicata <lb/>CE ad DF, hoc e&longs;t AC ad BD, &amp; ex reciproca filorum, <lb/>&longs;cilicet BD ad AC, qu&aelig; ratio, vti diximus, e&longs;t reciproc&atail; <lb/>primarum velocitatum, &longs;eu amplitudinum gene&longs;um &longs;impli&shy;<lb/>cium, ergo ip&longs;a &longs;patia in reditu filorum ab extremitatibus <lb/>&longs;olutis exacta, erunt vt AC ad BF, &longs;eu vt CE ad DF. <lb/><!-- KEEP S--></s>
          <s id="s.000798">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000799"><emph type="center"/>PROP. XXXIX. THEOR. XXXI.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000800">TEmpora &longs;implicium, &longs;imiliumque dictorum motuum <lb/>&longs;unt &aelig;qualia. </s>
        </p>
        <p type="main">
          <s id="s.000801">Nam cor. </s>
          <s id="s.000802">2. pr. <!-- REMOVE S-->8. huius primi demon&longs;tratum e&longs;t, tem&shy;<lb/>pora &longs;implicium, &longs;imiliumque motuum componi ex ratio&shy;<lb/>ne &longs;patiorum, &longs;eu altitudinum gene&longs;um, &amp; reciproca pri&shy;<lb/>marum, aut extremarum velocitatum, &longs;eu amplitudinum <lb/>gene&longs;um: &longs;unt autem altitudines gene&longs;um tractiones, &longs;eu <lb/>elongationes funiculorum, qu&aelig; &longs;unt vt longitudines funi&shy;<lb/>culorum, ergo tempora &aelig;qualia erunt. </s>
        </p>
        <p type="main">
          <s id="s.000803"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000804"><emph type="italics"/>Con&longs;tat, tempora a &longs;implicium gene&longs;um in tractionibus fu&shy;<lb/>niculorum, e&longs;&longs;e compo&longs;ita ex ratione elongationum funiculo&shy;<lb/>rum, &amp; ex reciproca primarum velocitatum.<emph.end type="italics"/></s>
        </p>
        <pb pagenum="81" xlink:href="022/01/087.jpg"/>
        <p type="main">
          <s id="s.000805"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000806"><emph type="italics"/>Superioris propo&longs;itionis veritas concordat cum prop.<emph.end type="italics"/> 37. <emph type="italics"/>hu&shy;<lb/>ius, in eo tant&ugrave;m variatur, quod ibi ponuntur data &longs;pati&atail; <lb/>elongitiones funiculorum, hic ver&ograve; tempora &longs;impliciu&mtail; <lb/>motuum, &amp; quia elongationes o&longs;ten&longs;&aelig; &longs;unt proportionales &longs;pa<lb/>tijs nunc exactis, manife&longs;tum e&longs;t, no&longs;tri iuris e&longs;&longs;e mod&ograve; &longs;patia <lb/>acceleratis motibus exact a ex temporibus &longs;implicium <expan abbr="motu&utilde;">motuum</expan> <lb/>datis concludere, mod&ograve; contr&agrave;, ex &longs;patijs altitudinibus gene&shy;<lb/>&longs;um proportionalibus, qua item data &longs;unt, tempora inuenire, <lb/>qua proinde methodus mihi videtur ampli&longs;&longs;ima.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000807"><emph type="center"/>PROP. XXXX. THEOR. XXXIII.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000808">SI eiu&longs;dem cra&longs;&longs;itiei funiculis pondera dependeant, qu&etail; <lb/>&longs;int in ratione reciproca longitudinum ip&longs;orum funi&shy;<lb/>culorum, &longs;patia temporibus gene&longs;um &longs;implicium motuum <lb/>exacta erunt in ratione duplicata elongationum. <lb/><arrow.to.target n="marg187"/></s>
        </p>
        <p type="margin">
          <s id="s.000809"><margin.target id="marg187"/><emph type="italics"/>Tab.<emph.end type="italics"/> 8. <emph type="italics"/>Fig.<emph.end type="italics"/> 6.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000810"><expan abbr="N&atilde;">Nam</expan> &longs;i &longs;it <expan abbr="p&otilde;dus">pondus</expan> E ad F &longs;icuti <expan abbr="l&otilde;gitudo">longitudo</expan> DB ad CA, &amp; &longs;int, <lb/>cra&longs;sities <expan abbr="funiculor&utilde;">funiculorum</expan> &aelig;quales erit &longs;an&egrave; ratio, qu&aelig; <expan abbr="c&otilde;poni-tur">componi&shy;<lb/>tur</expan> ex ratione <expan abbr="funiculor&utilde;">funiculorum</expan>, &amp; ex ea <expan abbr="p&otilde;derum">ponderum</expan>, &aelig;qualitatis; ob <lb/>idque gene&longs;es <expan abbr="&longs;implici&utilde;">&longs;implicium</expan> <expan abbr="motu&utilde;">motuum</expan>, <expan abbr="quar&utilde;">quarum</expan> altitudines CE, DF <lb/><expan abbr="habeb&utilde;t">habebunt</expan> amplitudines, <expan abbr="n&etilde;pe">nempe</expan> primas velocitates inter&longs;e &etail;qua<lb/>les &lpar;nam cum pondera erant &aelig;qualia, prim&aelig; velocitates <lb/>proportionabantur longitudinibus <expan abbr="funiculor&utilde;">funiculorum</expan>, ideo, cum </s>
        </p>
        <p type="main">
          <s id="s.000811"><arrow.to.target n="marg188"/><lb/>pondera reciprocantur longitudinibus ij&longs;dem, &longs;eu viribus <lb/>funiculorum, fit vt prim&aelig; velocitates &aelig;quales reddantur&rpar; <lb/>cum ergo ita &longs;it, &longs;patia recur&longs;uum temporibus imaginu&mtail; <lb/>&longs;implicium &amp; accelerato motu confecta erunt in ratione <lb/>duplicata elongationum. </s>
        </p>
        <pb pagenum="82" xlink:href="022/01/088.jpg"/>
        <p type="margin">
          <s id="s.000812"><margin.target id="marg188"/>28. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000813"><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000814"><emph type="italics"/>Cum ex eadem pr.<emph.end type="italics"/> 28. <emph type="italics"/>huius, eadem &longs;patia &longs;int vt quadra&shy;<lb/>ta temporum, erunt ip&longs;a tempera in ratione &longs;ubduplicat&atail; <lb/>elongationum.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000815"><emph type="center"/>PROP. XXXXI THEOR. XXXIV.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000816">SI funiculis &aelig;qualem cra&longs;&longs;itiem habentibus fuerint &longs;u&longs;&shy;<lb/><arrow.to.target n="marg189"/><lb/>pen&longs;a in&aelig;qualia pondera, &longs;patia, qu&aelig; acceleratis mo&shy;<lb/>tibus, ac temporibus gene&longs;um &longs;implicium recurruntur ne&shy;<lb/>ctentur ex ratione duplicata elongationum, &amp; ex duabus <lb/>reciproc&egrave; &longs;umptis rationibus, nempe longitudinum prima&shy;<lb/>rum funiculorum, antequam pondera &longs;u&longs;penderentur; &amp; <lb/>ip&longs;orum ponderum. </s>
