Viviani, Vincenzo, De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei

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          <p>
            <s xml:id="echoid-s3112" xml:space="preserve">Cum ſit BE ad EM, vt OH ad HE, erit
              <lb/>
              <figure xlink:label="fig-0116-01" xlink:href="fig-0116-01a" number="81">
                <image file="0116-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0116-01"/>
              </figure>
            componendo BM ad ME, vt OE ad EH,
              <lb/>
            vel vt RM ad MN; </s>
            <s xml:id="echoid-s3113" xml:space="preserve">quapropter rectan-
              <lb/>
            gulum BMN ſiue quadratum
              <note symbol="a" position="left" xlink:label="note-0116-01" xlink:href="note-0116-01a" xml:space="preserve">Coroll.
                <lb/>
              1. huius.</note>
            AM in Hyperbola ABC, æquale erit re-
              <lb/>
            ctangulo EMR, ſiue quadrato
              <note symbol="b" position="left" xlink:label="note-0116-02" xlink:href="note-0116-02a" xml:space="preserve">ibidem.</note>
            MP in Hyperbola PEQ; </s>
            <s xml:id="echoid-s3114" xml:space="preserve">ac ideò lineæ
              <lb/>
            MA, MP ſunt æquales, quare Hyperbo-
              <lb/>
            læ ABC, PEQ occurrunt ſimul in pun-
              <lb/>
            cto Q, in quo etiam ſe mutuò ſecabunt.
              <lb/>
            </s>
            <s xml:id="echoid-s3115" xml:space="preserve">Nã ſumpto in ſectione QEP infra P quo-
              <lb/>
            libet puncto S, per quod applicata STV,
              <lb/>
            ſectionem ABC, diametrum, ac regulas
              <lb/>
            ſecans in T, V, X, Y: </s>
            <s xml:id="echoid-s3116" xml:space="preserve">cum ſit EM minor
              <lb/>
            EV, habebit BE ad EM, vel OH ad HE,
              <lb/>
            vel YX ad XV, maiorem rationem quam
              <lb/>
            BE ad EV, & </s>
            <s xml:id="echoid-s3117" xml:space="preserve">componendo YV ad VX
              <lb/>
            maiorem rationem, quàm BV ad VE, vnde rectangulum YVE, ſiue
              <note symbol="c" position="left" xlink:label="note-0116-03" xlink:href="note-0116-03a" xml:space="preserve">Coroll.
                <lb/>
              1. huius.</note>
            dratum VS in Hyperbola EAS, maius erit rectangulo XVB, ſiue
              <note symbol="d" position="left" xlink:label="note-0116-04" xlink:href="note-0116-04a" xml:space="preserve">16. ſept.
                <lb/>
              Pappi.</note>
              <note symbol="e" position="left" xlink:label="note-0116-05" xlink:href="note-0116-05a" xml:space="preserve">Coroll.
                <lb/>
              1. huius.</note>
            to VT in Hyperbola ABC: </s>
            <s xml:id="echoid-s3118" xml:space="preserve">vnde punctum S cadit extra Hyperbolen ABC,
              <lb/>
            ac ideò ipſæ Hyperbolæ ſe mutuò ſecant, ſicuti in altero extremo Q, eiuſ-
              <lb/>
            dem applicatæ. </s>
            <s xml:id="echoid-s3119" xml:space="preserve">Erit ergo Hyperbole IEL, quæ datæ ABC ſimilis eſt, & </s>
            <s xml:id="echoid-s3120" xml:space="preserve">ad
              <lb/>
            eandem regulam, _MAXIMA_ inſcripta quæſita. </s>
            <s xml:id="echoid-s3121" xml:space="preserve">Quod erat, &</s>
            <s xml:id="echoid-s3122" xml:space="preserve">c.</s>
            <s xml:id="echoid-s3123" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s3124" xml:space="preserve">Verùm cum ad inueſtigationem MAXIMAE, & </s>
            <s xml:id="echoid-s3125" xml:space="preserve">MINIMAE in-
              <lb/>
            ſcriptæ, ac circumſcriptæ ſectionis, mlreferat ignorare quo nam in puncto,
              <lb/>
            maior, vel minor quæſitarum ſectionum, datæ ſectioni occurrat, ſufficit
              <lb/>
            enim oſtendere ipſas, vbicunq; </s>
            <s xml:id="echoid-s3126" xml:space="preserve">ſit earum occurſus, aliquando ſe mutuò ſeca-
              <lb/>
            re) ideò in proximè ſequentibus problematibus, hac omiſſa methodo per appli-
              <lb/>
            catarum potentias, tanquam prolixiori, & </s>
            <s xml:id="echoid-s3127" xml:space="preserve">minus concinna, hoc ipſum aliter
              <lb/>
            elegantiori induſtria demonſtr abimus, & </s>
            <s xml:id="echoid-s3128" xml:space="preserve">licet id pluribus, ac varijs aggreſ-
              <lb/>
            ſionibus conſequi poſsit, vt in hac, & </s>
            <s xml:id="echoid-s3129" xml:space="preserve">proxima propoſitione videre licet, ta-
              <lb/>
            men eas eligemus, quæ apportunæ magis nobis videbuntur.</s>
            <s xml:id="echoid-s3130" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div291" type="section" level="1" n="128">
          <head xml:id="echoid-head133" xml:space="preserve">ALITER.</head>
          <p>
            <s xml:id="echoid-s3131" xml:space="preserve">SEcetur igitur E F bifariam in 2, erit 2 centrum Hyperbolarum IEL,
              <lb/>
            PEQ, ex quo ductis harum aſumptotis, videlicet 2 3 inſcriptæ IEL,
              <lb/>
            & </s>
            <s xml:id="echoid-s3132" xml:space="preserve">2 4 circumſcriptæ PEQ, quæ cadet extra aſymptoton 2 3, ex D
              <note symbol="a" position="left" xlink:label="note-0116-06" xlink:href="note-0116-06a" xml:space="preserve">Exvlti-
                <lb/>
              ma parte
                <lb/>
              37. huius.</note>
            que agatur D 5 aſymptotos Hyperbolæ ABC.</s>
            <s xml:id="echoid-s3133" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3134" xml:space="preserve">Iam cum Hyperbolæ ABC, IEL ſimiles ſint, per diuerſos vertices, & </s>
            <s xml:id="echoid-s3135" xml:space="preserve">ad
              <lb/>
            eandem regulam FGH ſimul adſcriptæ, erunt earum aſymptoti D 5, 2 3
              <lb/>
            inter ſe parallelæ ſed 2 4 inter ipſas cadit, & </s>
            <s xml:id="echoid-s3136" xml:space="preserve">alteram 2 3 ſecat in 2,
              <note symbol="b" position="left" xlink:label="note-0116-07" xlink:href="note-0116-07a" xml:space="preserve">Coroll.
                <lb/>
              45. huius.</note>
            re ipſa 2 4 producta ad partes 4, ſecabit & </s>
            <s xml:id="echoid-s3137" xml:space="preserve">reliquam D 5; </s>
            <s xml:id="echoid-s3138" xml:space="preserve">ſed eſt 2 4
              <lb/>
            aſymptotos ſectionis SPEQ, & </s>
            <s xml:id="echoid-s3139" xml:space="preserve">quædam recta D 5 occurrit ei, ac </s>
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