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">
          <p>
            <s xml:id="echoid-s3072" xml:space="preserve">
              <pb o="91" file="0115" n="115" rhead=""/>
            Dico hanc eſſe _MINIMAM_ circumſcriptam quæſitam.</s>
            <s xml:id="echoid-s3073" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3074" xml:space="preserve">Cum ſint enim ipſæ Parabolæ congruentes, & </s>
            <s xml:id="echoid-s3075" xml:space="preserve">per diuerſos vertices ad-
              <lb/>
            ſcriptæ, erunt inter ſe nunquam coeuntes quare ABC datæ GDH erit
              <note symbol="a" position="right" xlink:label="note-0115-01" xlink:href="note-0115-01a" xml:space="preserve">42. h.</note>
            cumſcripta.</s>
            <s xml:id="echoid-s3076" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3077" xml:space="preserve">Præterea, quælibet alia Parabole per B adſcripta cum recto, quod exce-
              <lb/>
            dat BF, maior eſt ipſa ABC, quę verò cum recto BO, quod minus ſit ipſo BF,
              <lb/>
            qualis eſt PBQ, eſt quidem minor ipſa ABC, ſed omnino ſecat inſcriptam
              <lb/>
            GDH. </s>
            <s xml:id="echoid-s3078" xml:space="preserve">Quoniam ſi fiat vt FO ad OB, ita BD ad DE, ac per E applicetur
              <lb/>
            EGP ſecans DG in G, & </s>
            <s xml:id="echoid-s3079" xml:space="preserve">BP in P: </s>
            <s xml:id="echoid-s3080" xml:space="preserve">cum ſit BD ad DE, vt FO ad OB, erit com-
              <lb/>
            ponendo BE ad ED, vt FB ad BO; </s>
            <s xml:id="echoid-s3081" xml:space="preserve">vnde rectangulum ſub BE, & </s>
            <s xml:id="echoid-s3082" xml:space="preserve">BO
              <note symbol="b" position="right" xlink:label="note-0115-02" xlink:href="note-0115-02a" xml:space="preserve">1. Co-
                <lb/>
              roll. 1. h.</note>
            quadratum applicatæ EP in Parabola PBQ æquale erit rectangulo ſub me-
              <lb/>
            dijs ED, & </s>
            <s xml:id="echoid-s3083" xml:space="preserve">BF, ſiue DI, hoc eſt quadrato applicatę EG in Parabola GDH:</s>
            <s xml:id="echoid-s3084" xml:space="preserve">
              <note symbol="c" position="right" xlink:label="note-0115-03" xlink:href="note-0115-03a" xml:space="preserve">ibidem.</note>
            vnde EP, EG ſunt æquales. </s>
            <s xml:id="echoid-s3085" xml:space="preserve">Occurrit ergo Parabole BP, ſibi adſcriptæ DG
              <lb/>
            per diuerſos vertices, in puncto P, quare in eodem occurſu, & </s>
            <s xml:id="echoid-s3086" xml:space="preserve">ad alteram
              <lb/>
            partem ſe mutuò ſecant. </s>
            <s xml:id="echoid-s3087" xml:space="preserve">Quapropter congruens Parabole ABC erit
              <note symbol="d" position="right" xlink:label="note-0115-04" xlink:href="note-0115-04a" xml:space="preserve">50. h.</note>
            _NIMA_ circumſcripta quæſita.</s>
            <s xml:id="echoid-s3088" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div287" type="section" level="1" n="127">
          <head xml:id="echoid-head132" xml:space="preserve">PROBL. XVIII. PROP. LII.</head>
          <p>
            <s xml:id="echoid-s3089" xml:space="preserve">Datæ Hyperbolę, per punctum intra ipſam datum MAXIMAM
              <lb/>
            Hyperbolen inſcribere, quarum eadem ſit regula.</s>
            <s xml:id="echoid-s3090" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3091" xml:space="preserve">ESto data Hyperbole ABC, cuius centrum D; </s>
            <s xml:id="echoid-s3092" xml:space="preserve">& </s>
            <s xml:id="echoid-s3093" xml:space="preserve">punctum intra ipſam da-
              <lb/>
            tum ſit E. </s>
            <s xml:id="echoid-s3094" xml:space="preserve">Oportet per E Hyperbolen inſcribere, quæ ſit _MAXIMA_,
              <lb/>
            ſed tamen eius regula ſit quoque regula datæ ſectionis.</s>
            <s xml:id="echoid-s3095" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3096" xml:space="preserve">Iungatur ED ſecans datã ſectionem in
              <lb/>
              <figure xlink:label="fig-0115-01" xlink:href="fig-0115-01a" number="80">
                <image file="0115-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0115-01"/>
              </figure>
            B, & </s>
            <s xml:id="echoid-s3097" xml:space="preserve">producta ſumatur DF æqualis BD,
              <lb/>
            erit FB trãſuerſum ſectionis ABC,
              <note symbol="a" position="right" xlink:label="note-0115-05" xlink:href="note-0115-05a" xml:space="preserve">47. 1.