        </p>
        <p type="margin">
          <s id="s.000817"><margin.target id="marg189"/><emph type="italics"/>Tab.<emph.end type="italics"/> 8. <emph type="italics"/>Fig.<emph.end type="italics"/> 6.<!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000818">In antecedenti figura illud primum &longs;atis patet, qu&ograve;d &longs;i <lb/>loco ponderis F &longs;u&longs;pen&longs;um fui&longs;&longs;et pondus aliud grauius, <lb/>aut leuius, prior velocitas in a&longs;cen&longs;u fili, &longs;eu funiculi, aut <lb/>chord&aelig; aucta, vel imminuta fui&longs;&longs;et pro magnitudine pon&shy;<lb/>deris &longs;ub&longs;tituti; quamobrem priores velocitates ex in&aelig;qua <lb/>litate ponderum eidem chord&aelig; &longs;u&longs;pen&longs;orum dependentes <lb/>forent, vt ip&longs;a pondera; ver&ugrave;m cum &longs;uppo&longs;itis funiculis <lb/>&aelig;qualia pondera &longs;u&longs;pen&longs;a veniunt, prim&aelig; velocitates &longs;unt <lb/><arrow.to.target n="marg190"/><lb/>vt longitudines funiculorum, ergo velocitates prim&aelig;, cum <lb/>in&aelig;qualia &longs;unt pondera, qu&aelig; &longs;ubtrahuntur, nectentur ex <lb/>ratione longitudinum funiculorum, &amp; ex ea ponderum <lb/>in&aelig;qualium: qu&aelig;cumque igitur &longs;it tractio DF, gene&longs;es ha&shy;<lb/>bebimus &longs;imilium &longs;impliciumque motuum, vnam, cuius al&shy;<lb/>titudo CE, &amp; alteram habentem altitudinem DF, &amp; &longs;unt <lb/>earundem gene&longs;um amplitudines, &longs;eu prim&aelig; velocitates <lb/>in ratione compo&longs;ita funiculorum AC ad BD, &amp; ponderis <lb/><arrow.to.target n="marg191"/><lb/>pendentis ex E ad pondus &longs;u&longs;pen&longs;um in F; ergo &longs;patia ac&shy;<lb/>celeratis motibus tran&longs;acta temporibus gene&longs;um <expan abbr="&longs;implici&utilde;">&longs;implicium</expan> <pb pagenum="83" xlink:href="022/01/089.jpg"/>nectentur ex ratione dublicata elongationum, &longs;iue altitu&shy;<lb/>dinum gene&longs;um, &amp; ex duabus rationibus reciproc&egrave; &longs;um&shy;<lb/>ptis funiculorum AC ad BD, &amp; ponderum E ad F. <lb/><!-- KEEP S--></s>
          <s id="s.000819">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000820"><margin.target id="marg190"/><emph type="italics"/>Cor. <!-- KEEP S--></s>
          <s id="s.000821">pr.<emph.end type="italics"/> 37. <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000822"><margin.target id="marg191"/>30. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000823"><emph type="center"/>PROP. XXXXII. THEOR. XXXV.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000824">II&longs;dem po&longs;itis, &longs;i &longs;patia recur&longs;uum erunt ip&longs;&aelig; elongatio&shy;<lb/>nes, tempora, quibus ab extremitatibus &longs;olutis recur&shy;<lb/>runtur, erunt in ratione &longs;ubduplicata eorundem. </s>
          <s id="s.000825">Nam cum <lb/>gene&longs;es &longs;imilium, &longs;impliciumque motuum &longs;int &aelig;qu&egrave; am&shy;<lb/>pl&aelig;, erunt, tempora in ratione &longs;ubduplicata imaginum, <lb/>&longs;eu &longs;patiorum acceleratorum motuum, &longs;unt ver&ograve; &longs;patia <lb/>ip&longs;&aelig; elongationes; ergo &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000826"><emph type="center"/>PROP. XXXXIII. THEOR. XXXVI.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000827">CHord&aelig; non eiu&longs;dem cra&longs;&longs;itiei, eiu&longs;dem tamen mate&shy;<lb/>ri&aelig;, ac longitudinis, tunc &aelig;qu&egrave; trahentur vbi <expan abbr="&longs;u&longs;p&etilde;-&longs;a">&longs;u&longs;pen&shy;<lb/>&longs;a</expan> pondera cra&longs;&longs;itut inibus proportionalia fuerint. </s>
          <s id="s.000828">Nam <lb/>cra&longs;&longs;ior chorda pote&longs;t concipi compo&longs;ita ex funiculis eiu&longs; <lb/>dem cra&longs;&longs;itiei alterius chord&aelig;, &longs;i illa huius fuerit multiplex, <lb/>&amp; &longs;i partes exilior funiculus fuerit alterius cra&longs;&longs;ioris, erit <lb/>cra&longs;&longs;ities alicuius alterius funiculi, qu&aelig; pluries accept&atail; <lb/>con&longs;tituere poterit vtranque cra&longs;&longs;itiem funiculorum pro&shy;<lb/>po&longs;itorum &lpar;h&igrave;c enim non accidit enumerare cra&longs;&longs;ities in&shy;<lb/>ter&longs;e irrationales, quippe quia, quod de iam dictis o&longs;ten&shy;<lb/>derimus, de his quoque facil&egrave; e&longs;t iudicare, &longs;ec&ugrave;s e&longs;&longs;emus <lb/>longi, quam par e&longs;t, poti&longs;&longs;im&ugrave;m cum h&aelig;c pr&aelig;ter <expan abbr="in&longs;titut&utilde;">in&longs;titutum</expan> <lb/>adijciantur, &amp; quidem vt con&longs;tet, quomodo methodus i&longs;ta <lb/>no&longs;tra facilis &longs;it, ac vtili&longs;&longs;ima&rpar; quapropter &longs;i cuique acce&shy;<lb/>ptarum &aelig;qualium chordarum, pondera &aelig;qualia &longs;u&longs;pen&longs;a <lb/>&longs;int, porr&ograve; h&aelig;c omnes &aelig;qu&egrave; trahentur ab ip&longs;is &aelig;qualibus <lb/>ponderibus, &amp; &longs;ic etiam compo&longs;ita, nempe choid&aelig; pro-<pb pagenum="84" xlink:href="022/01/090.jpg"/>po&longs;it&aelig;; &longs;untque ita pondera in eadem ratione cra&longs;&longs;itierum, <lb/>&longs;icut propo&longs;uimus; ergo patet propo&longs;itum. </s>
        </p>
        <p type="main">
          <s id="s.000829"><emph type="center"/>PROP. XXXXIV. THEOR. XXXVII.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000830">SI fuerint eiu&longs;dem materi&aelig; funiculi, &amp; &longs;int illis &longs;u&longs;pen&longs;a <lb/>pondera cra&longs;&longs;itiebus proportionalia, ratio &longs;patiorum <lb/>in reditibus accelerato motu exactorum, <expan abbr="t&etilde;poribus">temporibus</expan> &longs;im&shy;<lb/><arrow.to.target n="marg192"/><lb/>plicium gene&longs;um, erit eadem ac funiculorum. </s>
        </p>
        <p type="margin">
