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

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        <div xml:id="echoid-div845" type="section" level="1" n="337">
          <head xml:id="echoid-head346" xml:space="preserve">LEMMA XV. PROP. LXXVI.</head>
          <p>
            <s xml:id="echoid-s8158" xml:space="preserve">Si in angulo A B C applicatæ fuerint quotcunque rectæ lineæ
              <lb/>
            A C, D E, &</s>
            <s xml:id="echoid-s8159" xml:space="preserve">c. </s>
            <s xml:id="echoid-s8160" xml:space="preserve">inter ſe parallelæ; </s>
            <s xml:id="echoid-s8161" xml:space="preserve">quæ a quacunque alia recta F
              <lb/>
            G vtrique lateri dati anguli occurrente in F, G, ſecentur in
              <lb/>
            H, I, &</s>
            <s xml:id="echoid-s8162" xml:space="preserve">c. </s>
            <s xml:id="echoid-s8163" xml:space="preserve">Dico vt rectangulum F H G ad D H E, ita eſſe rectan-
              <lb/>
            gulum FIG, ad AIC.</s>
            <s xml:id="echoid-s8164" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8165" xml:space="preserve">ETenim ratio rectanguli F
              <lb/>
              <figure xlink:label="fig-0292-01" xlink:href="fig-0292-01a" number="238">
                <image file="0292-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0292-01"/>
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            H G ad D H E, com-
              <lb/>
            ponitur ex ratione F H ad H
              <lb/>
            D, ſiue ex F I ad I A; </s>
            <s xml:id="echoid-s8166" xml:space="preserve">& </s>
            <s xml:id="echoid-s8167" xml:space="preserve">ex
              <lb/>
            ratione G H ad H E, ſiue ex
              <lb/>
            G I ad I C; </s>
            <s xml:id="echoid-s8168" xml:space="preserve">ſed & </s>
            <s xml:id="echoid-s8169" xml:space="preserve">ratio re-
              <lb/>
            ctanguli F I G ad A I C ex
              <lb/>
            ijſdem rationibus componi-
              <lb/>
            tur, quapropter rectangulum
              <lb/>
            F H G ad D H E, eſt vt re-
              <lb/>
            ctangulum F I G ad A I C.
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            </s>
            <s xml:id="echoid-s8170" xml:space="preserve">Quod erat, &</s>
            <s xml:id="echoid-s8171" xml:space="preserve">c. </s>
            <s xml:id="echoid-s8172" xml:space="preserve">Et permutan-
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            do, rectangulum F H G ad FIG, vt rectangulum D H E ad AIC.</s>
            <s xml:id="echoid-s8173" xml:space="preserve"/>
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        <div xml:id="echoid-div847" type="section" level="1" n="338">
          <head xml:id="echoid-head347" xml:space="preserve">THEOR. XLVIII. PROP. LXXVII.</head>
          <p>
            <s xml:id="echoid-s8174" xml:space="preserve">Si fuerint duæ æquales portiones de eodem angulo, vel de
              <lb/>
            eadem coni - ſectione, vel circulo, & </s>
            <s xml:id="echoid-s8175" xml:space="preserve">ex puncto medio baſis
              <lb/>
            vnius portionis applicata ſit in angulo, vel ſectione quædam
              <lb/>
            recta linea baſi alterius portionis æquidiſtans: </s>
            <s xml:id="echoid-s8176" xml:space="preserve">rectangulum ſub
              <lb/>
            ſegmentis huius applicatæ æquabitur quadrato ſemi - baſis eiuſ-
              <lb/>
            dem portionis, cui hæc ipſa applicata æquidiſtanter ducta fuit.</s>
            <s xml:id="echoid-s8177" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8178" xml:space="preserve">SInt de eodem angulo, vt in præſenti figura, vel de quacunque coni-ſe-
              <lb/>
            ctione, vt in figuris tertij Schematiſmi, abſciſſæ duæ æquales portio-
              <lb/>
            nes A B C, H E I (vti docuimus in præcedenti, atque ex quadrageſima
              <lb/>
            huius abſolui poſſe elicitur) quarum diametri ſint B F, E G baſes verò
              <lb/>
            ſint A C, H I ab ipſis diametris bifariam ſectæ in F, G. </s>
            <s xml:id="echoid-s8179" xml:space="preserve">Iam patet
              <note symbol="a" position="left" xlink:label="note-0292-01" xlink:href="note-0292-01a" xml:space="preserve">Schol.
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              75. h. &
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              per 1. Co-
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              roll. 40. h.</note>
            baſes omninò ſe mutuò ſecare, atque inter portionum diametros vt in
              <lb/>
            M, ſiue, punctum earum occurſus M differre à punctis medijs F, G.
              <lb/>
            </s>
            <s xml:id="echoid-s8180" xml:space="preserve">Itaque ſi per alterum ipſorum, vtputa per G puncto medio baſis H I, ap-
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            plicetur in angulo, vel ſectione recta S G T parallela alteri baſi A C, hæc
              <lb/>
            omnino ad vtranque partem cum anguli lateribus, vel cum ſectione conue-
              <lb/>
            niet, cumipſa ſit vna applicatarum ad diametrum B F. </s>
            <s xml:id="echoid-s8181" xml:space="preserve">Occurrat ergo in S,
              <lb/>
            T: </s>
            <s xml:id="echoid-s8182" xml:space="preserve">Dico rectangulum S G T quadrato dimidiæ baſis A C, ſiue quadrato F
              <lb/>
            C æquale eſſe.</s>
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