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

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        <div xml:id="echoid-div30" type="section" level="1" n="23">
          <head xml:id="echoid-head28" xml:space="preserve">PROBL. I. PROP. II.</head>
          <p>
            <s xml:id="echoid-s507" xml:space="preserve">Datæ Parabolæ per punctum in ea datum lineam contingentem
              <lb/>
            ducere.</s>
            <s xml:id="echoid-s508" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s509" xml:space="preserve">SIt Parabole, cuius diameter AB, & </s>
            <s xml:id="echoid-s510" xml:space="preserve">datum in ea punctum ſit C. </s>
            <s xml:id="echoid-s511" xml:space="preserve">Opor-
              <lb/>
            tet ex C Parabolæ contingentem rectam lineam ducere.</s>
            <s xml:id="echoid-s512" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">Prop. 33.
            <lb/>
          primi co-
            <lb/>
          nic.</note>
          <p>
            <s xml:id="echoid-s513" xml:space="preserve">Applicetur ordinatim recta CD, & </s>
            <s xml:id="echoid-s514" xml:space="preserve">diametri ſegmento DE æqualis po-
              <lb/>
            natur EA, iungaturque ACF. </s>
            <s xml:id="echoid-s515" xml:space="preserve">Dico ipſam eſſe tangentem quæſitam.</s>
            <s xml:id="echoid-s516" xml:space="preserve"/>
          </p>
          <figure number="6">
            <image file="0030-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0030-01"/>
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          <p>
            <s xml:id="echoid-s517" xml:space="preserve">Sumpto enim in ſectione quolibet puncto G, per eum applicetur BGF
              <lb/>
            rectam AC ſecans in F, diametrum verò in B, & </s>
            <s xml:id="echoid-s518" xml:space="preserve">iuncta DF ex E vertice.
              <lb/>
            </s>
            <s xml:id="echoid-s519" xml:space="preserve">ducatur EHM parallela ad AF ſecans DF in H, & </s>
            <s xml:id="echoid-s520" xml:space="preserve">CD in M, ſitque HL ipſi
              <lb/>
            FB æquidiſtans. </s>
            <s xml:id="echoid-s521" xml:space="preserve">Iam cum ſit AE æqualis ED, erit FH æqualis HD, ob pa-
              <lb/>
            rallelas AF, EH; </s>
            <s xml:id="echoid-s522" xml:space="preserve">itemque BL æqualis LB ob æquidiſtantes BF, LH: </s>
            <s xml:id="echoid-s523" xml:space="preserve">quare
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            fumpta EI media geometrica inter DE, & </s>
            <s xml:id="echoid-s524" xml:space="preserve">EB ipſa EI minor erit media. </s>
            <s xml:id="echoid-s525" xml:space="preserve">
              <lb/>
            arithmetica EL. </s>
            <s xml:id="echoid-s526" xml:space="preserve">Ampliùs quadratum GB ad CD eſt vt linea EB ad
              <note symbol="a" position="left" xlink:label="note-0030-02" xlink:href="note-0030-02a" xml:space="preserve">20. pri-
                <lb/>
              mi conic.</note>
            vel vt quadratum mediæ geometricæ EI ad quadratum ED, ergo & </s>
            <s xml:id="echoid-s527" xml:space="preserve">linea.
              <lb/>
            </s>
            <s xml:id="echoid-s528" xml:space="preserve">GB ad CD erit vt EI ad ED, cumque ſit EI minor EL, habebit EI ad ED: </s>
            <s xml:id="echoid-s529" xml:space="preserve">
              <lb/>
            ſiue GB ad CD, minorem rationem quam EL ad ED, vel quàm EH ad EM,
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            ſeu quam AF ad AC, vel quàm FB ad eandem CD, ergo GB minor eſt FB: </s>
            <s xml:id="echoid-s530" xml:space="preserve">
              <lb/>
            quare punctum F cadit extra Parabolen, & </s>
            <s xml:id="echoid-s531" xml:space="preserve">ſic de quolibet alio puncto rectæ
              <lb/>
            ACF. </s>
            <s xml:id="echoid-s532" xml:space="preserve">Vnde ipſa ACF Parabolen contingit in C. </s>
            <s xml:id="echoid-s533" xml:space="preserve">Quod faciendumerat.</s>
            <s xml:id="echoid-s534" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div32" type="section" level="1" n="24">
          <head xml:id="echoid-head29" xml:space="preserve">ALITER.</head>
          <p>
            <s xml:id="echoid-s535" xml:space="preserve">IIfdem poſitis, dico iterum punctum F cadere extra Parabolen. </s>
            <s xml:id="echoid-s536" xml:space="preserve">Nam ſe-
              <lb/>
            cta AB bifariam in H, cum eadem quoque in æqualiter ſecta ſit in E (nã
              <lb/>
            cum ſit DE æqualis EA, erit in prima figura BE maior EA, & </s>
            <s xml:id="echoid-s537" xml:space="preserve">in ſecunda BE
              <lb/>
            minor EA) erit rectangulum AHB maius rectangulo AEB, ac propterea.
              <lb/>
            </s>
            <s xml:id="echoid-s538" xml:space="preserve">quadratum EA ad rectangulum AHB, ſiue ad quadratum AH minorem </s>
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