          <s id="s.000831"><margin.target id="marg192"/><emph type="italics"/>Tab.<emph.end type="italics"/> 8. <emph type="italics"/>fig.<emph.end type="italics"/> 6. <lb/>42. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000832">Nam, vt in pr&aelig;cedenti figura, erit tractio CE ad DF ita <lb/><arrow.to.target n="marg193"/><lb/>AC ad BD, vel AE ad BF, &longs;unt autem prim&aelig; velocitates, <lb/>&longs;eu amplitudines gene&longs;um &longs;implicium, &longs;imiliumque <expan abbr="motu&utilde;">motuum</expan> <lb/>in ratione funiculorum, ergo decur&longs;uum &longs;patia motibus <lb/><arrow.to.target n="marg194"/><lb/>acceleratis exacta nectentur ex ratione duplicata altitu&shy;<lb/>dinum gene&longs;um &longs;implicium, nempe duplicata <expan abbr="funiculor&utilde;">funiculorum</expan>, <lb/>&amp; reciproca amplitudinum, &longs;untque ip&longs;&aelig; amplitudines <lb/>homolog&egrave; vt longitudines funiculorum, ergo relinquitur <lb/>vt ip&longs;a &longs;patia &longs;int in vnica ratione longitudinum funicu&shy;<lb/>lorum. </s>
        </p>
        <p type="margin">
          <s id="s.000833"><margin.target id="marg193"/><emph type="italics"/>Cor. <!-- KEEP S--></s>
          <s id="s.000834">pr.<emph.end type="italics"/> 37. <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000835"><margin.target id="marg194"/>27. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000836">Qu&ograve;d &longs;i &longs;patia recur&longs;uum ponantur ip&longs;&aelig; tractiones, vel <lb/>longitudines funiculorum, o&longs;tendetur tempora e&longs;&longs;e &aelig;qua&shy;<lb/>lia, quemadmodum &aelig;qualia &longs;unt tempora &longs;uperius pro&shy;<lb/>po&longs;ita &longs;implicium gene&longs;um. </s>
        </p>
        <p type="main">
          <s id="s.000837"><emph type="center"/>PROP. XXXXV. THEOR. XXXVIII.<emph.end type="center"/><lb/><arrow.to.target n="marg195"/><!-- KEEP S--></s>
        </p>
        <p type="margin">
          <s id="s.000838"><margin.target id="marg195"/><emph type="italics"/>Tab.<emph.end type="italics"/> 8. <emph type="italics"/>fig.<emph.end type="italics"/> 7.</s>
        </p>
        <p type="main">
          <s id="s.000839">SI eiu&longs;dem materiei quibu&longs;cunque funiculis aligentur <lb/>qu&aelig;cunque pondera, ijs &longs;ublatis a&longs;cen&longs;uum &longs;patia ab <lb/>extremitatibus &longs;olutis exacta temporibus gene&longs;um &longs;impli&shy;<lb/>cium, ijs nempe qu&aelig; impenderentur in motibus iuxta &longs;im&shy;<lb/>plices gene&longs;es, erunt in ratione compo&longs;ita quadratorum <lb/>elongationum chordarum, ex ea cra&longs;&longs;itierum, &amp; ex duabus <lb/>reciproc&egrave; &longs;umptis rationibus, nempe longitudinum fu&shy;<lb/>niculorum antequam traherentur; &amp; &longs;u&longs;pen&longs;orum ponde&shy;<lb/>rum. </s>
        </p>
        <pb pagenum="85" xlink:href="022/01/091.jpg"/>
        <p type="main">
          <s id="s.000840">Funiculi AB, GH trahantur &agrave; ponderibus quibu&longs;cunque <lb/>C, I in C, et I. <!-- KEEP S--></s>
          <s id="s.000841">Dico &longs;i exempta &longs;int pondera, fore, vt &longs;patia <lb/>qu&aelig; acceleratis motibus exiguntur ab extremitatibus &longs;o&shy;<lb/>lutis C, I &longs;int in ratione compo&longs;ita ex duplicata IH ad BC, <lb/>cra&longs;&longs;itudinis ad cra&longs;&longs;itudinem funiculorum AB, GH; dein&shy;<lb/>de ex funiculi longitudine HG ad longitudinem AB, pon&shy;<lb/>deri&longs;que I ad pondus C. <!-- KEEP S--></s>
          <s id="s.000842">Intelligatur funiculus, &longs;eu chor&shy;<lb/>da, &aelig;que cra&longs;&longs;a, ac &longs;imiliter cedens, qu&agrave;m GH &lpar;id quod <lb/>&longs;emper intelligimus quoties funiculi, inter&longs;e comparantur&rpar; <lb/>&longs;ed &aelig;qu&egrave; longa, ac AB, &longs;itque illi pondus F adiectum, ad <lb/>quod C eandem habeat rationem, ac cra&longs;&longs;ities AB ad cra&longs;&shy;<lb/>&longs;itiem DE, con&longs;tat elongationem EF &aelig;qualem fieri ip&longs;i <lb/>CB, &amp; cum prim&aelig; velocitates, &longs;eu amplitudines &aelig;qu&egrave; al&shy;<lb/>tarum gene&longs;um &longs;imilium, &longs;impliciumque motuum &longs;int <expan abbr="eti&atilde;">etiam</expan> <lb/>&aelig;quales, &longs;patia decur&longs;uum acceleratis motibus exacta <expan abbr="er&utilde;t">erunt</expan> <lb/>pror&longs;us &aelig;qualia; &longs;unt ver&ograve; funiculi DE, GH eiu&longs;dem cra&longs;&shy;<lb/>&longs;itiei, ei&longs;que &longs;unt &longs;u&longs;pen&longs;a duo&apos;pondera in&aelig;qualia F, I; ergo <lb/>decur&longs;uum &longs;patia ab extremitatibus &longs;olutis exacta <expan abbr="nect&etilde;-tur">necten&shy;<lb/>tur</expan> ex ratione duplicata elongationum FE, &longs;eu CB ad IH, <lb/>ex ratione, quam habent longitudines funiculorum HG ad <lb/>DE, &longs;eu AB, &amp; ex ea ponderum I ad F; ver&ugrave;m pondera I <lb/>ad F nectuntur ex rationibus ponderum I ad C et C ad F, <lb/>qu&aelig; po&longs;trema e&longs;t ratio cra&longs;&longs;itiei funiculi AB ad cra&longs;&longs;itiem <lb/>funiculi DE, &longs;eu GH; ergo vt propo&longs;uimus &longs;patia accele&shy;<lb/>ratis motibus exacta, nectentur ex rationibus <expan abbr="quadrator&utilde;">quadratorum</expan> <lb/>CB ad HI; cra&longs;&longs;itudinum funiculorum AB, GH; <expan abbr="ponder&utilde;">ponderum</expan> <lb/>I ad C, &amp; longitudinum HG ad AB. <!-- KEEP S--></s>
          <s id="s.000843">Quod &amp;c. <!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000844"><emph type="center"/>PROP. XXXXVI. THEOR. XXXIX.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000845">TEmpora gene&longs;um &longs;implicium, dum chordis &longs;u&longs;pen&shy;<lb/>duntur qu&aelig;cunque grauia, nectuntur, ex ratione <lb/>elongationum funiculorum, &amp; ex contrari&egrave; &longs;umptis ratio <lb/>nibus, cra&longs;&longs;itudinum, longitudinumque funiculorum, nec <pb pagenum="86" xlink:href="022/01/092.jpg"/>non ponderum funiculis &longs;u&longs;pen&longs;orum. </s>
        </p>
        <p type="main">
