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

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          <p>
            <s xml:id="echoid-s4502" xml:space="preserve">Iam in Parabola, quam exhibet prima huius ſchematis ſigura, cum ſint
              <lb/>
            BC, EF diametri ipſæ erunt inter ſe parallelæ, BA verò eas ſecat,
              <note symbol="a" position="left" xlink:label="note-0158-01" xlink:href="note-0158-01a" xml:space="preserve">cõuerſ.
                <lb/>
              46. pr. co-
                <lb/>
              nic.</note>
            angulus GBF æquatur angulo EFA, ſed eſt GBF obtuſus, cum GBE ſit re-
              <lb/>
            ctus (nam eſt CB axis Parabolæ) ergo angulus quoque EFA obtuſus erit,
              <lb/>
            ſiue maior conſequenti BFE.</s>
            <s xml:id="echoid-s4503" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4504" xml:space="preserve">In Hyperbola verò ſecundæ ſiguræ, cum angulus CBA externus triangu-
              <lb/>
            li DBF ſit acutus, (nam CBE rectus eſt) ſitque maior interno BFE, is quidem
              <lb/>
            acutus erit, & </s>
            <s xml:id="echoid-s4505" xml:space="preserve">qui ei deinceps EFA erit obtuſus, ſiue maior ipſo BFE.</s>
            <s xml:id="echoid-s4506" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4507" xml:space="preserve">In Ellipſi tandem tertiæ ſiguræ iuncta DA, cum in trianguiis DFB, DFA
              <lb/>
            ſit BF ęqualis AF, & </s>
            <s xml:id="echoid-s4508" xml:space="preserve">communis FD baſis verò BD maior DA, erit
              <note symbol="b" position="left" xlink:label="note-0158-02" xlink:href="note-0158-02a" xml:space="preserve">86. h.</note>
            BFD maior angulo DFA, & </s>
            <s xml:id="echoid-s4509" xml:space="preserve">eiad verticẽ EFA maior angulo ad verticẽ BFE.</s>
            <s xml:id="echoid-s4510" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4511" xml:space="preserve">In triangulis itaq; </s>
            <s xml:id="echoid-s4512" xml:space="preserve">AFE, BFE,
              <lb/>
              <figure xlink:label="fig-0158-01" xlink:href="fig-0158-01a" number="124">
                <image file="0158-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0158-01"/>
              </figure>
            cuiuslibet harum ſigurarum, cum
              <lb/>
            ſit latus A F æqualis FB, & </s>
            <s xml:id="echoid-s4513" xml:space="preserve">FE
              <lb/>
            commune, augulus verò E F A
              <lb/>
            demonſtratus ſit maior angulo
              <lb/>
            BFE, erit baſis A F maior baſi
              <lb/>
            BE. </s>
            <s xml:id="echoid-s4514" xml:space="preserve">Quare contingens B E ex
              <lb/>
            termino maioris axis, minor eſt
              <lb/>
            altera contingente A E. </s>
            <s xml:id="echoid-s4515" xml:space="preserve">Quod
              <lb/>
            primò probandum erat.</s>
            <s xml:id="echoid-s4516" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4517" xml:space="preserve">Si verò, in Ellipſi ABC, quar-
              <lb/>
            tæ ſiguræ, axis BC fuerit minor.
              <lb/>
            </s>
            <s xml:id="echoid-s4518" xml:space="preserve">Poſitis, & </s>
            <s xml:id="echoid-s4519" xml:space="preserve">conſtructis ijſdem. </s>
            <s xml:id="echoid-s4520" xml:space="preserve">Cum in triangulis AFD, BFD ſit latus AF æ-
              <lb/>
            qualle lateri BF, & </s>
            <s xml:id="echoid-s4521" xml:space="preserve">commune FD, baſis verò AD maior baſi DB (cum minor
              <lb/>
            ſemi-axis DB ſit _MINIMA_ ſemi-diametrorum) erit angulus AFD, ſiue
              <note symbol="c" position="left" xlink:label="note-0158-03" xlink:href="note-0158-03a" xml:space="preserve">ibidem.</note>
            maior angulo BFD, hoc eſt AFE, ſuntque in triangulis BFE, AFE latera BF,
              <lb/>
            AF inter ſe æqualia, & </s>
            <s xml:id="echoid-s4522" xml:space="preserve">latus FE commune: </s>
            <s xml:id="echoid-s4523" xml:space="preserve">quare baſis BE, erit maior
              <note symbol="d" position="left" xlink:label="note-0158-04" xlink:href="note-0158-04a" xml:space="preserve">30. ſec.
                <lb/>
              conic.</note>
            AE Quod ſuit vltimò demonſtrandum.</s>
            <s xml:id="echoid-s4524" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div443" type="section" level="1" n="185">
          <head xml:id="echoid-head190" xml:space="preserve">THEOR. XLIII. PROP. LXXXVIII.</head>
          <p>
            <s xml:id="echoid-s4525" xml:space="preserve">Si coni-ſectionem recta linea contingens cum axe conueniat, & </s>
            <s xml:id="echoid-s4526" xml:space="preserve">
              <lb/>
            à tactu erigatur contingenti perpendicularis, hæc neceſſariò cum
              <lb/>
            axe conueniet, in Ellipſi cum vtroque axe, ſed priùs cum maiori;
              <lb/>
            </s>
            <s xml:id="echoid-s4527" xml:space="preserve">parſque ipſius intercepta inter contactum, & </s>
            <s xml:id="echoid-s4528" xml:space="preserve">occurſum cum axe,
              <lb/>
            qui tamen in Ellipſi ſit axis maior, ſemper minor erit eo axis ſe-
              <lb/>
            gmento, quod inter occurſum, & </s>
            <s xml:id="echoid-s4529" xml:space="preserve">verticem intercipitur.</s>
            <s xml:id="echoid-s4530" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4531" xml:space="preserve">Cum autem in Ellipſi, contingens linea minori axi occurret,
              <lb/>
            tunc prædicta perpendicularis inter contactum, & </s>
            <s xml:id="echoid-s4532" xml:space="preserve">minorem axem
              <lb/>
            intercepta, maior ſemper erit ſegmento minoris axis, quod inter
              <lb/>
            occurſum, & </s>
            <s xml:id="echoid-s4533" xml:space="preserve">verticem intercipitur.</s>
            <s xml:id="echoid-s4534" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4535" xml:space="preserve">SIt coni-ſectio ABC, cuius axis BD, & </s>
            <s xml:id="echoid-s4536" xml:space="preserve">prima ſigura Parabolen, aut Hy-
              <lb/>
            perbolen repræſentet, ſecunda verò Ellipſim, cuius axis maior, ſit </s>
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