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

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          <head xml:id="echoid-head100" xml:space="preserve">THEOR. XV. PROP. XXXIV.</head>
          <p>
            <s xml:id="echoid-s2210" xml:space="preserve">Si à puncto, quod eſt in Hyperbola ducatur recta linea alteri
              <lb/>
            aſymptoton æquidiſtans, ipſa, ac ſectio, quæ inter has parallelas
              <lb/>
            intercipitur, in infinitum productę ſunt infra occurſum ſemper ma-
              <lb/>
            gis recedentes, ſed tamen nunquam perueniunt ad interuallum
              <lb/>
            æquale cuidam dato interuallo; </s>
            <s xml:id="echoid-s2211" xml:space="preserve">dum earum diſtantia metiatur per
              <lb/>
            interceptas æquidiſtantes cuilibet rectæ, quæ ducta ſit ex occurſu
              <lb/>
            vtramque aſymptoton ſecans.</s>
            <s xml:id="echoid-s2212" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2213" xml:space="preserve">SIt Hyperbole ABC, cuius aſymptoti ED, EF, ſitque ex quolibet ſectio-
              <lb/>
            nis puncto B recta BGN alteri aſymptoto ED æquidiſtans, quæ intra
              <lb/>
            lectionẽ cadens, in nullo alio pũcto quam
              <lb/>
              <figure xlink:label="fig-0087-01" xlink:href="fig-0087-01a" number="57">
                <image file="0087-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0087-01"/>
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            B cum ipſa conueniet. </s>
            <s xml:id="echoid-s2214" xml:space="preserve">Dico primùm (ſi
              <note symbol="a" position="right" xlink:label="note-0087-01" xlink:href="note-0087-01a" xml:space="preserve">Coroll.
                <lb/>
              11. huius.</note>
            B ducatur quæcunque HBF vtranq; </s>
            <s xml:id="echoid-s2215" xml:space="preserve">aſym-
              <lb/>
            ptoton ſecans) ipſam, & </s>
            <s xml:id="echoid-s2216" xml:space="preserve">ſectionem IAM
              <lb/>
            infra BI eſſe sẽper magis inter ſerecedẽtes.</s>
            <s xml:id="echoid-s2217" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s2218" xml:space="preserve">Applicentur quotcunque DAG, LMN
              <lb/>
            infra HB, ipſi æquidiſtantes: </s>
            <s xml:id="echoid-s2219" xml:space="preserve">patet has om-
              <lb/>
            nes LN, DG, HB inter ſe æquales eſſe, ſed
              <lb/>
            eſt DA minor HI, ergo AG maior
              <note symbol="b" position="right" xlink:label="note-0087-02" xlink:href="note-0087-02a" xml:space="preserve">10. h.</note>
            IB, eſtque LM minor DA, quare & </s>
            <s xml:id="echoid-s2220" xml:space="preserve">MN
              <lb/>
            maior AG, & </s>
            <s xml:id="echoid-s2221" xml:space="preserve">hoc ſemper ſi in infinitum
              <lb/>
            producantur; </s>
            <s xml:id="echoid-s2222" xml:space="preserve">ergo linea BGN, & </s>
            <s xml:id="echoid-s2223" xml:space="preserve">ſectio
              <lb/>
            IAM ſunt ſemper ſimul recedentes. </s>
            <s xml:id="echoid-s2224" xml:space="preserve">Quod
              <lb/>
            primò, &</s>
            <s xml:id="echoid-s2225" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2226" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2227" xml:space="preserve">Et quoniam earum interuallum, per eaſ-
              <lb/>
            dem interceptas metitum, ſemper minus
              <lb/>
            eſt HB interuallo parallelarum BN, HL,
              <lb/>
            cum ſit GA minos GD, NM minos NL, & </s>
            <s xml:id="echoid-s2228" xml:space="preserve">
              <lb/>
            omnes GD, NL, &</s>
            <s xml:id="echoid-s2229" xml:space="preserve">c. </s>
            <s xml:id="echoid-s2230" xml:space="preserve">ipſi BH equales: </s>
            <s xml:id="echoid-s2231" xml:space="preserve">qua-
              <lb/>
            propter, licet huiuſmodi lineæ ſint ſemper magis recedentes, non tamen
              <lb/>
            perueniunt ad interuallum æquale interuallo BH. </s>
            <s xml:id="echoid-s2232" xml:space="preserve">Quod erat tandem, &</s>
            <s xml:id="echoid-s2233" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2234" xml:space="preserve"/>
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