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

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          <pb o="36" file="0056" n="56" rhead=""/>
          <p>
            <s xml:id="echoid-s1280" xml:space="preserve">Iam applicata quacunque OPQR, tùm in Parabola AED, tùm in ſemi-
              <lb/>
            Parabola DHC; </s>
            <s xml:id="echoid-s1281" xml:space="preserve">cum ſit quadratum AD ad OP vt linca GF ad FS, vel vt DH
              <lb/>
            ad HQ, vel vt quadratum DC ad QR, ſintque antecedentia AD, DC ęqua-
              <lb/>
            lia, erunt & </s>
            <s xml:id="echoid-s1282" xml:space="preserve">conſequentia OP, QR æqualia, nempè applicata OP æqualis ap-
              <lb/>
            plicatæ QR, & </s>
            <s xml:id="echoid-s1283" xml:space="preserve">ita de omnibus &</s>
            <s xml:id="echoid-s1284" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1285" xml:space="preserve">quare integra Parabole AED æquatur
              <lb/>
            ſemi-Parabolæ DHC.</s>
            <s xml:id="echoid-s1286" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1287" xml:space="preserve">Amplius ducta quacunque TVX
              <lb/>
            parallela ad BD, erit BD ad TX, vt
              <lb/>
              <figure xlink:label="fig-0056-01" xlink:href="fig-0056-01a" number="32">
                <image file="0056-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0056-01"/>
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            rectangulum ADC ad AXC, vel vt
              <lb/>
            HD ad VX, & </s>
            <s xml:id="echoid-s1288" xml:space="preserve">permutando, cum ſit
              <lb/>
            BD dupla DH, & </s>
            <s xml:id="echoid-s1289" xml:space="preserve">TX erit dupla XV,
              <lb/>
            & </s>
            <s xml:id="echoid-s1290" xml:space="preserve">ſic de omnibus interceptis, & </s>
            <s xml:id="echoid-s1291" xml:space="preserve">æ-
              <lb/>
            quidiſtantibus in ſemi-Parabola DB
              <lb/>
            C, & </s>
            <s xml:id="echoid-s1292" xml:space="preserve">in ſemi-Parabola DHC, vnde
              <lb/>
            tota ſemi-Parabole DBC dupla eſt
              <lb/>
            totius ſemi-Parabolæ DHC, & </s>
            <s xml:id="echoid-s1293" xml:space="preserve">ſum-
              <lb/>
            ptis æqualibus; </s>
            <s xml:id="echoid-s1294" xml:space="preserve">ſemi-Parabole ABD
              <lb/>
            dupla Parabolæ AFD, ſiue trilineum
              <lb/>
            ANBDFA, æquale erit Parabolæ
              <lb/>
            AFD.</s>
            <s xml:id="echoid-s1295" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1296" xml:space="preserve">Tandé, ſi ſit AE contingens ABC
              <lb/>
            in A, erit EB æqualis BD, & </s>
            <s xml:id="echoid-s1297" xml:space="preserve">ducta
              <lb/>
            in trilineo AEBDFA quacunque IKZ parallela ad ED, erit IK æqualis
              <note symbol="a" position="left" xlink:label="note-0056-01" xlink:href="note-0056-01a" xml:space="preserve">3. Co-
                <lb/>
              roll. 13. h.</note>
            & </s>
            <s xml:id="echoid-s1298" xml:space="preserve">ſic de omnibus alijs interceptis in trilineis AEBNA, & </s>
            <s xml:id="echoid-s1299" xml:space="preserve">ANBDFA quare
              <lb/>
            totum trilineum AEBNA æquabitur toto trilineo ANBDFA, ſed hoc, modò
              <lb/>
            oſtenſum fuit æquale Parabolæ AFD, quapropter totum triangulum AED
              <lb/>
            erit ſeſquialterum ſemi-Parabolæ ABD, vel erit vt 6 ad 4, ſed ad triangulum
              <lb/>
            ABD eſt vt 6 ad 3; </s>
            <s xml:id="echoid-s1300" xml:space="preserve">quare ſemi - Parabole ABD ad inſcriptum triangulum
              <lb/>
            ABD erit vt 4 ad 3, & </s>
            <s xml:id="echoid-s1301" xml:space="preserve">duplum ad duplum, hoc eſt Parabole ABC ad trian-
              <lb/>
            gulum ABC, ſuper eadem baſi AC, & </s>
            <s xml:id="echoid-s1302" xml:space="preserve">eiuſdem altitudinis cum Parabola,
              <lb/>
            erit vt 4, ad 3, nempe ſeſquitertium. </s>
            <s xml:id="echoid-s1303" xml:space="preserve">Quod erat demonſtrandum.</s>
            <s xml:id="echoid-s1304" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s1305" xml:space="preserve">Sed iam tempus eſt vt ſuſceptum opus aggrediamur, initio facto à defini-
              <lb/>
            tionibus.</s>
            <s xml:id="echoid-s1306" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div101" type="section" level="1" n="59">
          <head xml:id="echoid-head64" xml:space="preserve">Definitiones Secundæ.
            <lb/>
          I.</head>
          <p>
            <s xml:id="echoid-s1307" xml:space="preserve">CONI SECTIONES ÆQVALITER INCLINAT Æ
              <lb/>
              <figure xlink:label="fig-0056-02" xlink:href="fig-0056-02a" number="33">
                <image file="0056-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0056-02"/>
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            vocentur illę, quarum ordinatim ductæ æquales inuicem
              <lb/>
            angulos cum earum diametris efficiunt.</s>
            <s xml:id="echoid-s1308" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1309" xml:space="preserve">Videlicet coni - ſectio ABC vocabitur æqualiter incli-
              <lb/>
            nata, vel eiuſdem inclinationis, ac ſectio conica DEF, cum
              <lb/>
            vtriuſque ordinatim ductæ AGC, DHF, earum diametros
              <lb/>
            BG, EH, ad æquales diuidunt angulos, hoc eſt cum an-
              <lb/>
            gulus AGB, angulo DHE, & </s>
            <s xml:id="echoid-s1310" xml:space="preserve">qui ei deinceps CGB reli-
              <lb/>
            quo FHE æqualis fuerit.</s>
            <s xml:id="echoid-s1311" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div103" type="section" level="1" n="60">
          <head xml:id="echoid-head65" xml:space="preserve">II.</head>
          <p>
            <s xml:id="echoid-s1312" xml:space="preserve">Coni-ſectio vel circulus, coni-ſectioni, vel circulo </s>
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