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

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        <div xml:id="echoid-div267" type="section" level="1" n="121">
          <head xml:id="echoid-head126" xml:space="preserve">THEOR. XXVI. PROP. XXXXVIII.</head>
          <p>
            <s xml:id="echoid-s2804" xml:space="preserve">Similes Hyperbolæ per diuerſos vertices ſimul adſcriptæ habent
              <lb/>
            aſymptotos parallelas, & </s>
            <s xml:id="echoid-s2805" xml:space="preserve">quando centrum interioris cadat vltra
              <lb/>
            centrum exterioris, tunc huius aſymptotos interiorem Hyperbolen
              <lb/>
            ſecabit, ac ipſæ Hyperbolæ neceſſariò ſe mutuò ſecabunt. </s>
            <s xml:id="echoid-s2806" xml:space="preserve">Cum
              <lb/>
            verò centrum interioris idem ſit cum centro exterioris, tunc vnius
              <lb/>
            aſymptotos erit aſymptotos alterius; </s>
            <s xml:id="echoid-s2807" xml:space="preserve">& </s>
            <s xml:id="echoid-s2808" xml:space="preserve">ſectiones erunt ſimul nun-
              <lb/>
            quam coeuntes. </s>
            <s xml:id="echoid-s2809" xml:space="preserve">Et ſi interioris centrum cadat infra centrum ex-
              <lb/>
            terioris, tunc eædem ſectiones erunt inter ſe nunquam coeuntes; </s>
            <s xml:id="echoid-s2810" xml:space="preserve">& </s>
            <s xml:id="echoid-s2811" xml:space="preserve">
              <lb/>
            aſymptotos inſcriptæ ſecabit Hyperbolen circumſcriptam.</s>
            <s xml:id="echoid-s2812" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2813" xml:space="preserve">SInt, vt in vtraque figura huius propoſitionis, duæ ſimiles Hyperbolæ
              <lb/>
            ABC, DEF per diuerſos vertices B, E ſimul adſcriptæ, quarum centra
              <lb/>
            ſint G, H, & </s>
            <s xml:id="echoid-s2814" xml:space="preserve">ſectionis ABC aſymptoti ſint GI, GO; </s>
            <s xml:id="echoid-s2815" xml:space="preserve">ſectionis verò DEF ſint
              <lb/>
            HM, HR; </s>
            <s xml:id="echoid-s2816" xml:space="preserve">Dico has aſymptotos eſſe inter ſe æquidiſtantes.</s>
            <s xml:id="echoid-s2817" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2818" xml:space="preserve">Nam in ſimilibus Hyperbolis ABC, DEF, anguli IGE, MHE, ab earum
              <lb/>
            aſymptotis, & </s>
            <s xml:id="echoid-s2819" xml:space="preserve">diametris ad homologas partes facti ſunt æquales,
              <note symbol="a" position="left" xlink:label="note-0108-01" xlink:href="note-0108-01a" xml:space="preserve">Coroll.
                <lb/>
              40. huius.</note>
            alterni, quare ipſæ aſymptoti inter ſe æquidiſtabunt. </s>
            <s xml:id="echoid-s2820" xml:space="preserve">Quod primò, &</s>
            <s xml:id="echoid-s2821" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2822" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2823" xml:space="preserve">Iam in hac prima figura, (in
              <lb/>
              <figure xlink:label="fig-0108-01" xlink:href="fig-0108-01a" number="75">
                <image file="0108-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0108-01"/>
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            qua centrum H interioris DEF
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            remotius eſt à verticibus B, E,
              <lb/>
            quàm ſit centrum G exterioris
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            Hyperbolæ ABC) cum ſint HM,
              <lb/>
            HR aſymptoti Hyperbolæ DEF,
              <lb/>
            & </s>
            <s xml:id="echoid-s2824" xml:space="preserve">in loco ab eis, & </s>
            <s xml:id="echoid-s2825" xml:space="preserve">ſectione ter-
              <lb/>
            minato ducta ſit GI alteri aſym-
              <lb/>
            ptoton
              <unsure/>
            HM æquidiſtans, ipſa
              <lb/>
            omnino ſectionem DEF ſecabit.
              <lb/>
            </s>
            <s xml:id="echoid-s2826" xml:space="preserve">Quod ſecundò, &</s>
            <s xml:id="echoid-s2827" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2828" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2829" xml:space="preserve">Sed ipſa GI, cum ſit aſympto-
              <lb/>
            tos ſectionis ABC, tota cadit ex-
              <lb/>
            tra ipſam BA, quare occurſus
              <lb/>
            prædictæ aſymptoton GI cum ſe-
              <lb/>
            ctione ED, erit extra ſectionem
              <lb/>
            BA, vnde ipſa interior ſectio ED
              <lb/>
            neceſſariò ſecabit priùs exterio-
              <lb/>
            rem BA. </s>
            <s xml:id="echoid-s2830" xml:space="preserve">Quod tertiò, &</s>
            <s xml:id="echoid-s2831" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2832" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2833" xml:space="preserve">Ad pleniorem autem doctrinam, ſi quæratur, quo nam in puncto huiuſ-
              <lb/>
            modi Hyperbolæ ſe mutuò ſecent, ita id conſequetur.</s>
            <s xml:id="echoid-s2834" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2835" xml:space="preserve">Sumpta enim GS æquali GB, erit tota BS tranſuerſum exterioris ABC;
              <lb/>
            </s>
            <s xml:id="echoid-s2836" xml:space="preserve">item ſumpta HT æquali HE, erit tota TE tranſuerſum interioris DEF.</s>
            <s xml:id="echoid-s2837" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2838" xml:space="preserve">Iam, vel H centrum interioris cadit in ipſo puncto S; </s>
            <s xml:id="echoid-s2839" xml:space="preserve">vel ſupra inter S, & </s>
            <s xml:id="echoid-s2840" xml:space="preserve">
              <lb/>
            T, vel infra inter G, &</s>
            <s xml:id="echoid-s2841" xml:space="preserve">S.</s>
            <s xml:id="echoid-s2842" xml:space="preserve"/>
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