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

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        <div xml:id="echoid-div281" type="section" level="1" n="126">
          <head xml:id="echoid-head131" xml:space="preserve">PROBL. XVII. PROP. LI.</head>
          <p>
            <s xml:id="echoid-s3036" xml:space="preserve">Datæ Parabolæ per punctum intra ipſam datum MAXIMAM
              <lb/>
            Parabolen inſcribere, & </s>
            <s xml:id="echoid-s3037" xml:space="preserve">è contra.</s>
            <s xml:id="echoid-s3038" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3039" xml:space="preserve">Datæ Parabolæ per punctum extra ipſam datum MINIMAM
              <lb/>
            Parabolen circumſcribere.</s>
            <s xml:id="echoid-s3040" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3041" xml:space="preserve">SIt data Parabole ABC, & </s>
            <s xml:id="echoid-s3042" xml:space="preserve">oporteat primò per punctum D intra ipſam da-
              <lb/>
            tum _MAXIMAM_ Parabolen inſcribere.</s>
            <s xml:id="echoid-s3043" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3044" xml:space="preserve">Ducatur diameter
              <lb/>
              <figure xlink:label="fig-0114-01" xlink:href="fig-0114-01a" number="79">
                <image file="0114-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0114-01"/>
              </figure>
            BDE, cuius rectum
              <lb/>
            latus ſit BF, (quod
              <lb/>
            in poſterùm intelli-
              <lb/>
            gatur ſemper ex ver-
              <lb/>
            tice cõtingenter ap-
              <lb/>
            plicatum ſectioni,
              <lb/>
            prout in prę;</s>
            <s xml:id="echoid-s3045" xml:space="preserve">cedenti-
              <lb/>
            bus factum eſt, & </s>
            <s xml:id="echoid-s3046" xml:space="preserve">in
              <lb/>
            quinta primarũ defi-
              <lb/>
            nitionũ monuimus)
              <lb/>
            & </s>
            <s xml:id="echoid-s3047" xml:space="preserve">per verticem D,
              <lb/>
            circa diametrũ D E
              <lb/>
            adſcribatur datæ Parabolæ ABC Parabole GDH, cuius rectum DI
              <note symbol="a" position="left" xlink:label="note-0114-01" xlink:href="note-0114-01a" xml:space="preserve">5. huius.</note>
            le ſit recto BF; </s>
            <s xml:id="echoid-s3048" xml:space="preserve">nam ipſa erit congruens datæ Dico hanc eſſe _MAXIMAM_
              <lb/>
            in ſcriptam quæſitam.</s>
            <s xml:id="echoid-s3049" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3050" xml:space="preserve">Nam cum ipſæ ſint congruentes Parabolæ per diuerſos vertices ſimul ad-
              <lb/>
            ſcriptæ erunt inter ſe nunquam coeuntes: </s>
            <s xml:id="echoid-s3051" xml:space="preserve">quare GDH datæ ABC erit
              <note symbol="b" position="left" xlink:label="note-0114-02" xlink:href="note-0114-02a" xml:space="preserve">42. h.</note>
            datum punctum D inſcripta.</s>
            <s xml:id="echoid-s3052" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3053" xml:space="preserve">Ampliùs, quælibet alia Parabole per verticem D adſcripta cum recto,
              <lb/>
            quod minus ſit ipſo DI minor eſt Parabola GDH, quæ verò cum recto
              <note symbol="c" position="left" xlink:label="note-0114-03" xlink:href="note-0114-03a" xml:space="preserve">2. Co-
                <lb/>
              roll. 19. h.</note>
            quod excedat ipſum DI, qualis eſt MDN, eſt quidem maior GDH,
              <note symbol="d" position="left" xlink:label="note-0114-04" xlink:href="note-0114-04a" xml:space="preserve">ibidem.</note>
            omnino ſecat circumſcriptam ABC. </s>
            <s xml:id="echoid-s3054" xml:space="preserve">Nam ſi fiat vt LI ad ID, ita BD ad DE,
              <lb/>
            & </s>
            <s xml:id="echoid-s3055" xml:space="preserve">per E applicetur EMA ſecans BA in A, & </s>
            <s xml:id="echoid-s3056" xml:space="preserve">DM in M. </s>
            <s xml:id="echoid-s3057" xml:space="preserve">Cum ſit BD ad DE,
              <lb/>
            vt Li ad ID, erit componendo BE ad ED, vt LD ad DI; </s>
            <s xml:id="echoid-s3058" xml:space="preserve">vnde rectangulum
              <lb/>
            ſub extremis BE, & </s>
            <s xml:id="echoid-s3059" xml:space="preserve">DI, ſiue BF, hoc eſt quadratum applicatæ AE in
              <note symbol="e" position="left" xlink:label="note-0114-05" xlink:href="note-0114-05a" xml:space="preserve">Coroll.
                <lb/>
              1. huius.</note>
            rabola ABC, æquale erit rectangulo ſub medijs ED, DL ſiue quadrato
              <note symbol="f" position="left" xlink:label="note-0114-06" xlink:href="note-0114-06a" xml:space="preserve">ibidem.</note>
            plicatę; </s>
            <s xml:id="echoid-s3060" xml:space="preserve">ME in Parabola MDN, ac ideò AE, ME ſunt æquales, quapropter
              <lb/>
            Parabole DN occurrit ſibi adſcriptæ BA, per diuerſos vertices in puncto M,
              <lb/>
            & </s>
            <s xml:id="echoid-s3061" xml:space="preserve">ob id in eodem occurſu, & </s>
            <s xml:id="echoid-s3062" xml:space="preserve">ad alteram quoque partem ſe mutuò ſecabũt:</s>
            <s xml:id="echoid-s3063" xml:space="preserve">
              <note symbol="g" position="left" xlink:label="note-0114-07" xlink:href="note-0114-07a" xml:space="preserve">50. h.</note>
            Itaque congruens Parabole GDH erit _MAXIMA_ inſcripta quæſita. </s>
            <s xml:id="echoid-s3064" xml:space="preserve">Quod
              <lb/>
            primò, &</s>
            <s xml:id="echoid-s3065" xml:space="preserve">c.</s>
            <s xml:id="echoid-s3066" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3067" xml:space="preserve">IAM datæ Parabolæ GDH, oporteat per punctum B extra ipſam datam
              <lb/>
            _MINIMAM_ Parabolen circumſcribere.</s>
            <s xml:id="echoid-s3068" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3069" xml:space="preserve">Ducatur BDE diameter datę; </s>
            <s xml:id="echoid-s3070" xml:space="preserve">GDH, cuius rectum ſit DI, & </s>
            <s xml:id="echoid-s3071" xml:space="preserve">ei adſcribatur
              <lb/>
            per B, cum recto BF, quod æquet ipſum DI, congruens Parabole ABC:</s>
            <s xml:id="echoid-s3072" xml:space="preserve"/>
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