          <s id="s.000846">Nam Cor: 2. pr. <!-- REMOVE S-->8. primi demon&longs;tratum e&longs;t, tempor&atail; <lb/>&longs;implicium &longs;imiliumque motuum componi ex ratione &longs;pa&shy;<lb/>tiorum, &longs;eu altitudinum gene&longs;um, &amp; reciproca primarum <lb/>velocitatum, &longs;eu amplitudinum gene&longs;um, &longs;unt autem alti&shy;<lb/>tudines gene&longs;um tractiones, &longs;eu elongationes funiculorum; <lb/>velocitatesver&ograve; prim&aelig; nectuntur ex rationibus cra&longs;&longs;itudi&shy;<lb/>num, &amp; ex ea longitudinum funiculorum antequam tra&shy;<lb/>herentur &lpar;hoc enim &longs;ubinde o&longs;tendemus&rpar; ergo tempora <lb/>propo&longs;ita &longs;implicium gene&longs;um, dum chordis <expan abbr="allig&atilde;tur">alligantur</expan> qu&etail;&shy;<lb/>cunque in&aelig;qualia pondera, componentur ex rationibus <lb/>elongationum funiculorum, &amp; ex contrari&egrave; &longs;umptis cra&longs;&longs;i&shy;<lb/>tudinum, longitudinumque funiculorum, &amp; ponderum. </s>
        </p>
        <p type="main">
          <s id="s.000847"><emph type="center"/><emph type="italics"/>A&szlig;umptum.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000848"><arrow.to.target n="marg196"/></s>
        </p>
        <p type="margin">
          <s id="s.000849"><margin.target id="marg196"/><emph type="italics"/>Tab.<emph.end type="italics"/> 8. <emph type="italics"/>fig.<emph.end type="italics"/> 7.</s>
        </p>
        <p type="main">
          <s id="s.000850">VEr&ugrave;m prim&aelig; velocitates in ij&longs;dem chordis componi <lb/>ex ratione cra&longs;&longs;itudinum, longitudinum <expan abbr="funiculor&utilde;">funiculorum</expan>, <lb/>&amp; &longs;u&longs;pen&longs;orum ponderum, &longs;ic o&longs;tendemus, </s>
        </p>
        <p type="main">
          <s id="s.000851">Quoniam in eadem po&longs;trema figura velocitas, qua&mtail; <lb/>haberet funiculus AB ex liberatione ponderis e&longs;t &aelig;qualis <lb/>velocitati, quam haberet alius funiculus, vbi hic etiam li&shy;<arrow.to.target n="marg197"/><lb/>beraretur &agrave; pondere, &longs;cilicet cum pondera cra&longs;&longs;itiebus fu&shy;<lb/>niculorum proportionalia &longs;unt, &amp; ip&longs;i funiculi &aelig;qu&egrave; longi; <lb/>velocitas funiculi DE &agrave; pondere F ad velocitatem <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> <lb/><arrow.to.target n="marg198"/><lb/>funiculi, &longs;i loco ponderis F &longs;ub&longs;titutum e&longs;&longs;et aliud &aelig;quale <lb/>ip&longs;i I, e&longs;&longs;et vt pondus F ad &longs;ub&longs;titutum, &longs;eu ad I, e&longs;t autem <lb/>velocitas eiu&longs;dem funiculi DE, dum fui&longs;&longs;et pondus ei &longs;u&longs;&shy;<lb/>pen&longs;um &aelig;quale I ad velocitatem funiculi GH a pondere I <lb/>vt longitudo DE ad GH; ergo patet propo&longs;itum. </s>
        </p>
        <pb pagenum="87" xlink:href="022/01/093.jpg"/>
        <p type="margin">
          <s id="s.000853"><margin.target id="marg197"/>43. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="margin">
          <s id="s.000854"><margin.target id="marg198"/><emph type="italics"/>Ex<emph.end type="italics"/> 41. <emph type="italics"/>huius.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000855"><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000856"><emph type="italics"/>Quod hucu&longs;que ostendimus in funiculis ponderibus de&shy;<lb/>grauatis, non ab&longs;imili modo pr&aelig;st abimus in chordis ad <expan abbr="vtr&atilde;-que">vtran&shy;<lb/>que</expan> extremitatem firmatis, &amp; adductis, hoc tantum di&longs;crimi&shy;<lb/>ne, vt &longs;i in ijs pondere &longs;ublato, motus extremitatis &longs;olut&aelig; at&shy;<lb/>tendebatur, h&igrave;c media parte attract&acirc; chord&acirc;, &amp; &longs;ubinde &longs;ui <lb/>iuris relict&acirc;, vibrationem eius ob&longs;eruamus, &amp; equidem illa <lb/>omnia in hunc finem o&longs;tendimus, quippe ab hac re, plurima <lb/>vtili&longs;&longs;im&aelig;que veritates manere po&longs;&longs;unt. </s>
          <s id="s.000857">Nam de arcubus po&longs;&longs;es <lb/>pulcherrima in&longs;titui ratio, &amp; qui vellet armonicorum &longs;ono&shy;<lb/>rum, vel vocum per chordarum vibrationes editarum, tempo&shy;<lb/>ra, cum &longs;oni ad aures perueniunt, inue&longs;tigare, reor non aliam <lb/>viam, qu&agrave;m hanc ingredi nos debere, atque ind&egrave; con&longs;onantia&shy;<lb/>rum forta&longs;&longs;e naturam percipere po&longs;&longs;e, vt primus <gap/>ui<gap/> Gal&shy;<lb/>lileus quamquam vibrationes ten&longs;arum chordarum <expan abbr="differ&atilde;t">differant</expan> <lb/>ab ijs pendulorum.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000858"><emph type="center"/><emph type="italics"/>FINIS.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <pb pagenum="88" xlink:href="022/01/094.jpg"/>
        <p type="main">
          <s id="s.000859"><emph type="center"/>SPIEGATIONE<emph.end type="center"/></s>
        </p>
        <p type="main">
          <s id="s.000860"><emph type="center"/>di vna nuoua &longs;pecie di Bale&longs;tra.<emph.end type="center"/><lb/><arrow.to.target n="marg199"/></s>
        </p>
        <p type="margin">
          <s id="s.000861"><margin.target id="marg199"/><emph type="italics"/>Tab.<emph.end type="italics"/> 9.</s>
        </p>
        <p type="main">
          <s id="s.000862">IN que&longs;ta figura &longs;i e&longs;prime vna nuoua <expan abbr="inu&etilde;-tione">inuen&shy;<lb/>tione</expan> di Bale&longs;tra, la quale, ma&longs;&longs;imamente <lb/>in grande, per tirar granate, &ograve; &longs;a&longs;&longs;i pu&ograve; e&longs;&shy;<lb/>&longs;ere di gran con&longs;eguenze nella militare, co&shy;<lb/>me dimo&longs;trera&longs;&longs;i. </s>
        </p>
        <p type="main">
          <s id="s.000863">Dalle &longs;ue parti &longs;i verr&agrave; in cognitione del <lb/>modo di fabbricarta, e &longs;ono le &longs;eguenti. </s>
        </p>
        <p type="main">
          <s id="s.000864">AM, MN &longs;ono amendue le braccia. </s>
          <s id="s.000865">Il punto M &egrave; il cen&shy;<lb/>tro della machina. </s>
          <s id="s.000866">Per la cauit&agrave; M deue pa&longs;lar la pall&atail; <lb/>&longs;cagliata dalla corda; e per di &longs;otto M &longs;i ferma &amp; inca&longs;tra <lb/>nel manico, al modo delle bale&longs;tre communi. </s>
          <s id="s.000867">Ai due capi, <lb/>&ograve; &longs;iano e&longs;tremit&agrave; A, N &longs;i annette la fune. </s>
