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

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            <s xml:id="echoid-s2894" xml:space="preserve">
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            ipſa HM producta omnino ſecabit ſectionem CBA, vel ſupra contingentem
              <lb/>
            CEA, vt in N, vel in ipſo occurſu A, vel infra ad partes AL; </s>
            <s xml:id="echoid-s2895" xml:space="preserve">ſi in N, vel in
              <lb/>
            A, patet interiorem ſectionem totam cadere infra applicatas ex N, vel ex A,
              <lb/>
            & </s>
            <s xml:id="echoid-s2896" xml:space="preserve">nunquam infra N, vel A ſectioni BNA occurrere, ne priùs ſecet propriam
              <lb/>
            aſymptoton HM.</s>
            <s xml:id="echoid-s2897" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2898" xml:space="preserve">Si verò HM ſecet exteriorẽ
              <lb/>
              <figure xlink:label="fig-0110-01" xlink:href="fig-0110-01a" number="76">
                <image file="0110-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0110-01"/>
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            BA infra contingentem CAE,
              <lb/>
            vt in hac ipſa figura; </s>
            <s xml:id="echoid-s2899" xml:space="preserve">item pa-
              <lb/>
            tet ſectiones BA, ED infra LD
              <lb/>
            nunquam ſimul conuenire, ſin
              <lb/>
            aliter propriam aſymptoton
              <lb/>
            ſecaret.</s>
            <s xml:id="echoid-s2900" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2901" xml:space="preserve">Præterea indirectũ produ-
              <lb/>
            cta communi diametro EBG,
              <lb/>
            ſumptiſque in ea punctis V, T,
              <lb/>
            quæ ſint extrema tranſuerſo-
              <lb/>
            rum laterum datarum ſectio-
              <lb/>
            num, ductiſque regulis TX,
              <lb/>
            VZ; </s>
            <s xml:id="echoid-s2902" xml:space="preserve">ipſę vti ſuperiùs oſtenſum
              <lb/>
            fuit, inter ſe æquidiſtabunt.
              <lb/>
            </s>
            <s xml:id="echoid-s2903" xml:space="preserve">Ampliùs ſumpto in portione
              <lb/>
            ED quolibet puncto K, per ip-
              <lb/>
            ſum applicetur KY vtranque Hyperbolen ſecans in P, K; </s>
            <s xml:id="echoid-s2904" xml:space="preserve">regulas verò in
              <lb/>
            Z, X.</s>
            <s xml:id="echoid-s2905" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2906" xml:space="preserve">Iam, vel recta VZ eſt regula interioris ſectionis DEF, & </s>
            <s xml:id="echoid-s2907" xml:space="preserve">TX exterioris;
              <lb/>
            </s>
            <s xml:id="echoid-s2908" xml:space="preserve">vel ipſæ ſimul congruunt, ſi tamen puncta V, T, in vnum conueniant; </s>
            <s xml:id="echoid-s2909" xml:space="preserve">vel è
              <lb/>
            contra VZ eſt regula exterioris, TX verò interioris. </s>
            <s xml:id="echoid-s2910" xml:space="preserve">Si primum, cum in ſe-
              <lb/>
            ctione ABC quadratum applicatæ PY æquale ſit rectangulo BYX, & </s>
            <s xml:id="echoid-s2911" xml:space="preserve">
              <note symbol="a" position="left" xlink:label="note-0110-01" xlink:href="note-0110-01a" xml:space="preserve">Coroll.
                <lb/>
              1. huius.</note>
            Hyperbola DEF quadratum KY ſit æquale rectangulo EYZ, ſitque rectan-
              <lb/>
            gulum BYX maius EYZ, cum ſub maioribus lateribus contineatur, erit quo-
              <lb/>
            que quadratum PY, maius quadrato KY: </s>
            <s xml:id="echoid-s2912" xml:space="preserve">vnde punctum K eſt intra ſectio-
              <lb/>
            nem ABC. </s>
            <s xml:id="echoid-s2913" xml:space="preserve">Si ſecundum nempe ſit VZ vtriuſque ſectionis communis regu-
              <lb/>
            la, erit quadratum PY æquale rectangulo BYZ, & </s>
            <s xml:id="echoid-s2914" xml:space="preserve">quadratum KY æquale
              <lb/>
            rectangulo EYZ, ſed rectangulum BYZ maius eſt EYZ, cum altitudo BY
              <lb/>
            maior ſit altitudine EY, quare quadratum PY maius eſt quadrato KY, ſiue
              <lb/>
            punctum K eſt intra ſectionem ABC. </s>
            <s xml:id="echoid-s2915" xml:space="preserve">Si denique interior recta VZ fuerit re-
              <lb/>
            gula exterioris ſectionis ABC, & </s>
            <s xml:id="echoid-s2916" xml:space="preserve">exterior TX, regula interioris DEF, erit
              <lb/>
            HE ipſi HT æqualis, ſed eſt ablata HB minor ablata GV (cum ponatur GB,
              <lb/>
            quæ maior eſt HB, æqualis GV) ergo reliqua BE maior erit reſiduis ſegmen-
              <lb/>
            tis GH, VT, & </s>
            <s xml:id="echoid-s2917" xml:space="preserve">eò maior vnico ſegmento VT, ſed eſt EY minor VY, quare
              <lb/>
            BE ad EY maiorem habebit rationẽ quàm TV ad VY, vel quàm XZ ad ZY,
              <lb/>
            & </s>
            <s xml:id="echoid-s2918" xml:space="preserve">componendo BY ad YE maiorem habebit rationem quàm XY ad YZ, vn-
              <lb/>
            de rectangulum BYZ ſiue quadratum PY maius erit rectangulo EYX
              <note symbol="b" position="left" xlink:label="note-0110-02" xlink:href="note-0110-02a" xml:space="preserve">ibidem.</note>
              <note symbol="c" position="left" xlink:label="note-0110-03" xlink:href="note-0110-03a" xml:space="preserve">16. ſept.
                <lb/>
              Pappi.</note>
            quadrato KY, hoc eſt punctum K incidet intra ſectionem ABC, & </s>
            <s xml:id="echoid-s2919" xml:space="preserve">ſic
              <note symbol="d" position="left" xlink:label="note-0110-04" xlink:href="note-0110-04a" xml:space="preserve">Coroll.
                <lb/>
              1. huius.</note>
            quolibet alio puncto portionis DEF; </s>
            <s xml:id="echoid-s2920" xml:space="preserve">Quare huiuſmodi ſimiles Hyperbolæ,
              <lb/>
            neque infra applicatam LF, neque inter lineas LF, AC ſimul conueniunt,
              <lb/>
            vnde ſunt in totum nunquam coeuntes. </s>
            <s xml:id="echoid-s2921" xml:space="preserve">Quod ſextò, &</s>
            <s xml:id="echoid-s2922" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2923" xml:space="preserve"/>
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