Viviani, Vincenzo, De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei
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            plicentur. </s>
            <s xml:id="echoid-s3411" xml:space="preserve">Dico ipſam eſſe quæſitam. </s>
            <s xml:id="echoid-s3412" xml:space="preserve">Cum enim ſit CB media proportio
              <lb/>
              <note symbol="a" position="left" xlink:label="note-0126-01" xlink:href="note-0126-01a" xml:space="preserve">Coroll.
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              prop. 1. h.</note>
            nalis inter altitudinem BA, & </s>
            <s xml:id="echoid-s3413" xml:space="preserve">latitudinem BE, erit BC, itemque ei æqua- lis BD, deſcriptæ ſectionis ſemi-applicata, nempe ſectio per C, & </s>
            <s xml:id="echoid-s3414" xml:space="preserve">D omni-
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            no tranſibit; </s>
            <s xml:id="echoid-s3415" xml:space="preserve">eſtque A vertex, AB diameter, & </s>
            <s xml:id="echoid-s3416" xml:space="preserve">AF tranſuerſum Hyperbole,
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            aut Ellipſis, ex conſtructione: </s>
            <s xml:id="echoid-s3417" xml:space="preserve">quare factum eſt quod erat propoſitum.</s>
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        <div xml:id="echoid-div324" type="section" level="1" n="140">
          <head xml:id="echoid-head145" xml:space="preserve">COROLL.</head>
          <p>
            <s xml:id="echoid-s3419" xml:space="preserve">EX hac conſtat quomodo, magnitudine, & </s>
            <s xml:id="echoid-s3420" xml:space="preserve">poſitione datis tranſuerſo la-
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            tere, aut diametro AF, & </s>
            <s xml:id="echoid-s3421" xml:space="preserve">vna applicatarum CD, per terminos, A, C,
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            F, D, Ellipſis deſcribi poſſit.</s>
            <s xml:id="echoid-s3422" xml:space="preserve"/>
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        <div xml:id="echoid-div325" type="section" level="1" n="141">
          <head xml:id="echoid-head146" xml:space="preserve">THEOR. XXIX. PROP. LIIX.</head>
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            <s xml:id="echoid-s3423" xml:space="preserve">Si coni-ſectionem, vel circuli circumferentiam duæ rectæ lineæ
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            contingant, ipſæ productæ conuenient ſimul extra ſectionem; </s>
            <s xml:id="echoid-s3424" xml:space="preserve">ſed
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            in Parabola, vel Hyperbola ſibiipſis occurrent ad partes periphe-
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            riæ à contactibus terminatę: </s>
            <s xml:id="echoid-s3425" xml:space="preserve">In Ellipſi verò ad partes ſui ipſius por-
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            tionis à linea tactus iungente abſciſſæ, in qua centrum nõ reperitur.</s>
            <s xml:id="echoid-s3426" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3427" xml:space="preserve">ESto Parabole, vel Hyperbole ABC (nam de circulo, & </s>
            <s xml:id="echoid-s3428" xml:space="preserve">Ellipſi id ab
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            Apollonio oſtenſum fuit in vigeſima ſeptima ſecũdi conicorum) quàm
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            in punctis A, C tangant rectæ AD, CE.
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            </s>
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              <figure xlink:label="fig-0126-01" xlink:href="fig-0126-01a" number="91">
                <image file="0126-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0126-01"/>
              </figure>
            Dico, ſi producantur ad partes ſectionis
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            ABC à contactibus A, C terminatæ ipſas
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            inter ſe conuenire.</s>
            <s xml:id="echoid-s3430" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3431" xml:space="preserve">Si enim per alterum contactuum, vt per
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            C, intelligatur fectro ris diameter H C G,
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              <note symbol="b" position="left" xlink:label="note-0126-02" xlink:href="note-0126-02a" xml:space="preserve">24. 25.
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              pr. conic.</note>
            certum eſt contingentem A D, ſi produ- catur, cum diametro HG extra ſectionem
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            conuenire, hoc eſt ad partes G; </s>
            <s xml:id="echoid-s3432" xml:space="preserve">ſiergo AD
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            ſecat CG, neceſſariò ſecabit priùs tangen-
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            tem CE, quæ cadit inter ſectionis periphe-
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            riam ABC, & </s>
            <s xml:id="echoid-s3433" xml:space="preserve">diametrum HC: </s>
            <s xml:id="echoid-s3434" xml:space="preserve">quare tan-
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            gentes AD, CE ſibi ipſis occurrunt. </s>
            <s xml:id="echoid-s3435" xml:space="preserve">Quod
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            erat, &</s>
            <s xml:id="echoid-s3436" xml:space="preserve">c.</s>
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          <head xml:id="echoid-head147" xml:space="preserve">ALITER.</head>
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            <s xml:id="echoid-s3438" xml:space="preserve">CVm recta CE ſectioni occurrat, & </s>
            <s xml:id="echoid-s3439" xml:space="preserve">producta ex vtraque parte extra ſe-
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            ctionem cadat, ſi ex puncto A, quod eſt in ſectione, ducta ſit AF, ipſi
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            EC æquidiſtans, producta ex vtraque parte ſectioni occurret; </s>
            <s xml:id="echoid-s3440" xml:space="preserve">ſed AD
              <note symbol="c" position="left" xlink:label="note-0126-03" xlink:href="note-0126-03a" xml:space="preserve">18. pri-
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              mi conic.</note>
            cadit extra ſectionem, cum ſit contingens; </s>
            <s xml:id="echoid-s3441" xml:space="preserve">quare AD non congruit cum AF,
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            ſed ipſę ſe mutuò ſecant. </s>
            <s xml:id="echoid-s3442" xml:space="preserve">Cum ergo DA ſecet alteram æquidiſtantium AF, ſi
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            producatur, ſecabit, & </s>
            <s xml:id="echoid-s3443" xml:space="preserve">reliquam CE ad partes peripheriæ ABC à contacti-
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            bus A, C, terminatæ. </s>
            <s xml:id="echoid-s3444" xml:space="preserve">Quod demonſtrandum erat.</s>
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