          <s id="s.000868">I punti A, E, F, <lb/>G, I, K &longs;ono in vna linea retta. </s>
          <s id="s.000869">Gl&apos; interualli AE, EF, FG, <lb/>GI, IK, &longs;ono, benche non di nece&longs;&longs;it&agrave;, eguali. </s>
          <s id="s.000870">Le altezze, <lb/>&ograve; comme&longs;&longs;ure KL, IH, GD, FC, EB perpendiculari, nell&apos; <lb/>incuruar&longs;i dell&apos; arco, &longs;i aprono intorno a&apos; centri K, I, G, F, <lb/>E. <!-- KEEP S--></s>
          <s id="s.000871">Donde ne &longs;iegue, che prendendo la corda dal &longs;uo mez&shy;<lb/>zo, e tirandola ver&longs;o O; amendue le braccia &longs;i aprono nel&shy;<lb/>le predette comme&longs;&longs;ure, come compare nell&apos; vno d&apos;e&longs;&longs;i &longs;e&shy;<lb/>gnato a punti con le lettere corri&longs;pondenti. </s>
          <s id="s.000872">Cia&longs;cuna <lb/>delle predette comme&longs;&longs;ure viene &longs;trettamente rin&longs;errat&atail; <lb/>da vna molla, come &longs;i vede in L, H, D, C, B; e que&longs;te mol&shy;<lb/>le, quanto pi&ugrave; &longs;i auuicinano al centro M, deuono e&longs;&longs;ere pi&ugrave; <lb/>grandi e pi&ugrave; ma&longs;&longs;iccie, in modo che, per cagione della <expan abbr="gr&atilde;-dezza">gran&shy;<lb/>dezza</expan> opportuna, vengano ad aprir&longs;i con egual facilit&agrave; <lb/>dell&apos;altre, e per cagione della gro&longs;&longs;ezza, habbiano nel &longs;er&shy;<lb/>rar&longs;i maggior forza, &ograve; &longs;ia momento, per la ragione, che &longs;ot&shy;<lb/>to &longs;i dir&agrave;. </s>
        </p>
        <p type="main">
          <s id="s.000873">Ci&ograve; pre&longs;uppo&longs;to, &egrave; facil co&longs;a dimo&longs;trare i vantaggi di <pb pagenum="89" xlink:href="022/01/095.jpg"/>que&longs;ta machina &longs;opra le ordinarie. </s>
        </p>
        <p type="main">
          <s id="s.000874">Primieramente nel triangolo ALK, e&longs;&longs;endo le altezz&etail; <lb/>EB, FC, GD, IH, KL perpendicolari, e perci&ograve; paralelle; <lb/>ne &longs;iegue che le proportioni di AE ad EB, di AF ad FC, di <lb/>AG a GD, di AI ad IH, di AK a KL &longs;ieno tutte eguali; e <lb/>douendo e&longs;&longs;ere parimente eguali le re&longs;i&longs;tenze delle molle <lb/>in B, C, D, H, L, che &longs;i &longs;uppongono di egual neruo nell&apos; <lb/>aprir&longs;i; ne &longs;iegue &lpar;&longs;econdo i principij della Meccanica&rpar; che <lb/>attraendo&longs;i con la fune l&apos;e&longs;tremit&agrave; A, nel mede&longs;imo tempo <lb/>e con la mede&longs;ima facilit&agrave; vincera&longs;&longs;i l&apos;equilibrio di tutte le <lb/>molle; la re&longs;i&longs;tenza delle quali &longs;i con&longs;idera in ragione di <lb/>pe&longs;o, &longs;i come le linee AE, EB; AF, FC; AG, GD; &amp;c. <!-- REMOVE S-->&longs;i con&shy;<lb/>&longs;iderano come vetti, &ograve; lieue, che hanno i loro ippomoclij, &ograve; <lb/>&longs;iano centri in E, F, G, I, K, e la potenza in A, la quale &egrave; <lb/>comune a tutte. </s>
        </p>
        <p type="main">
          <s id="s.000875">In &longs;econdo luogo, hauendo il braccio AE al braccio EB <lb/>&lpar;il &longs;imile dica&longs;i degli altri&rpar; hauendo, dico, gran proportio&shy;<lb/>ne, re&longs;ter&agrave; molto ageuolato il moto. </s>
        </p>
        <p type="main">
          <s id="s.000876">Terzo e&longs;&longs;endo molte le molle, e a prendo&longs;i tutte, ne deue <lb/>&longs;eguire vn notabile incuruamento d&apos;amendue le braccia; <lb/>onde la&longs;ciando l&apos;arco in libert&agrave;, e chiudendo&longs;i tutte le &longs;u&shy;<lb/>det te molle nel mede&longs;imo tempo, cio&egrave; qua&longs;i in vn&apos;attimo; <lb/>dour&agrave; la corda, che era tirata ver&longs;o O, pa&longs;&longs;are qua&longs;i in <expan abbr="i&longs;t&atilde;-te">i&longs;tan&shy;<lb/>te</expan> ver&longs;o M; il che non potendo&longs;i fare &longs;e non con &longs;omma ve&shy;<lb/>locit&agrave;, per la grandezza dello &longs;patio; e a que&longs;ta corri&longs;pon&shy;<lb/>dendo la forza, ne &longs;eguir&agrave; vn colpo molto con&longs;iderabile, e <lb/>vantaggio&longs;o, come cia&longs;cuno pu&ograve; arguire. </s>
        </p>
        <p type="main">
          <s id="s.000877">Re&longs;tano hora a &longs;cior&longs;i alcune difficolt&agrave;. </s>
          <s id="s.000878">La prima &egrave;, <lb/>che, quantunque &longs;ia vero, che quella forza ba&longs;tante in A <lb/>per vincer l&apos;equilibrio della molla B, quella mede&longs;ima al&shy;<lb/>tre&longs;i &longs;ia &longs;ufficiente a vincer l&apos; equilibrio di tutte l&apos;altre, per <lb/>e&longs;&longs;ere eguali le proportioni delle vetti; ci&ograve; non o&longs;tante, <expan abbr="c&otilde;-&longs;iderando&longs;i">con&shy;<lb/>&longs;iderando&longs;i</expan> il braccio incuruato, come &longs;i vede nell&apos; arco <lb/>KLA &longs;egnato a punti, le proportioni rie&longs;cono alterate; do-<pb pagenum="90" xlink:href="022/01/096.jpg"/>uendo&longs;i prendere per le lunghezze delle vetti &longs;udette, non <lb/>pi&ugrave; le lunghezze di prima, ma bensi le applicate di detto <lb/>arco, cioe af, ag, ai, ak; delle quali aK, e l&apos; altre a lei pi&ugrave; <lb/>vicine &longs;i abbreuiano molto pi&ugrave; quando l&apos; arco &egrave; incurua&shy;<lb/>to, che quando non &egrave;: Onde per tal ragione dourebbero <lb/>le parti pi&ugrave; vicine al centro M aprir&longs;i meno dell&apos;altre pi&ugrave; <lb/>vicine alle e&longs;tremit&agrave;. </s>
          <s id="s.000879">A ci&ograve; &longs;i ri&longs;ponde, che per e&longs;&longs;er la <lb/>corda a o pi&ugrave; obliqua alla lunghezza a e di quel che &longs;ia all&apos; <lb/>altre pi&ugrave; vicine al centro M, quindi ne &longs;iegue, che per quel&apos; <lb/>altra cagione s&apos; aprono pi&ugrave; ageuolmente le parti vicine al <lb/>centro; onde, temperata vna ragione con l&apos; altra &lpar;quando <lb/>l&apos; arco non &longs;ia e&longs;tremamente incuruato&rpar; &longs;i con&longs;egui&longs;ce vno <lb/>&longs;tato d&apos;apertura opportuna. </s>
        </p>
        <p type="main">
