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

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        <div xml:id="echoid-div420" type="section" level="1" n="176">
          <head xml:id="echoid-head181" xml:space="preserve">THEOR. XXXIX. PROP. LXXXIII.</head>
          <p>
            <s xml:id="echoid-s4328" xml:space="preserve">Si binarum Ellipſium ſimul adſcriptarum altera alteri fuerit in-
              <lb/>
            ſcripta, & </s>
            <s xml:id="echoid-s4329" xml:space="preserve">per terminos communis applicatæ ſe mutuò contingant;
              <lb/>
            </s>
            <s xml:id="echoid-s4330" xml:space="preserve">quælibet alia Ellipſis datis adſcripta, cum eadem applicata, & </s>
            <s xml:id="echoid-s4331" xml:space="preserve">
              <lb/>
            cum æquali tranſuerſo latere, inſcriptam Ellipſim omnino ſecabit.</s>
            <s xml:id="echoid-s4332" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4333" xml:space="preserve">SInt duæ Ellipſes ABCD, AECF ſimul adſcriptæ, circa communem ap-
              <lb/>
            plicatam AC, ſitque ipſarum altera, nempe AECF, alteri inſcripta, ita
              <lb/>
            vt in extremis tantùm A, C, ſe mutuò contingant: </s>
            <s xml:id="echoid-s4334" xml:space="preserve">dico, ſi his alia adſcriba-
              <lb/>
            tur Ellipſis ALCI, circa eandem applicatam AC, & </s>
            <s xml:id="echoid-s4335" xml:space="preserve">cum tranſuerſo LI,
              <lb/>
            quod æquale ſit ipſo BD tranſuerſo circumſcriptæ, ipſam ALCI omnino ſe-
              <lb/>
            care inſcriptam AECF.</s>
            <s xml:id="echoid-s4336" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4337" xml:space="preserve">Nam ſi alterum extremum tranſuerſæ
              <lb/>
              <figure xlink:label="fig-0153-01" xlink:href="fig-0153-01a" number="120">
                <image file="0153-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0153-01"/>
              </figure>
            diametri LI, quale eſt punctum L, cadat
              <lb/>
            intra inſcriptam, vt inter E, & </s>
            <s xml:id="echoid-s4338" xml:space="preserve">O, tunc aliud
              <lb/>
            extremum I neceſſariò cadet extra, infra F,
              <lb/>
            cum ſit LI, ſiue BD maior EF, ex quo ma-
              <lb/>
            nifeſtè patet, Ellipſim ALCI ſecare inſcri-
              <lb/>
            ptam AECF.</s>
            <s xml:id="echoid-s4339" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4340" xml:space="preserve">Si verò vtrunque extremum L, I, cadit
              <lb/>
            extra inſcriptam, vti exhibetur ab hac figu-
              <lb/>
            ra; </s>
            <s xml:id="echoid-s4341" xml:space="preserve">tunc ducta AG, quæ circumſcriptam
              <lb/>
            ABCD contingat in A, ipſa, vtrinque pro-
              <lb/>
            ducta, ad alteram partium ſecabit
              <note symbol="a" position="right" xlink:label="note-0153-01" xlink:href="note-0153-01a" xml:space="preserve">81. h.</note>
            no ALCI; </s>
            <s xml:id="echoid-s4342" xml:space="preserve">vnde, quæ ex A cõtingit ALCI,
              <lb/>
            diuerſa erit ab AG, & </s>
            <s xml:id="echoid-s4343" xml:space="preserve">ſit ipſa AM.</s>
            <s xml:id="echoid-s4344" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4345" xml:space="preserve">Iam ſi ALCI non ſecat inſcriptã AECF;
              <lb/>
            </s>
            <s xml:id="echoid-s4346" xml:space="preserve">contingat in A, C, ſi poſſibile eſt. </s>
            <s xml:id="echoid-s4347" xml:space="preserve">Cum ergo recta AM contingat Ellipſim
              <lb/>
            ALCI, atque hæc contingat inſcriptam Ellipſim AECF, eadem recta AM,
              <lb/>
            in A quoque continget AECF, ſed etiam AG eandem AECF contingit in
              <lb/>
            A: </s>
            <s xml:id="echoid-s4348" xml:space="preserve">quare ex eodem puncto A ductæ erunt binæ rectę lineę eandem Ellipſim
              <lb/>
            contingentes; </s>
            <s xml:id="echoid-s4349" xml:space="preserve">quod eſt impoſſibile. </s>
            <s xml:id="echoid-s4350" xml:space="preserve">Non igitur Ellipſis ALCI
              <note symbol="b" position="right" xlink:label="note-0153-02" xlink:href="note-0153-02a" xml:space="preserve">ex 32.
                <lb/>
              pr. conic.</note>
            inſcriptam AECF, quapropter in occurſibus A, C neceſſariò eam ſecabit,
              <lb/>
            Quod oſtendere propoſitum fuit. </s>
            <s xml:id="echoid-s4351" xml:space="preserve">Sed hoc idem</s>
          </p>
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        <div xml:id="echoid-div424" type="section" level="1" n="177">
          <head xml:id="echoid-head182" xml:space="preserve">ALITER affirmatiuè.</head>
          <p>
            <s xml:id="echoid-s4352" xml:space="preserve">CVm recta AG contingat ad A circumſcriptam Ellipſim ABCD, atque
              <lb/>
            hæc ad idem A contingat inſcriptam AECF, ipſa AG omninò con-
              <lb/>
            tinget ad A inſcriptam AECF; </s>
            <s xml:id="echoid-s4353" xml:space="preserve">ſed MA (quę vt ſupra oſtendimus, diuerſa eſt
              <lb/>
            à GA) hanc ſecat in A; </s>
            <s xml:id="echoid-s4354" xml:space="preserve">quare MA producta, ad alteram partium omninò
              <lb/>
            ſecabit inſcriptam AECF, & </s>
            <s xml:id="echoid-s4355" xml:space="preserve">eò magis Ellipſis ALCI, quam contingit ad A
              <lb/>
            recta MA, ad eandem partem ſecabit inſcriptam AECF. </s>
            <s xml:id="echoid-s4356" xml:space="preserve">Quod erat, &</s>
            <s xml:id="echoid-s4357" xml:space="preserve">c.</s>
            <s xml:id="echoid-s4358" xml:space="preserve"/>
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