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

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            <s xml:id="echoid-s6753" xml:space="preserve">
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            G, H cum ſit vna applicatarum, &</s>
            <s xml:id="echoid-s6754" xml:space="preserve">c.) </s>
            <s xml:id="echoid-s6755" xml:space="preserve">interiorem ſecans in D, F. </s>
            <s xml:id="echoid-s6756" xml:space="preserve">Dico re-
              <lb/>
            ctangulum ſub ſegmentis G D, D H æquari quadrato ſemi-tangentis A E.</s>
            <s xml:id="echoid-s6757" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6758" xml:space="preserve">Nam iuncta D E, & </s>
            <s xml:id="echoid-s6759" xml:space="preserve">bifariam ſecta in N, ducatur eius diameter N O P,
              <lb/>
            quæ erit vtriuſque ſectionis diameter (cum ipſę ponantur parallelę, vel con-
              <lb/>
            centricæ) eas ſecans in O, P. </s>
            <s xml:id="echoid-s6760" xml:space="preserve">Patet, ſi ex O, P concipiantur contingentes
              <lb/>
            ſectiones O V, P Q has inter ſe æquidiſtare, ſed O V ipſi D E
              <note symbol="a" position="left" xlink:label="note-0244-01" xlink:href="note-0244-01a" xml:space="preserve">43. 44.
                <lb/>
              huius.</note>
            cum hæc ſit vna applicatarum in ſectione D E F ad diametrum R N O, qua-
              <lb/>
            re, & </s>
            <s xml:id="echoid-s6761" xml:space="preserve">P Q ipſi D E æquidiſtabit, hoc eſt D E erit vna applicatarum in ſe-
              <lb/>
            ctione A B C ad diametrum R N P; </s>
            <s xml:id="echoid-s6762" xml:space="preserve">ex quo N E producta ad vtranque par-
              <lb/>
            tem exteriori ſectioni A B C occurret, vt in L, M, & </s>
            <s xml:id="echoid-s6763" xml:space="preserve">à diametro P N bifa-
              <lb/>
            riam ſecabitur in N, ſed D E quoque bifariam ſecta fuit in N, quare inter-
              <lb/>
            ceptæ L D, E M inter ſe ſunt æquales, hoc eſt rectangulum L D M æquale
              <lb/>
            eſt rectangulo L E M.</s>
            <s xml:id="echoid-s6764" xml:space="preserve"/>
          </p>
          <figure number="201">
            <image file="0244-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0244-01"/>
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          <p>
            <s xml:id="echoid-s6765" xml:space="preserve">Iam cum ſit applicata A C bifariam ſecta in E, ducta eius diametro B
              <lb/>
            E, hæc quoque bifariam ſecabit aliam applicatam G H, vt in I, eritque
              <lb/>
            etiam diameter ſectionis parallelæ, vel concentricæ D E F; </s>
            <s xml:id="echoid-s6766" xml:space="preserve">& </s>
            <s xml:id="echoid-s6767" xml:space="preserve">cum A C
              <lb/>
            contingat ſectionem D E F in E, ſitque D F ei æquidiſtans, hæc item bi-
              <lb/>
            fariam ſecabitur à diametro E I, vt in I. </s>
            <s xml:id="echoid-s6768" xml:space="preserve">Cum ſit ergo G I æqualis I H, & </s>
            <s xml:id="echoid-s6769" xml:space="preserve">
              <lb/>
            ablata D I æqualis ablatæ I F, erit reliqua G D reliquæ F H æqualis, ſiue
              <lb/>
            rectangulum G D H æquale rectangulo G F H.</s>
            <s xml:id="echoid-s6770" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6771" xml:space="preserve">Tandem ex B ducatur contingens B Q alteri contingenti P Q conue-
              <lb/>
            niens in Q. </s>
            <s xml:id="echoid-s6772" xml:space="preserve">Erit ergo rectangulum G D H ad L D M, vt quadratum B
              <note symbol="b" position="left" xlink:label="note-0244-02" xlink:href="note-0244-02a" xml:space="preserve">17. tertij
                <lb/>
              conic.</note>
            ad P Q; </s>
            <s xml:id="echoid-s6773" xml:space="preserve">eademque ratione rectangulum A E C ad L E M, vt quadratum
              <lb/>
            B Q ad P Q: </s>
            <s xml:id="echoid-s6774" xml:space="preserve">quapropter rectangulum G D H ad L D M, erit vt A E C ad
              <lb/>
            L E M, & </s>
            <s xml:id="echoid-s6775" xml:space="preserve">permutando G D H ad A E C, vel ad quadratum A C, (cum
              <lb/>
            A E, E C ſint æquales) vt rectangulum L D M ad L E M, ſed L D M ipſi
              <lb/>
            L E M æquale oſtenſum fuit, quare, & </s>
            <s xml:id="echoid-s6776" xml:space="preserve">rectangulum G D H, vel G F H
              <lb/>
            æquale erit quadrato ſemi-tangentis A E. </s>
            <s xml:id="echoid-s6777" xml:space="preserve">Quod erat demonſtrandum:</s>
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