          <s id="s.000880">La &longs;econda difficolt&agrave; &egrave; che cia&longs;cuna molla nel &longs;uo re&shy;<lb/>&longs;tringer&longs;i, par che cagioni qualche effetto contrario all&apos;in&shy;<lb/>tento. </s>
          <s id="s.000881">Imperoche, per e&longs;empio, nella molla B il mezzo <lb/>anello, che ri&longs;guarda l&apos;e&longs;tremit&agrave; A, nello &longs;tringer&longs;i f&agrave; ben&longs;i <lb/>il &longs;uo douere, perche il &longs;uo moto &egrave; ver&longs;o il centro M; ma l&apos; <lb/>altra met&agrave;, che ri&longs;guarda il &longs;udetto centro M, nello &longs;trin&shy;<lb/>ger&longs;i, hauendo il &longs;uo moto ver&longs;o A, &longs;i oppone al chiudi&shy;<lb/>mento della molla &longs;eguente C; e il &longs;imile dica&longs;i dell&apos; altre. <lb/></s>
          <s id="s.000882">A ci&ograve; &longs;i &egrave; po&longs;to rimedio col far pi&ugrave; grandi, e pi&ugrave; ma&longs;&longs;iccie <lb/>le molle pi&ugrave; vicine al centro M, accre&longs;cendole, e ingro&longs;&longs;an&shy;<lb/>dole di mano in mano opportunamente. </s>
          <s id="s.000883">Quindi ne &longs;egue <lb/>che per la maggior grandezza <expan abbr="c&otilde;&longs;entono">con&longs;entono</expan> egualmente all&apos; <lb/>aprir&longs;i con facilit&agrave;; ma all&apos; incontro nel &longs;errar&longs;i, per e&longs;&longs;ere <lb/>pi&ugrave; ma&longs;&longs;iccie, e di maggior corpo, vengono ad hauere <lb/>maggior momento delle men corpulenti, &longs;uperando co&ntail; <lb/>ci&ograve; non &longs;olo il detto moto oppo&longs;to, ma etiandio impri&shy;<lb/>mendo maggior moto al ferro dell&apos;arco, con cui &longs;i acco&shy;<lb/>muna il moto. </s>
        </p>
        <p type="main">
          <s id="s.000884">Auuerta&longs;i, che quanto &longs;aranno di maggior numero le <lb/><expan abbr="c&otilde;me&longs;lure">comme&longs;ture</expan>, le molle di maggior pe&longs;o, e l&apos;arco pi&ugrave; pouero di <lb/>corpo, tanto riu&longs;cir&agrave; il colpo a di&longs;mi&longs;ura maggiore, per l&apos; <pb pagenum="91" xlink:href="022/01/097.jpg"/>incuruamento notabile delle braccia, e per il maggior mo&shy;<lb/>mento delle molle; e ci&ograve; con adoperare la mede&longs;ima <lb/>forza. </s>
        </p>
        <p type="main">
          <s id="s.000885">Auuerta&longs;i parimente, che il braccio AE, &egrave; il &longs;uo corri&longs;&shy;<lb/>pondente deuono e&longs;&longs;ere alquanto pi&ugrave; corti, cio&egrave; A vna <lb/>delle e&longs;tremit&agrave; dell&apos;arco deue e&longs;&longs;ere pi&ugrave; ver&longs;o il centro di <lb/>quel che &longs;ia il concor&longs;o delle linee LB, KE, come pure dall&apos; <lb/>altra parte; perche &longs;i vede che aprendo&longs;i meno le parti vi&shy;<lb/>cine ad A, l&apos;altre molle fanno miglior effetto. </s>
        </p>
        <p type="main">
          <s id="s.000886">Finalmente la &longs;perienza ha mo&longs;trato, che e&longs;&longs;endo&longs;i la&shy;<lb/>uorata vna tal machina con pochi&longs;&longs;imi nodi, ageuoli&longs;&longs;ima <lb/>ad aprir&longs;i, e &longs;enza hauer ingrandite e ingro&longs;late le molle, <lb/>che pi&ugrave; &longs;i vanno auuicinando al centro M, come &longs;i &egrave; det&shy;<lb/>to; con tutto ci&ograve; l&apos; ordigno &egrave; riu&longs;cito di forza molto &longs;upe&shy;<lb/>riore a vna bale&longs;tra grande, e difficilif&longs;ima a inarcar&longs;i. </s>
          <s id="s.000887">On&shy;<lb/>de non dubito, che, facendo&longs;i con tutte le regole accenna&shy;<lb/>te, non debba riu&longs;cire vna machina di effetto marauiglio&longs;o <lb/>aggiungendo che per tirar granate dourebbero i bracci <lb/>e&longs;&longs;er di legno, armati di ferro &longs;ol doue &longs;i richiede. </s>
        </p>
        <pb pagenum="92" xlink:href="022/01/098.jpg"/>
        <p type="main">
          <s id="s.000888"><emph type="center"/>Nouum genus Bali&longs;t&aelig; <lb/>Explicatio.<emph.end type="center"/><!-- KEEP S--></s>
        </p>
        <p type="main">
          <s id="s.000889"><arrow.to.target n="marg200"/></s>
        </p>
        <p type="margin">
          <s id="s.000890"><margin.target id="marg200"/><emph type="italics"/>Tab.<emph.end type="italics"/> 9.</s>
        </p>
        <p type="main">
          <s id="s.000891">IN hac figura exprimitur nouum genus Bali&shy;<lb/>&longs;t&aelig;, qu&aelig; machina pr&aelig;&longs;ertim in mole maio&shy;<lb/>ri, non parum vtilitatis afferre pote&longs;t rei mi&shy;<lb/>litari ad eiaculanda mi&longs;&longs;ilia, vt demon&longs;tra&shy;<lb/>bitur. </s>
          <s id="s.000892">Ex eius ver&ograve; partibus, quas &longs;ubinde <lb/>recen&longs;eo, etiam modus &longs;tructur&aelig; apparebit. </s>
        </p>
        <p type="main">
          <s id="s.000893">AM, MX &longs;unt brachia. </s>
          <s id="s.000894">Punctum M centrum machi&shy;<lb/>n&aelig;. </s>
          <s id="s.000895">Per cauitatem M tran&longs;it telum emi&longs;&longs;um. </s>
          <s id="s.000896">Infra M in&shy;<lb/>&longs;eritur manubrium, vt in bali&longs;tis vulgaribus. </s>
          <s id="s.000897">Extremis <lb/>capitibus A, N adnectitur funis. </s>
          <s id="s.000898">Puncta A, E, F, G, I, K <lb/>&longs;unt in linea recta. </s>
          <s id="s.000899">Interualla AE, EF, FG, GI, IK &longs;unt &lpar;li&shy;<lb/>c&egrave;t non nece&longs;&longs;ari&ograve;&rpar; &aelig;qualia. </s>
          <s id="s.000900">Altitudinis, &longs;eu commi&longs;&longs;ur&aelig; <lb/>KL, IH, GD, FC, EB &longs;unt perpendiculares rect&aelig; occult&aelig; <lb/>KA. </s>
          <s id="s.000901">Singul&aelig; autem, dum curuatur arcus, aperiuntur cer&shy;<lb/>ca centra K, I, G, F, E. <!-- KEEP S--></s>
          <s id="s.000902">Hinc &longs;equitur vt funis ex medio <lb/>dum attrahitur in O, aperiantur pr&aelig;dict&aelig; commi&longs;&longs;ur&aelig;, &longs;eu <lb/>nodi, &amp; curuentur vtraque brachia, vt in eorum altero ap&shy;<lb/>paret punctis notato. </s>
          <s id="s.000903">Quilibet ex his nodis arcti&longs;&longs;im&egrave; &longs;trin&shy;<lb/>gitur &longs;upern&egrave;, a &longs;uo elaterio, vt videre e&longs;t in L, H, D, C, B. <lb/><!-- KEEP S--></s>
