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

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          <p>
            <s xml:id="echoid-s836" xml:space="preserve">Nam ſit quæcunque recta DBE ſectionem contingens in B: </s>
            <s xml:id="echoid-s837" xml:space="preserve">patet per 3.
              <lb/>
            </s>
            <s xml:id="echoid-s838" xml:space="preserve">ſec. </s>
            <s xml:id="echoid-s839" xml:space="preserve">conic. </s>
            <s xml:id="echoid-s840" xml:space="preserve">ipſam DE cum vtraque aſymptoto conuenire, & </s>
            <s xml:id="echoid-s841" xml:space="preserve">ad tactum B ſe-
              <lb/>
            cari bifariam, & </s>
            <s xml:id="echoid-s842" xml:space="preserve">quadratum vtriuſque portionis DB, BE æquale eſſe quarte
              <lb/>
            parti figuræ, quæ ad diametrum CB per tactum ducta conſtituitur; </s>
            <s xml:id="echoid-s843" xml:space="preserve">quare ſi
              <lb/>
            fiat CA æqualis CB, appliceturque quælibet GIH ipſi DB æquidiſtans,
              <lb/>
            aſymptoton, ſectionem, ac diametrum ſecans in G, I, H, & </s>
            <s xml:id="echoid-s844" xml:space="preserve">per I ducatur
              <lb/>
            IP parallela ad CD, ſecans diametrum in P infra C (nam punctum I eſt intra
              <lb/>
            angulum GCH) erit vt in præcedenti oſtenſum fuit rectangulum AHB ad
              <lb/>
            quadratum HI vt quadratum CB ad quadratum BD, vel vt quadratum PH
              <lb/>
            ad quadratum HI; </s>
            <s xml:id="echoid-s845" xml:space="preserve">vnde rectangulum AHB æquale erit quadrato HP, ſiue
              <lb/>
            recta HP erit media proportionalis inter AH & </s>
            <s xml:id="echoid-s846" xml:space="preserve">HB; </s>
            <s xml:id="echoid-s847" xml:space="preserve">hoc eſt punctum P ca-
              <lb/>
            det inter C & </s>
            <s xml:id="echoid-s848" xml:space="preserve">B; </s>
            <s xml:id="echoid-s849" xml:space="preserve">quare IP, quæ ipſi GC æquidiſtat contingentem BD ſeca-
              <lb/>
            bit in Q, eritque BD maior DQ, ſiue maior intercepta GI.</s>
            <s xml:id="echoid-s850" xml:space="preserve"/>
          </p>
          <figure number="18">
            <image file="0042-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0042-01"/>
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          <p>
            <s xml:id="echoid-s851" xml:space="preserve">Iam applicata infra G qualibet alia RN diametro occurrent in O, ex N du-
              <lb/>
            cta ſit NS parallela ad RC, quæ contingentem BD, ac diametrum ſecabit vt
              <lb/>
            ſupra in T & </s>
            <s xml:id="echoid-s852" xml:space="preserve">S. </s>
            <s xml:id="echoid-s853" xml:space="preserve">Cumque rectangulum AHB ſit æquale quadrato HP, vt mo-
              <lb/>
            dò oſtendimus, ſitque in directum ipſi AH addita quædam HO, erit, per
              <lb/>
            præcedens Lemma, rectangulum AOB maius quadrato OP, ſed rectangu-
              <lb/>
            lum AOB eadem ratione, vt ſupra, oſtenditur æquale quadrato OS; </s>
            <s xml:id="echoid-s854" xml:space="preserve">quare
              <lb/>
            quadratum OS maius eſt quadrato OP, hoc eſt punctum S cadit inter C, & </s>
            <s xml:id="echoid-s855" xml:space="preserve">
              <lb/>
            P, ſiue CP eſt maior CS, vel DQ maior DT, hoc eſt GI maior RN. </s>
            <s xml:id="echoid-s856" xml:space="preserve">Quare
              <lb/>
            aſymptoton
              <unsure/>
            CD, & </s>
            <s xml:id="echoid-s857" xml:space="preserve">ſectio BIN quæ in infinitum productæ, nunquam ſimul
              <lb/>
            conueniunt, ad ſe propiùs accedunt; </s>
            <s xml:id="echoid-s858" xml:space="preserve">idemque de aſymptoto CE. </s>
            <s xml:id="echoid-s859" xml:space="preserve">Quod erat
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            primò &</s>
            <s xml:id="echoid-s860" xml:space="preserve">c.</s>
            <s xml:id="echoid-s861" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s862" xml:space="preserve">Præterea dico ipſas ad interuallum peruenire minus dato interuallo M.</s>
            <s xml:id="echoid-s863" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s864" xml:space="preserve">Sumatur DT ex cõtingente BD, quę ſit minor interuallo M, & </s>
            <s xml:id="echoid-s865" xml:space="preserve">per T aga-
              <lb/>
            tur STN parallela ad CD diametro occurrens in S, ſeceturq; </s>
            <s xml:id="echoid-s866" xml:space="preserve">SV æqualis </s>
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