                <lb/>
              conic.</note>
            vertex B, ſitque BG eius rectum latus, & </s>
            <s xml:id="echoid-s3098" xml:space="preserve">
              <lb/>
            regula FG, quæ producatur, & </s>
            <s xml:id="echoid-s3099" xml:space="preserve">per E ſit
              <lb/>
            ducta EH parallela ad BG, & </s>
            <s xml:id="echoid-s3100" xml:space="preserve">per verticẽ
              <lb/>
            E, circa communem diametrum BE, da-
              <lb/>
            tę ſectioni ABC adſcribatur
              <note symbol="b" position="right" xlink:label="note-0115-06" xlink:href="note-0115-06a" xml:space="preserve">7. huius.</note>
            IEL, cuius latera ſint FE, EH, hoc eſt
              <lb/>
            eadem ſit regula FGH: </s>
            <s xml:id="echoid-s3101" xml:space="preserve">patet ipſam IEL
              <lb/>
            datæ ABC eſſe inſcriptam, cum in infini-
              <lb/>
            tum productæ ſint inter ſe
              <note symbol="c" position="right" xlink:label="note-0115-07" xlink:href="note-0115-07a" xml:space="preserve">45. h.</note>
            coeuntes.</s>
            <s xml:id="echoid-s3102" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3103" xml:space="preserve">Dico ampliùs ipſam IEL eſſe _MAXI-_
              <lb/>
            _MAM_. </s>
            <s xml:id="echoid-s3104" xml:space="preserve">Quoniam quęlibet alia adſcripta
              <lb/>
            per verticem E, cum eodem tranſuerſo
              <lb/>
            FE, ſed cum recto, quod minus ſit recto
              <lb/>
            EH, minor eſt ipſa IEL; </s>
            <s xml:id="echoid-s3105" xml:space="preserve">quæ verò cum recto EO, quod excedat EH,
              <note symbol="d" position="right" xlink:label="note-0115-08" xlink:href="note-0115-08a" xml:space="preserve">2. Co-
                <lb/>
              roll. 19. h.</note>
            lis eſt Hyperbole PEQ, eſt quidem maior ipſa IEL; </s>
            <s xml:id="echoid-s3106" xml:space="preserve">ſed omnino ſecat
              <note symbol="e" position="right" xlink:label="note-0115-09" xlink:href="note-0115-09a" xml:space="preserve">ibidem.</note>
            ABC. </s>
            <s xml:id="echoid-s3107" xml:space="preserve">Nam ſi fiat vt OH ad HE, ita BE ad EM, & </s>
            <s xml:id="echoid-s3108" xml:space="preserve">per M applicetur MPA
              <lb/>
            Hyperbolen PEQ ſecans in P, BA verò in A, & </s>
            <s xml:id="echoid-s3109" xml:space="preserve">producta ſecet regulam
              <lb/>
            FH, in N, & </s>
            <s xml:id="echoid-s3110" xml:space="preserve">iunctam regulam FO deſcriptæ Hyperbolæ PEQ in R.</s>
            <s xml:id="echoid-s3111" xml:space="preserve"/>
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