          <s id="s.000904">Elateria autem qu&ograve; propinquiora centro M tanto maiora, <lb/>&amp; cra&longs;&longs;iora debent e&longs;&longs;e remotioribus: Hinc fit vt, propter <lb/>molem opportun&egrave; auctam, &aelig;qu&egrave; facil&egrave; aperiantur, ac c&aelig;&shy;<lb/>tera; &amp; vice ver&longs;a, propter cra&longs;&longs;itiem maiorem, &longs;ibi relicta <lb/>validi&ugrave;s re&longs;tringantur. </s>
          <s id="s.000905">Cuius rei paulo infra rationem <lb/>dabimus. </s>
        </p>
        <p type="main">
          <s id="s.000906">His po&longs;itis facile e&longs;t o&longs;tendere, quantum pr&aelig;&longs;tet hu&shy;<lb/>iu&longs;cemodi machina vulgaribus &amp; communibus bali&longs;tis. </s>
        </p>
        <p type="main">
          <s id="s.000907">Prim&ugrave;m, in Triangulo ALK c&ugrave;m altitudines EB, FC, <lb/>GD, IH, KL &longs;int perpendiculares, ideoque parallel&aelig;, hinc <pb pagenum="93" xlink:href="022/01/099.jpg"/>&longs;it vtrationes AE ad EB, AF ad FC, AG ad GD, AI ad <lb/>IH, AK ad KL &longs;int &aelig;quales. </s>
          <s id="s.000908">Sunt pariter &aelig;quales re&longs;i&shy;<lb/>&longs;tenti&aelig; elateriorum in B, C, D, H, L &lpar;po&longs;uimus enim ela&shy;<lb/>teria ita opportun&egrave; aucta vt &aelig;qu&egrave; facile &longs;ingula aperian&shy;<lb/>tur&rpar; ergo &lpar;ex primis principijs mechanicorum&rpar; dum attra&shy;<lb/>huntur fune extrema capita A, N, eodem tempore, eadem&shy;<lb/>que facili ate vincetur &aelig;quilibrium omnium elateriorum, <lb/>quorum re&longs;i&longs;tentia in &longs;ingulis con&longs;ideratur in ratione pon&shy;<lb/>deris, quemadmodum line&aelig; AE, EB; AF, FC; AG, GD <lb/>&amp;c. <!-- REMOVE S-->con&longs;iderantur vt vectes, quorum hippomoclia &longs;eu <expan abbr="c&etilde;-tra">cen&shy;<lb/>tra</expan> &longs;unt in E, F, G, I, K, potentia autem con&longs;ideratur in A <lb/>communis omnibus. </s>
        </p>
        <p type="main">
          <s id="s.000909">Secund&ograve;, c&ugrave;m AE ad EB &lpar;idem die de c&aelig;teris&rpar; habeant <lb/>magnam proportionem, facil&egrave; aperientur nodi, &amp; curuabi&shy;<lb/>tur arcus; quantumuis augeatur numerus nodorum. </s>
        </p>
        <p type="main">
          <s id="s.000910">Terti&ograve; Cum &longs;int plures nodi, atque omnes aperiantur, <lb/>nece&longs;&longs;e e&longs;t vt brachia arcus vald&egrave; incuruentur; <expan abbr="quamobr&etilde;">quamobrem</expan> <lb/>&longs;i idem arcus &longs;ibi relinquatur, pr&aelig;dicti nodi omnes, vi ela&shy;<lb/>teriorum, ictu oculi claudentur; eodemque puncto tempo&shy;<lb/>ris corda ex O percurret totum &longs;patium v&longs;que ad M: Qu&ograve;d <lb/>c&ugrave;m fieri nequeat ni&longs;i &longs;umma velocitate, propter magni&shy;<lb/>tudinem pr&aelig;dicti &longs;patij, &amp; velocitati re&longs;pondent vis, atque <lb/>impetus, nece&longs;&longs;e e&longs;t vt hinc &longs;equatur ictus valde notabilis, <lb/>vt facil&egrave; e&longs;t vnicuique conijcere. </s>
        </p>
        <p type="main">
          <s id="s.000911">Super &longs;unt nunc difficultates nonnull&aelig; &longs;oluend&aelig;. </s>
          <s id="s.000912">Prima <lb/>e&longs;t, qu&ograve;d lic&egrave;t vis &longs;ufficiens in A ad vincendum <expan abbr="&aelig;quilibri&utilde;">&aelig;quilibrium</expan> <lb/>elaterij B, illa eadem quoque &longs;ufficiat ad vincendum &aelig;qui&shy;<lb/>librium c&aelig;terorum, propter &aelig;quales proportiones <expan abbr="vecti&utilde;">vectium</expan>; <lb/>his tamen non ob&longs;tantibus, &longs;i con&longs;ideretur brachium iam <lb/>incuruatum, vt apparet in KLA punctis notato, proportio&shy;<lb/>nes ill&aelig; cernuntur notabiliter variat&aelig;. </s>
          <s id="s.000913">Neque enim pro <lb/>longitudinibus vectium &longs;umi po&longs;&longs;unt longitudines priores, <lb/>&longs;ed loco ip&longs;arum accipiend&aelig; &longs;unt applicat&aelig; arcus, videli&shy;<lb/>cet af, ag, ai, ak quarum ak, eidemque propinquiores, <expan abbr="qu&atilde;-">quan-</expan><pb pagenum="94" xlink:href="022/01/100.jpg"/>do arcus incuruatur, breuiores fiunt, qu&agrave;m e&longs;&longs;ent ante&atail;. <lb/></s>
          <s id="s.000914">Re&longs;pondeo, qu&ograve;d corda ao c&ugrave;m &longs;it obliquior re&longs;pectu <lb/>longitudinis ae, qu&agrave;m re&longs;pectu c&aelig;terarum centro propin&shy;<lb/>quiorum, hinc fit vt, quant&ugrave;m e&longs;t ex hac ratione, facili&ugrave;s <lb/>aperiantur partes propinquiores centro; quamobrem, vtra&shy;<lb/>que ratione inuicem temperata, dummodo arcus non &longs;it <lb/>&longs;umm&egrave; incuruatus omnes partes aperientur, quantum &longs;a&shy;<lb/>tis e&longs;t ad intentum. </s>
        </p>
        <p type="main">
          <s id="s.000915">Altera difficultas e&longs;t, quod elaterium quodlibet dum <lb/>re&longs;tringitur videtur ob&longs;tare motui elaterij &longs;equentis. </s>
          <s id="s.000916">Nam, <lb/>exempli gratia, in elaterio B &longs;emiannulus re&longs;piciens extre&shy;<lb/>mum A, dum &longs;tringitur, optim&egrave; pr&aelig;&longs;tat &longs;uum effectum, <expan abbr="c&utilde;">cum</expan> <lb/>eius motus &longs;it vers&ugrave;s centrum M At &egrave; contrario reliqua <lb/>pars, &longs;eu &longs;emiannulus re&longs;piciens pr&aelig;dictum centrum M, <expan abbr="c&utilde;">cum</expan> <lb/>habeat &longs;uum motum ver&longs;us A videtur ob&longs;tare, quo minus <lb/>liber&egrave; claudatur &longs;equens elaterium C. <!-- KEEP S--></s>
          <s id="s.000917">Aque idem de c&aelig;te&shy;<lb/>ris dicendum. </s>
          <s id="s.000918">Huic incommodo con&longs;ultum e&longs;t augendo <lb/>magnitudinem, &amp; cra&longs;&longs;itiem elateriorum, qu&ograve; magis acce&shy;<lb/>dunt ad centrum M. </s>
          <s id="s.000919">Hinc enim &longs;equitur vt propter ma&shy;<lb/>gnitudinem facil&egrave; con&longs;entiant arcui, dum incuruatur; at <lb/>dum idem arcus liber&egrave; &longs;ibi relinquitur, cum &longs;int corpulen&shy;<lb/>tiora &amp; cra&longs;&longs;iora habent maius momentum, qu&agrave;m c&aelig;tera <lb/>graciliora, ideoque non modo vincunt motum illum op&shy;<lb/>po&longs;itum, &longs;ed etiam imprimunt maiorem motum ferro ar&shy;<lb/>cus, cui ille motus communicatur. </s>
