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

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            <s xml:id="echoid-s766" xml:space="preserve">
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            quoniam harum quoque habemus demonſtrationes breuiores, & </s>
            <s xml:id="echoid-s767" xml:space="preserve">affirmati-
              <lb/>
            uas, non indirectas, quales ab Apollonio exhibentur in prima, ſecunda, ac
              <lb/>
            decima tertia, nè noſtri libelli molem aliundè tranſcriptis demonſtratiombus
              <lb/>
            augere velle videamur, apponemus hic proprias, ita procedendo.</s>
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        <div xml:id="echoid-div55" type="section" level="1" n="34">
          <head xml:id="echoid-head39" xml:space="preserve">THEOR. II. PROP. VIII.</head>
          <p>
            <s xml:id="echoid-s769" xml:space="preserve">Si Hyperbolen recta linea ad verticem contingat, & </s>
            <s xml:id="echoid-s770" xml:space="preserve">ab ipſa ex
              <lb/>
            vertice ad vtramque partem diametri ſumatur æqualis ei, quæ po-
              <lb/>
              <note position="left" xlink:label="note-0040-01" xlink:href="note-0040-01a" xml:space="preserve">Prop. 1. 2
                <lb/>
              ſecundi
                <lb/>
              con ic.</note>
            teſt quartam figuræ partem, quæ à ſectionis centro ad ſumptos ter-
              <lb/>
            minos contingentis ducuntur cum ſectione non conuenient; </s>
            <s xml:id="echoid-s771" xml:space="preserve">(quæ
              <lb/>
            in poſterum cum Apollonio vocentur ASYMPTOTI) nec erit al-
              <lb/>
            tera aſymptoton, quæ diuidat angulum ab ipſis factum.</s>
            <s xml:id="echoid-s772" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s773" xml:space="preserve">SIt Hyperbole, cuius diameter, & </s>
            <s xml:id="echoid-s774" xml:space="preserve">tranſuerſum latus AB, centrum C, & </s>
            <s xml:id="echoid-s775" xml:space="preserve">
              <lb/>
            rectum figuræ latus B F, linea verò D E ſectionem contingat in B, & </s>
            <s xml:id="echoid-s776" xml:space="preserve">
              <lb/>
              <figure xlink:label="fig-0040-01" xlink:href="fig-0040-01a" number="16">
                <image file="0040-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0040-01"/>
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            quartæ parti figuræ, quæ à lateribus
              <lb/>
            AB, BF continetur æquale ſit quadra-
              <lb/>
            tum vtriuſque ipſarum BD, BE, & </s>
            <s xml:id="echoid-s777" xml:space="preserve">iun-
              <lb/>
            ctæ CD, CE producantur. </s>
            <s xml:id="echoid-s778" xml:space="preserve">Dico pri-
              <lb/>
            mum eas cum ſectione numquam con-
              <lb/>
            uenire.</s>
            <s xml:id="echoid-s779" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s780" xml:space="preserve">Nam in altera ipſarum, vt in CD,
              <lb/>
            infra contingentem, ſumpto quolibet
              <lb/>
            puncto G, ab eo ordinatim applicetur
              <lb/>
            GIH ſectionem, ac diametrum ſecans
              <lb/>
            in I, H, quæ ipſi D B æquidiſtabit. </s>
            <s xml:id="echoid-s781" xml:space="preserve">Et
              <lb/>
            quoniam eſt vt latus AB ad BF, ita
              <lb/>
            quadratum AB ad rectangulum ABF,
              <lb/>
            vel ſumptis horum ſub-quadruplis, ita
              <lb/>
            quadratum CB ad quadratum BD, vel quadratum CH ad quadratum HG,
              <lb/>
            & </s>
            <s xml:id="echoid-s782" xml:space="preserve">vt idem latus AB ad BF ita eſt rectangulum AHB ad quadratum HI,
              <note symbol="a" position="left" xlink:label="note-0040-02" xlink:href="note-0040-02a" xml:space="preserve">21. pri-
                <lb/>
              mi conic.</note>
            quadratum CH ad HG, vt rectangulum AHB ad quadratum HI, & </s>
            <s xml:id="echoid-s783" xml:space="preserve">permu-
              <lb/>
            tando quadratum CH ad rectangulum AHB, vt quadratum GH, ad HI,
              <lb/>
            ſed quadratum CH maius eſt rectangulo AHB (cum eius exceſſus ſit qua-
              <lb/>
            dratum CB, nam eſt AB ſecta bifariam in C, & </s>
            <s xml:id="echoid-s784" xml:space="preserve">ei adiecta eſt quædam B H)
              <lb/>
            quare & </s>
            <s xml:id="echoid-s785" xml:space="preserve">quadratum GH quadrato IH maius erit, hoc eſt punctum G cadet
              <lb/>
            extra Hy perbolen, & </s>
            <s xml:id="echoid-s786" xml:space="preserve">hoc ſemper de omnibus punctis rectarum CDG, CEL
              <lb/>
            quamuis in infinitum productarum. </s>
            <s xml:id="echoid-s787" xml:space="preserve">Sunt igitur lineæ CD; </s>
            <s xml:id="echoid-s788" xml:space="preserve">CE ſectioni nun-
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            quam occurrentes. </s>
            <s xml:id="echoid-s789" xml:space="preserve">Quod erat primò demonſtrandum, taleſque lineæ vo-
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            centur ASYMPTOTI.</s>
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          </p>
          <p>
            <s xml:id="echoid-s791" xml:space="preserve">Amplius, ijſdem manentibus, dico quamlibet aliam CM, quæ diuidat
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            angulum DCE, neceſſariò Hyperbolen ſecare. </s>
            <s xml:id="echoid-s792" xml:space="preserve">Ducta enim BM, ex vertice
              <lb/>
            B, parallcla ad CD, conueniet cum CM; </s>
            <s xml:id="echoid-s793" xml:space="preserve">nam & </s>
            <s xml:id="echoid-s794" xml:space="preserve">ipſa CM cum altera æqui-
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            diſtantium CD conuenit in C: </s>
            <s xml:id="echoid-s795" xml:space="preserve">occurrat ergo in M, per quod ordinatim </s>
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