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

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          <p>
            <s xml:id="echoid-s703" xml:space="preserve">
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            IDH: </s>
            <s xml:id="echoid-s704" xml:space="preserve">quare rectangulum ADB ad rectangulum EDB, erit vt idem ADB ad
              <lb/>
            IDH, ſed ADB ad EDB, eſt vt AD ad DE, vel vt AB ad BC, ergo rectan-
              <lb/>
            gulum quoque ADB ad rectangulum IDH, erit vt AB ad BC. </s>
            <s xml:id="echoid-s705" xml:space="preserve">Sequitur er-
              <lb/>
            go vt AB ſit tranſuerſum latus, & </s>
            <s xml:id="echoid-s706" xml:space="preserve">BC rectum deſcriptæ Hyperbolæ, vt in
              <lb/>
            prima huius oſtenſum eſt. </s>
            <s xml:id="echoid-s707" xml:space="preserve">Quod erat faciendum.</s>
            <s xml:id="echoid-s708" xml:space="preserve"/>
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        <div xml:id="echoid-div51" type="section" level="1" n="32">
          <head xml:id="echoid-head37" xml:space="preserve">PROBL. V. PROP. VII.</head>
          <p>
            <s xml:id="echoid-s709" xml:space="preserve">Duabus datis in ſubiecto plano rectis lineis terminatis, inuenire
              <lb/>
            in eodem plano circa ipſarum alteram, tanquam circà diametrum,
              <lb/>
              <note position="left" xlink:label="note-0038-01" xlink:href="note-0038-01a" xml:space="preserve">Prop. 54.
                <lb/>
              pri. con.</note>
            coni - ſectionem, quæ Ellipſis appellatur, cuius tranſuerſum latus
              <lb/>
            ſit prædicta diameter, rectum verò latus ſit altera data linea, & </s>
            <s xml:id="echoid-s710" xml:space="preserve">ad dia-
              <lb/>
            metrō
              <unsure/>
            ordinatim ductæ in dato angulo applicentur.</s>
            <s xml:id="echoid-s711" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s712" xml:space="preserve">SInt datæ in ſubiecto plano terminate rectæ lineæ AB, BC, quæ ad datum
              <lb/>
            angulum P componantur. </s>
            <s xml:id="echoid-s713" xml:space="preserve">Oportet in ſubiecto plano Ellipſim deſcribe-
              <lb/>
            re, cuius diameter ſit AB, vertex B, tranſuerſum latus AB, rectum BC, & </s>
            <s xml:id="echoid-s714" xml:space="preserve">ad
              <lb/>
            diametrō
              <unsure/>
            AB ordinatim ductæ conſtituant angulos dato, angulo P æquales.</s>
            <s xml:id="echoid-s715" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s716" xml:space="preserve">Iungatur AC, ſumaturque in
              <lb/>
              <figure xlink:label="fig-0038-01" xlink:href="fig-0038-01a" number="15">
                <image file="0038-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0038-01"/>
              </figure>
            AB quodcunque punctum D, à
              <lb/>
            quo ducatur, in ſubiecto plano,
              <lb/>
            recta GDFE ipſi BC parallela, è
              <lb/>
            qua ex vtraque parte abſcindan-
              <lb/>
            tur DF, DG mediæ proportiona-
              <lb/>
            les inter BD, & </s>
            <s xml:id="echoid-s717" xml:space="preserve">DE; </s>
            <s xml:id="echoid-s718" xml:space="preserve">erit vtriuſq;
              <lb/>
            </s>
            <s xml:id="echoid-s719" xml:space="preserve">ipſarum quadratum ęquale rectã-
              <lb/>
            gulo EDB: </s>
            <s xml:id="echoid-s720" xml:space="preserve">per rectam autem FG
              <lb/>
            intelligatur ſecans planum IFGHG
              <lb/>
            ad vtramque partem ſubiecti pla-
              <lb/>
            ni productum, quorum commu-
              <lb/>
            nis ſectio ſit recta FG, cui per D
              <lb/>
            in plano ſecante IFHG, perpen-
              <lb/>
            dicularis ducatur IDH hic inde
              <lb/>
            producta.</s>
            <s xml:id="echoid-s721" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s722" xml:space="preserve">Iam, veleſt CB non maior BA, vel maior. </s>
            <s xml:id="echoid-s723" xml:space="preserve">Si non maior, erit quoque ED
              <lb/>
            non maior ipſa DA. </s>
            <s xml:id="echoid-s724" xml:space="preserve">Itaque ex educta IDH infra ſubiectum planum dema-
              <lb/>
            tur DI, quæ maior ſit ipſa DB, iungatur I B, & </s>
            <s xml:id="echoid-s725" xml:space="preserve">ex A ducatur AO parallela
              <lb/>
            ad I B, ſecans IDH in O, & </s>
            <s xml:id="echoid-s726" xml:space="preserve">fiat vt ID ad DF, ita D F ad aliam DH; </s>
            <s xml:id="echoid-s727" xml:space="preserve">erit re-
              <lb/>
            ctangulum IDH æquale quadrato D F, ſiue rectangulo EDB, ſed rectangu-
              <lb/>
            lum EDB eſt non maius rectangulo ADB (nam eſt ED non maior recta DA)
              <lb/>
            ergo rectangulum IDH erit non maius rectangulo BDA, ſed rectangulum
              <lb/>
            IDO maius eſt rectangulo BDA (nam cum ſit vt ID ad DB, ita OD ad DA,
              <lb/>
            ſitque I D maior D B ex conſtructione, erit quoque DO maior DA) quare
              <lb/>
            IDH rectangulum minus erit rectangulo IDO, hoc eſt linea DH minor DO;
              <lb/>
            </s>
            <s xml:id="echoid-s728" xml:space="preserve">vnde punctum H eſt inter D, & </s>
            <s xml:id="echoid-s729" xml:space="preserve">O, ſiue inter parallelas I B, AO; </s>
            <s xml:id="echoid-s730" xml:space="preserve">quare iun-
              <lb/>
            cta AH, & </s>
            <s xml:id="echoid-s731" xml:space="preserve">producta ſecabit productam IB ad partes B, L, vt putà in L.</s>
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