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

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        <div xml:id="echoid-div383" type="section" level="1" n="161">
          <p>
            <s xml:id="echoid-s3974" xml:space="preserve">
              <pb o="118" file="0142" n="142" rhead=""/>
            circulum dico in punctis D, E, quæſitum ſoluere: </s>
            <s xml:id="echoid-s3975" xml:space="preserve">nempe AE, & </s>
            <s xml:id="echoid-s3976" xml:space="preserve">AD eſſe
              <lb/>
            quæſitas extremas. </s>
            <s xml:id="echoid-s3977" xml:space="preserve">Nam cum ſit BD æqualis BE, erit data AB media ari-
              <lb/>
            thmetica inter inuentas EA, AD. </s>
            <s xml:id="echoid-s3978" xml:space="preserve">Cumque ſit BF radius circuli EFD, & </s>
            <s xml:id="echoid-s3979" xml:space="preserve">an-
              <lb/>
            gulus BFA rectus, erit FA ipſi circulo contingens, quare rectangulum EAD
              <lb/>
            æquabitur quadrato AF, ſiue quadrato AC, vnde data AC erit media geo-
              <lb/>
            metrica inter eaſdem inuentas EA, AD. </s>
            <s xml:id="echoid-s3980" xml:space="preserve">Quare ignotæ extremæ, ſunt in-
              <lb/>
            uentæ, vti quærebantur. </s>
            <s xml:id="echoid-s3981" xml:space="preserve">Quod, &</s>
            <s xml:id="echoid-s3982" xml:space="preserve">c.</s>
            <s xml:id="echoid-s3983" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div385" type="section" level="1" n="162">
          <head xml:id="echoid-head167" xml:space="preserve">PROBL. XXX. PROP. LXXV.</head>
          <p>
            <s xml:id="echoid-s3984" xml:space="preserve">Datæ Parabolæ, per punctum intra ipſam datum, MAXIMAM
              <lb/>
            Ellipſim inſcribere, cuius latera datam habeant rationem: </s>
            <s xml:id="echoid-s3985" xml:space="preserve">& </s>
            <s xml:id="echoid-s3986" xml:space="preserve">è
              <lb/>
            contra.</s>
            <s xml:id="echoid-s3987" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3988" xml:space="preserve">Datæ Ellipſi, per punctum extra ipſam datum, MINIMAM
              <lb/>
            Parabolen circumſcribere.</s>
            <s xml:id="echoid-s3989" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3990" xml:space="preserve">ESto data Parabole ABC, & </s>
            <s xml:id="echoid-s3991" xml:space="preserve">datum intra ipſam punctum ſit E; </s>
            <s xml:id="echoid-s3992" xml:space="preserve">oportet per
              <lb/>
            E _MAXIMAM_ Ellipſim inſcribere, cuius rectum latus ad tranſuerſum
              <lb/>
            rationem habeat R ad S.</s>
            <s xml:id="echoid-s3993" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3994" xml:space="preserve">Ducatur ex E Parabolę diameter BED,
              <lb/>
              <figure xlink:label="fig-0142-01" xlink:href="fig-0142-01a" number="110">
                <image file="0142-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0142-01"/>
              </figure>
            & </s>
            <s xml:id="echoid-s3995" xml:space="preserve">applicetur EF, & </s>
            <s xml:id="echoid-s3996" xml:space="preserve">ſumpta V media pro-
              <lb/>
            portionali inter S, & </s>
            <s xml:id="echoid-s3997" xml:space="preserve">R; </s>
            <s xml:id="echoid-s3998" xml:space="preserve">fiat vt R ad V, ita
              <lb/>
            FE ad ED, iunctaque FD, quæ producta
              <lb/>
            ſectioni occurrat in G, ex quo
              <note symbol="a" position="left" xlink:label="note-0142-01" xlink:href="note-0142-01a" xml:space="preserve">27. pri-
                <lb/>
              mi conic.</note>
            GHI, circa tranſuerſum latus EH, & </s>
            <s xml:id="echoid-s3999" xml:space="preserve">ter-
              <lb/>
            minos applicatæ AC deſcribatur
              <note symbol="b" position="left" xlink:label="note-0142-02" xlink:href="note-0142-02a" xml:space="preserve">Coroll.