        </p>
        <p type="main">
          <s id="s.000920">Aduerte quod commi&longs;&longs;ur&aelig; &longs;eu nodi, qu&ograve; plures fuerint, <lb/>elateria autem maioris ponderis, arcus denique corporis <lb/>gracilioris equ&aelig; expeditioris, tanto ictus longat pr&aelig;&longs;tan&shy;<lb/>tior &longs;equetur, tum propter notabilem curuaturam brachio&shy;<lb/>rum, tum propter momentum maius elateriorum, &amp; <expan abbr="quid&etilde;">quidem</expan> <lb/>po&longs;ita eadem potentia, aut etiam minori, pro vt longitudi&shy;<lb/>nes vectium &longs;tatuuntur. </s>
        </p>
        <p type="main">
          <s id="s.000921">Aduerte etiam, longitudinem brachij AE, eiu&longs;demqu&etail; <lb/>corre&longs;pondentis debere c&aelig;teris paribus nonnihil imminui, <pb pagenum="95" xlink:href="022/01/101.jpg"/>quod fiet &longs;i A, alterum extremum arcus, &longs;it propri&ugrave;s cen&shy;<lb/>tro M, qu&agrave;m &longs;it concur&longs;us linearum LB, KE. <!-- KEEP S--></s>
          <s id="s.000922">Idem dicen&shy;<lb/>dum de altero extremo N. <!-- KEEP S--></s>
          <s id="s.000923">Nam c&ugrave;m minus aperiantur <lb/>partes propinquiores puncto A, c&aelig;tera elateria, vt com&shy;<lb/>pertum e&longs;t, meliorem effectum pr&aelig;&longs;tant. </s>
        </p>
        <p type="main">
          <s id="s.000924">Fauet denique experientia. </s>
          <s id="s.000925">Nam huiu&longs;cemodi machi&shy;<lb/>na pauci&longs;&longs;imis nodis con&longs;tructa, facillim&aelig; curuatur&aelig;, cum <lb/>elaterijs eiu&longs;dem pror&longs;us molis &amp; cra&longs;&longs;itudinis; nihilomi&shy;<lb/>nus long&egrave; &longs;uperauit vim bali&longs;t&aelig; communis maxim&aelig;, &amp; dif<lb/>ficillim&aelig; flexionis. </s>
          <s id="s.000926">Quamobrem non dubito quin, &longs;i pr&aelig;&shy;<lb/>cepta &longs;uperi&ugrave;s data exact&egrave; &longs;eruentur, elaborari po&longs;&longs;it ma&shy;<lb/>china mir&aelig; vtilitatis. </s>
          <s id="s.000927">Adde po&longs;tremo ad iacienda <expan abbr="qu&aelig;d&atilde;">qu&aelig;dam</expan> <lb/>mi&longs;&longs;ilia, vt e&longs;t genus quoddam bolidum, vulgo <emph type="italics"/>granate,<emph.end type="italics"/> op&shy;<lb/>portuniora e&longs;&longs;e brachia lignea, tantummodo, vbi nece&longs;&longs;i&shy;<lb/>tas po&longs;tulat, armata ferro. </s>
        </p>
        <p type="main">
          <s id="s.000928"><emph type="center"/><emph type="italics"/>FINIS.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s>
        </p>
      </chap>
    </body>
    <back>
      <pb xlink:href="022/01/102.jpg"/>
      <section>
        <p type="main">
          <s id="s.000929"><emph type="italics"/>Vid. <!-- REMOVE S-->D. <!-- REMOVE S-->Bernardus Marchellus Re&shy;<lb/>ctor P&oelig;nitent. <!-- REMOVE S-->in Metropol. <!-- REMOVE S-->Bonon. <lb/><!-- REMOVE S-->pro Illu&longs;tri&longs;s. <!-- REMOVE S-->&amp; Reverendi&longs;s. <!-- REMOVE S-->Domino <lb/>D. <!-- REMOVE S-->Iacobo Boncompagno Archiepi&longs;&shy;<lb/>copo, &amp; Principe.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000930"><emph type="italics"/>Vid. <!-- REMOVE S-->Silue&longs;ter Bonfiliolus Inqui&longs;itionis <lb/>reui&longs;or, &amp; imprimi po&longs;&longs;e cen&longs;uit.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000931"><emph type="italics"/>Stante atte&longs;tatione.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000932"><emph type="center"/><emph type="italics"/>Imprimatur.<emph.end type="italics"/><emph.end type="center"/></s>
        </p>
        <p type="main">
          <s id="s.000933"><emph type="italics"/>F. <!-- REMOVE S-->Io&longs;eph Maria Agudius Vicarius <lb/>Sancti Offi c ij Bononi&aelig;.<emph.end type="italics"/></s>
        </p>
        <p type="main">
          <s id="s.000934"><emph type="center"/>8 00 57<emph.end type="center"/></s>
        </p>
        <pb xlink:href="022/01/103.jpg"/>
        <p type="caption">
          <s id="s.000935">TABVLA I.<lb/><figure id="id.022.01.103.1.jpg" xlink:href="022/01/103/1.jpg"/><!-- KEEP S--></s>
        </p>
        <pb xlink:href="022/01/104.jpg"/>
        <p type="caption">
          <s id="s.000936">TABVLA II.<lb/><figure id="id.022.01.104.1.jpg" xlink:href="022/01/104/1.jpg"/><!-- KEEP S--></s>
        </p>
        <pb xlink:href="022/01/105.jpg"/>
        <p type="caption">
          <s id="s.000937">TABVLA III.<lb/><figure id="id.022.01.105.1.jpg" xlink:href="022/01/105/1.jpg"/><!-- KEEP S--></s>
        </p>
        <pb xlink:href="022/01/106.jpg"/>
        <p type="caption">
          <s id="s.000938">TABVLA VI.<lb/><figure id="id.022.01.106.1.jpg" xlink:href="022/01/106/1.jpg"/><!-- KEEP S--></s>
        </p>
        <pb xlink:href="022/01/107.jpg"/>
        <p type="caption">
          <s id="s.000939">TABVLA V.<lb/><figure id="id.022.01.107.1.jpg" xlink:href="022/01/107/1.jpg"/><!-- KEEP S--></s>
        </p>
        <pb xlink:href="022/01/108.jpg"/>
        <p type="caption">
          <s id="s.000940">TABVLA IV.<lb/><figure id="id.022.01.108.1.jpg" xlink:href="022/01/108/1.jpg"/><!-- KEEP S--></s>
        </p>
        <pb xlink:href="022/01/109.jpg"/>
        <p type="caption">
          <s id="s.000941">TABVLA VII.<lb/><figure id="id.022.01.109.1.jpg" xlink:href="022/01/109/1.jpg"/><!-- KEEP S--></s>
        </p>
        <pb xlink:href="022/01/110.jpg"/>
        <p type="caption">
          <s id="s.000942">TABVLA VIII.<lb/><figure id="id.022.01.110.1.jpg" xlink:href="022/01/110/1.jpg"/><pb xlink:href="022/01/111.jpg"/><figure id="id.022.01.111.1.jpg" xlink:href="022/01/111/1.jpg"/><!-- KEEP S--></s>
        </p>
        <pb xlink:href="022/01/112.jpg"/>
        <figure id="id.022.01.112.1.jpg" xlink:href="022/01/112/1.jpg"/>
        <p type="caption">
          <s id="s.000943">TABVLA VIIII.<!-- KEEP S--></s>
        </p>
      </section>
    </back>
  </text>
</archimedes>