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

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          <p>
            <s xml:id="echoid-s4461" xml:space="preserve">
              <pb o="133" file="0157" n="157" rhead=""/>
            cum duplo rectanguli LGM, ſed duplum rectanguli K F I, maius eſt duplo
              <lb/>
            rectanguli LGM, vt ſuperiùs oſtenſum fuit; </s>
            <s xml:id="echoid-s4462" xml:space="preserve">quare his demptis, erunt reli-
              <lb/>
            qua quadrata EF, FI ſimul, maiora reliquis ſimul EG, GM, ſed quadratum
              <lb/>
            FI minus eſt quadrato GM, cum ſit linea FI minor GM, ergo reliquum qua-
              <lb/>
            dratum EF maius erit reliquo EG, ſiue ſemi-diameter EF maior ſemi-dia-
              <lb/>
            metro EG. </s>
            <s xml:id="echoid-s4463" xml:space="preserve">Quod ſecundò erat, &</s>
            <s xml:id="echoid-s4464" xml:space="preserve">c.</s>
            <s xml:id="echoid-s4465" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4466" xml:space="preserve">Dico tandem, in vno quoque Ellipſeos rectangulo quadrante, nempe
              <lb/>
            in quadrante ABE, reperiri aliam ſemi-diametrum ipſi EG æqualem.</s>
            <s xml:id="echoid-s4467" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4468" xml:space="preserve">Producta enim applicata GL ad N, & </s>
            <s xml:id="echoid-s4469" xml:space="preserve">iuncta EN, in triangulis ELG,
              <lb/>
            ELN, eſt NL æqualis LG, & </s>
            <s xml:id="echoid-s4470" xml:space="preserve">LE communis, & </s>
            <s xml:id="echoid-s4471" xml:space="preserve">anguli ad L recti, quare ba-
              <lb/>
            ſes EG, GN æquales erunt, & </s>
            <s xml:id="echoid-s4472" xml:space="preserve">ſic in quolibet alio quadrante; </s>
            <s xml:id="echoid-s4473" xml:space="preserve">& </s>
            <s xml:id="echoid-s4474" xml:space="preserve">vnaquequę
              <lb/>
            ſemi-diametrorum vnica eſt in eodem quadrante, cum quę ad partes maio-
              <lb/>
            ris axis ducuntur, maiores ſint, & </s>
            <s xml:id="echoid-s4475" xml:space="preserve">quæ ad partes minoris, minores: </s>
            <s xml:id="echoid-s4476" xml:space="preserve">quapro-
              <lb/>
            pter à centro Ellipſeos quatuor tantùm (in rectangulis quadrantibus inter ſe-
              <lb/>
            mi-axes) ſemi-diametri æquales ad eius peripheriam duci poterunt. </s>
            <s xml:id="echoid-s4477" xml:space="preserve">Quod
              <lb/>
            vltimò demonſtrandum erat.</s>
            <s xml:id="echoid-s4478" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div434" type="section" level="1" n="182">
          <head xml:id="echoid-head187" xml:space="preserve">COROLL. I.</head>
          <p>
            <s xml:id="echoid-s4479" xml:space="preserve">HInc eſt, quod _MAXIMA_ diametrorum, in Ellipſi, eſt axis maior, _MINI-_
              <lb/>
            _MA_ verò axis minor; </s>
            <s xml:id="echoid-s4480" xml:space="preserve">eadem enim eſt ratio de duplis, ac de ſubduplis.</s>
            <s xml:id="echoid-s4481" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div435" type="section" level="1" n="183">
          <head xml:id="echoid-head188" xml:space="preserve">COROLL. II.</head>
          <p>
            <s xml:id="echoid-s4482" xml:space="preserve">PAtet etiam, quod ſi ex Ellipſeos centro ad interuallum cuiuſcunque ſe-
              <lb/>
            mi-diametri, quæ non ſit axis, circulus deſcribatur, ipſum ad partes
              <lb/>
            maioris axis cadere intra, ad partes verò minoris cadere extra, & </s>
            <s xml:id="echoid-s4483" xml:space="preserve">in quatuor
              <lb/>
            tantùm punctis Ellipſim ſecare.</s>
            <s xml:id="echoid-s4484" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div436" type="section" level="1" n="184">
          <head xml:id="echoid-head189" xml:space="preserve">THEOR. XLII. PROP. LXXXVII.</head>
          <p>
            <s xml:id="echoid-s4485" xml:space="preserve">Si ad extremum axis datæ coni-ſectionis ducta fuerit contingens
              <lb/>
            linea, quæ cum alia ad alterum ſectionis punctum contingente
              <lb/>
            conueniat; </s>
            <s xml:id="echoid-s4486" xml:space="preserve">ſemper ea, quæ inter occurſum, & </s>
            <s xml:id="echoid-s4487" xml:space="preserve">axem intercipitur
              <lb/>
            (qui tamen in ſectione Ellipſis, ſit axis maior) minor eſt altera con-
              <lb/>
            tingente: </s>
            <s xml:id="echoid-s4488" xml:space="preserve">& </s>
            <s xml:id="echoid-s4489" xml:space="preserve">in Ellipſi tantùm, contingens ex minori axe altera
              <lb/>
            contingente maior eſt.</s>
            <s xml:id="echoid-s4490" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4491" xml:space="preserve">SIt coni-ſectio AB, cuius axis BC (quitamen in Ellipſi ſit axis maior) & </s>
            <s xml:id="echoid-s4492" xml:space="preserve">in
              <lb/>
            Hyperbola, ac Ellipſi ſit centrum D, ſitque ex vertice B contingens li-
              <lb/>
            nea BE; </s>
            <s xml:id="echoid-s4493" xml:space="preserve">ſumptoque in ſectione quolibet alio puncto A (quod tamen in Elli-
              <lb/>
            pſi non ſit alterum axis extremum; </s>
            <s xml:id="echoid-s4494" xml:space="preserve">nam ipſæ contingentes, per 27. </s>
            <s xml:id="echoid-s4495" xml:space="preserve">ſecundi
              <lb/>
            conic. </s>
            <s xml:id="echoid-s4496" xml:space="preserve">inter ſe æquidiſtarent) ab eo ducatur contingens AE quæ
              <note symbol="a" position="right" xlink:label="note-0157-01" xlink:href="note-0157-01a" xml:space="preserve">58. h.</note>
            cum BE conueniet in E. </s>
            <s xml:id="echoid-s4497" xml:space="preserve">Dico tangentem BE ipſa AE minorem eſſe.</s>
            <s xml:id="echoid-s4498" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4499" xml:space="preserve">Ducatur per occurſum E diameter DEF, iungaturque AB. </s>
            <s xml:id="echoid-s4500" xml:space="preserve">Patet ipſam
              <lb/>
              <note symbol="b" position="right" xlink:label="note-0157-02" xlink:href="note-0157-02a" xml:space="preserve">30. fecú-
                <lb/>
              di conic.</note>
            diametrum, cumtranſeat per occurſum tangentium, ſecare AB tactus iun-
              <lb/>
            gentem biſariam in F.</s>
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