                <lb/>
              57. h.</note>
            AECH. </s>
            <s xml:id="echoid-s4000" xml:space="preserve">Hanc dico eſſe quæſitam.</s>
            <s xml:id="echoid-s4001" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4002" xml:space="preserve">Cum enim in Parabola ſint diametri
              <note symbol="c" position="left" xlink:label="note-0142-03" xlink:href="note-0142-03a" xml:space="preserve">Coroll.
                <lb/>
              1. 13. h.</note>
            gmenta BH, BD, BE proportionalia, ſint-
              <lb/>
            que quadrata applicatarum IH, AD, FE
              <lb/>
            in eadem ratione ipſorum
              <note symbol="d" position="left" xlink:label="note-0142-04" xlink:href="note-0142-04a" xml:space="preserve">20. pri-
                <lb/>
              mi conic.</note>
            erunt quoq; </s>
            <s xml:id="echoid-s4003" xml:space="preserve">ipſæ applicatæ continuæ pro-
              <lb/>
            portionales, quapropter rectangulum ſub
              <lb/>
            IH, vel ſub HG, & </s>
            <s xml:id="echoid-s4004" xml:space="preserve">FE æquabitur quadrato AD, ac proinde quadratum
              <lb/>
            AD, ad rectangulum HDE, erit vt rectangulum ſub GH, EF, ad idem re-
              <lb/>
            ctangulum HDE, ſed rectangulum ſub GH, EF, ad ſibi ſimile rectangulum
              <lb/>
            HDE, (habent enim circa rectos angulos latera proportionalia, cum ſit GH
              <lb/>
            ad HD, vt FE ad ED, & </s>
            <s xml:id="echoid-s4005" xml:space="preserve">permutando GH ad FE, vt HD ad DE) eſt vt qua-
              <lb/>
            dratum FE ad ED (vtraque enim proportio, duplicata eſt proportionis linee
              <lb/>
            FE ad ED) quo circa, & </s>
            <s xml:id="echoid-s4006" xml:space="preserve">quadratum AD ad rectangulum HDE, hoc eſt in
              <lb/>
            Ellipſi, @@ rectum latus ad tranſuerſum, erit vt quadratum FE ad ED,
              <note symbol="e" position="left" xlink:label="note-0142-05" xlink:href="note-0142-05a" xml:space="preserve">22. pri-
                <lb/>
              mi conic.</note>
            vt quadratum R ad V, vel vt data linea R ad S. </s>
            <s xml:id="echoid-s4007" xml:space="preserve">Deſcripta eſt ergo Ellipſis
              <lb/>
            AECH, cuius latera habent datam rationem R ad S. </s>
            <s xml:id="echoid-s4008" xml:space="preserve">& </s>
            <s xml:id="echoid-s4009" xml:space="preserve">eſt datæ
              <note symbol="f" position="left" xlink:label="note-0142-06" xlink:href="note-0142-06a" xml:space="preserve">Schol.
                <lb/>
              62. h.</note>
            ABC inſcripta. </s>
            <s xml:id="echoid-s4010" xml:space="preserve">Amplius dico, ipſam eſſe _MAXIMAM_ Ellipſium quarum
              <lb/>
            latera ſint in ratione R ad S, ſiue eſſe _MAXIMAM_ ſibi ſimilium: </s>
            <s xml:id="echoid-s4011" xml:space="preserve">nam, quæ
              <lb/>
            cum minoribus lateribus datæ Parabolæ per E adſcribitur ad partes H, </s>
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