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

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        <div xml:id="echoid-div665" type="section" level="1" n="265">
          <head xml:id="echoid-head274" xml:space="preserve">LEMMA X. PROP. XXXV.</head>
          <p>
            <s xml:id="echoid-s6351" xml:space="preserve">Si duæ rectæ lineæ terminatæ A B, C D bifariam ſectæ fue-
              <lb/>
            rint in E, F, & </s>
            <s xml:id="echoid-s6352" xml:space="preserve">proportionaliter producantur, vt in prima figu-
              <lb/>
            ra; </s>
            <s xml:id="echoid-s6353" xml:space="preserve">vel diuidantur, vt in ſecunda, in G, H, ita vt ſit A B ad B G,
              <lb/>
            vt C D ad D H, parteſq; </s>
            <s xml:id="echoid-s6354" xml:space="preserve">adiectæ, vel demptæ B G, D H iterum
              <lb/>
            proportionaliter ſecentur in I, L, ita vt B I ad I G, ſit vt D L ad
              <lb/>
            L H. </s>
            <s xml:id="echoid-s6355" xml:space="preserve">Dico rectangulum A G B ad rectangulum A I B, eſſe vt re-
              <lb/>
            ctangulum C H D ad rectangulum C L D.</s>
            <s xml:id="echoid-s6356" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6357" xml:space="preserve">NAm cum ſit A B ad B G, vt C D ad D H, erit in prima figura com-
              <lb/>
            ponendo, in ſecunda verò diuidendo A G ad G B, vt C H ad H D,
              <lb/>
              <figure xlink:label="fig-0227-01" xlink:href="fig-0227-01a" number="190">
                <image file="0227-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0227-01"/>
              </figure>
            & </s>
            <s xml:id="echoid-s6358" xml:space="preserve">eſt B G ad G I, vt D H ad H L (cum diui-
              <lb/>
            dendo factum ſit B I ad I G, vt D L ad L H)
              <lb/>
            ergo ex æquo A G ad G I erit vt C H ad H L,
              <lb/>
            & </s>
            <s xml:id="echoid-s6359" xml:space="preserve">in prima figura per conuerſionem rationis,
              <lb/>
            in ſecunda verò, componendo, per conuer-
              <lb/>
            ſionem rationis, & </s>
            <s xml:id="echoid-s6360" xml:space="preserve">conuertendo, erit G A ad
              <lb/>
            A I, vt H C ad C L: </s>
            <s xml:id="echoid-s6361" xml:space="preserve">& </s>
            <s xml:id="echoid-s6362" xml:space="preserve">cum ſuperiùs demon-
              <lb/>
            ſtratum ſit eſſe B G ad G I, vt D H ad H L,
              <lb/>
            erit, per conuerſionem rationis, G B ad B I,
              <lb/>
            vt H D ad D L. </s>
            <s xml:id="echoid-s6363" xml:space="preserve">Iam rectangulum A G B ad
              <lb/>
            A I B habet rationem compoſitam ex ratione
              <lb/>
            G A ad A I, vel ex H C ad C L, & </s>
            <s xml:id="echoid-s6364" xml:space="preserve">ex ratio-
              <lb/>
            ne G B ad B I, vel ex H D ad D L, ſed ex ijſ-
              <lb/>
            dem rationibus H C ad C D, & </s>
            <s xml:id="echoid-s6365" xml:space="preserve">H D ad D L
              <lb/>
            componitur ratio rectanguli C H D ad rectan-
              <lb/>
            gulum C L D, quare vt rectangulum A G B ad
              <lb/>
            A I B, ita rectangulum C H D ad C L D. </s>
            <s xml:id="echoid-s6366" xml:space="preserve">Quod erat, &</s>
            <s xml:id="echoid-s6367" xml:space="preserve">c.</s>
            <s xml:id="echoid-s6368" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div667" type="section" level="1" n="266">
          <head xml:id="echoid-head275" xml:space="preserve">THEOR. XXI. PROP. XXXVI.</head>
          <p>
            <s xml:id="echoid-s6369" xml:space="preserve">Quælibet Portiones eiuſdem, vel diuerſarum Parabolarum
              <lb/>
            ſunt Acuminata Proportionalia.</s>
            <s xml:id="echoid-s6370" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6371" xml:space="preserve">Item, Portiones eiuſdem, vel diuerſarum Hyperbolarum,
              <lb/>
            Ellipſium, aut Circulorum; </s>
            <s xml:id="echoid-s6372" xml:space="preserve">quarum tamen ſegmenta diametro-
              <lb/>
            rum in ijſdem portionibus intercepta ad ſuas ſemi-diametros
              <lb/>
            eandem homologam habeant rationem, ſunt pariter inter ſe
              <lb/>
            Acuminata proportionalia.</s>
            <s xml:id="echoid-s6373" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6374" xml:space="preserve">SInt primò duę portiones A B C, D E F eiuſdem, vel diuerſarum Pa-
              <lb/>
            rabolarum in prima figura, quarum baſes ſint A C, D F. </s>
            <s xml:id="echoid-s6375" xml:space="preserve">Dico </s>
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