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

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        <div xml:id="echoid-div380" type="section" level="1" n="160">
          <head xml:id="echoid-head165" xml:space="preserve">PROBL. XXIX. PROP. LXXIII.</head>
          <p>
            <s xml:id="echoid-s3944" xml:space="preserve">Dato angulo rectilineo, per punctum in qualibet eius diametro
              <lb/>
            datum, MAXIMAM Ellipſim inſcribere, cuius latera datam ha-
              <lb/>
            beant rationem.</s>
            <s xml:id="echoid-s3945" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3946" xml:space="preserve">SIt datus angulus ABC, diameter BD, & </s>
            <s xml:id="echoid-s3947" xml:space="preserve">datum punctum D, per quod
              <lb/>
            oporteat Ellipſim inſcribere, cuius tranſuerſum latus ad rectum, datam
              <lb/>
            quamcunque habeat rationem E ad F, & </s>
            <s xml:id="echoid-s3948" xml:space="preserve">ſit _MAXIMA_.</s>
            <s xml:id="echoid-s3949" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s3950" xml:space="preserve">Applicetur per D, ordinatim GDH, &</s>
            <s xml:id="echoid-s3951" xml:space="preserve">
              <note symbol="a" position="right" xlink:label="note-0141-01" xlink:href="note-0141-01a" xml:space="preserve">66. h.</note>
              <figure xlink:label="fig-0141-01" xlink:href="fig-0141-01a" number="108">
                <image file="0141-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0141-01"/>
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            per H ducatur HIL diametrum ſecans in I, & </s>
            <s xml:id="echoid-s3952" xml:space="preserve">
              <lb/>
            BA in L, ita vt ex I, & </s>
            <s xml:id="echoid-s3953" xml:space="preserve">L ductis A I, LM ipſi
              <lb/>
            DH parallelis, rectangulum DIM, ad quadra-
              <lb/>
            tum AI, rationem habeat E ad F, & </s>
            <s xml:id="echoid-s3954" xml:space="preserve">cum
              <note symbol="b" position="right" xlink:label="note-0141-02" xlink:href="note-0141-02a" xml:space="preserve">72. h.</note>
            uerſo DM, per extrema applicatæ AC, Elli-
              <lb/>
            pſis deſcribatur DAMC. </s>
            <s xml:id="echoid-s3955" xml:space="preserve">Dico hanc
              <note symbol="c" position="right" xlink:label="note-0141-03" xlink:href="note-0141-03a" xml:space="preserve">Coroll.
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              57. h.</note>
            _MAXIMAM_ quæſitam.</s>
            <s xml:id="echoid-s3956" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s3957" xml:space="preserve">Eſt enim LB ad BG, ſiue MB ad BD,
              <note symbol="d" position="right" xlink:label="note-0141-04" xlink:href="note-0141-04a" xml:space="preserve">71. h.</note>
            LA ad AG, ſiue vt MI ad ID, quare BA, BC
              <lb/>
            Ellipſim contingent, ideoque ipſa erit
              <note symbol="e" position="right" xlink:label="note-0141-05" xlink:href="note-0141-05a" xml:space="preserve">34. pri.
                <lb/>
              conic.</note>
            lo inſcripta, eritque _MAXIMA_, vt in præce-
              <lb/>
            dentibus oſtenſum fuit. </s>
            <s xml:id="echoid-s3958" xml:space="preserve">Quod erat, &</s>
            <s xml:id="echoid-s3959" xml:space="preserve">c.</s>
            <s xml:id="echoid-s3960" xml:space="preserve"/>
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        <div xml:id="echoid-div383" type="section" level="1" n="161">
          <head xml:id="echoid-head166" xml:space="preserve">LEMMA X. PROP. LXXIV.</head>
          <p>
            <s xml:id="echoid-s3961" xml:space="preserve">Datis medijs proportionalibus, Arithmetica nempe, & </s>
            <s xml:id="echoid-s3962" xml:space="preserve">Geo-
              <lb/>
            metrica inter eaſdem ignotas extremas; </s>
            <s xml:id="echoid-s3963" xml:space="preserve">ipſas extremas inuenire.</s>
            <s xml:id="echoid-s3964" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3965" xml:space="preserve">SIt AB media arithmetica, & </s>
            <s xml:id="echoid-s3966" xml:space="preserve">AC media
              <lb/>
              <figure xlink:label="fig-0141-02" xlink:href="fig-0141-02a" number="109">
                <image file="0141-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0141-02"/>
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            geometrica inter duas eaſdem ignotas
              <lb/>
            extremas, quarum idem ſit terminus A, & </s>
            <s xml:id="echoid-s3967" xml:space="preserve">
              <lb/>
            ſimul congruere intelligantur: </s>
            <s xml:id="echoid-s3968" xml:space="preserve">patet primò
              <lb/>
            AB ſuperare ipſam AC, cum media ari-
              <lb/>
            thmetica ſit maior media geometrica. </s>
            <s xml:id="echoid-s3969" xml:space="preserve">Iam
              <lb/>
            oporteat datis AC, AB ignotas extremas
              <lb/>
            proportionales inuenire.</s>
            <s xml:id="echoid-s3970" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3971" xml:space="preserve">Fiat centro A interuallo A C circulus
              <lb/>
            CF, cui ex puncto B contingens ducatur
              <lb/>
            BF, quæ eum radio FA rectum efficiet an-
              <lb/>
            gulum, vnde ſubtenſa BA erit maior ipſa
              <lb/>
            BF; </s>
            <s xml:id="echoid-s3972" xml:space="preserve">ſi ergo cum centro B, interuallo BF
              <lb/>
            deſcribatur ſemi- circulus DFE, ipſæ ſeca-
              <lb/>
            bit BA infra A, ſed tamen vltra C (cum ſit
              <lb/>
            BC minor BF, eo quod AC æquatur AF,
              <lb/>
            &</s>
            <s xml:id="echoid-s3973" xml:space="preserve">tota AB minor eſt duobus AF, FB) ſecabitque productam AB in E; </s>
            <s xml:id="echoid-s3974" xml:space="preserve"/>
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