Apollonius <Pergaeus>, Apollonii Pergaei Conicorvm Lib. V. VI. VII. paraphraste Abalphato Asphahanensi : nunc primum editi ; additvs in calce Archimedis assvmptorvm liber, ex codibvs arabicis mss Abrahamus Ecchellensis Maronita latinos reddidit, Jo. Alfonsvs Borellvs curam in geometricis versione contulit & [et] notas vberiores in vniuersum opus adiecit

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        <div xml:id="echoid-div1039" type="section" level="1" n="329">
          <p>
            <s xml:id="echoid-s12125" xml:space="preserve">
              <pb o="349" file="0387" n="388" rhead="Conicor. Lib. VII."/>
            tri figuræ P Q minus eſt quadrato diametri figuræ I L, & </s>
            <s xml:id="echoid-s12126" xml:space="preserve">quadratum
              <lb/>
            diametri figuræ I L minus eſt quadrato diametri figuræ A C. </s>
            <s xml:id="echoid-s12127" xml:space="preserve">Ponãtur
              <lb/>
            poſtea diametri S T, & </s>
            <s xml:id="echoid-s12128" xml:space="preserve">γ O vltra diametrum P Q, ſitque A X ordinatim
              <lb/>
            applicata ad diametrum S T, & </s>
            <s xml:id="echoid-s12129" xml:space="preserve">V X ad axim perpendicularis ſit, oſten-
              <lb/>
            detur (quemadmodum in præcedentibus dictum eſt) quod diameter fi-
              <lb/>
            guræ P Q minor ſit diametro figuræ S T, & </s>
            <s xml:id="echoid-s12130" xml:space="preserve">diameter figuræ S T minor
              <lb/>
            ſit diametro figuræ γ O, vbicunque ſecet ad axim perpendicularis X V
              <lb/>
            ipſam A C. </s>
            <s xml:id="echoid-s12131" xml:space="preserve">Et hoc erat oſtendendum.</s>
            <s xml:id="echoid-s12132" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div1041" type="section" level="1" n="330">
          <head xml:id="echoid-head408" xml:space="preserve">In Sectionem IX. Propoſit. XXXXI.
            <lb/>
          XXXXVII. & XXXXVIII.</head>
          <head xml:id="echoid-head409" xml:space="preserve">LEMMA. XIII.</head>
          <p style="it">
            <s xml:id="echoid-s12133" xml:space="preserve">Sl recta linea G H ſecetur bifariam in D, & </s>
            <s xml:id="echoid-s12134" xml:space="preserve">non bifariam in O,
              <lb/>
            E, atque fiat G a æqualis H E; </s>
            <s xml:id="echoid-s12135" xml:space="preserve">ſi quidem proportio dupli O H
              <lb/>
            ad H G eadem fuerit proportioni G H ad H E, erit duplum rectan-
              <lb/>
            guli ex differentia ipſarum E H, G O in H O æquale quadratis ex G
              <lb/>
            O, & </s>
            <s xml:id="echoid-s12136" xml:space="preserve">ex O H: </s>
            <s xml:id="echoid-s12137" xml:space="preserve">ſi verò proportio illa maior fueri erit rectangulum ma-
              <lb/>
            ius quadratis; </s>
            <s xml:id="echoid-s12138" xml:space="preserve">& </s>
            <s xml:id="echoid-s12139" xml:space="preserve">ſi eadem proportio fuerit minor, idipſum rectangulum
              <lb/>
            quadratis minus erit.</s>
            <s xml:id="echoid-s12140" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s12141" xml:space="preserve">Et primo quia duplum O H
              <lb/>
              <figure xlink:label="fig-0387-01" xlink:href="fig-0387-01a" number="458">
                <image file="0387-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0387-01"/>
              </figure>
            ad H G eſt vt G H ad H E,
              <lb/>
            ergo duplum rectanguli O H
              <lb/>
            E æquale erit quadrato ex G
              <lb/>
            H; </s>
            <s xml:id="echoid-s12142" xml:space="preserve">auferatur cõmuniter du-
              <lb/>
            plum rectanguli H O G, quia
              <lb/>
            H O eſt communis rectangulo-
              <lb/>
            rum altitudo, remanet duplũ
              <lb/>
            rectanguli ex differentia ipſa-
              <lb/>
            rum E H, G O, ſeu ex diffe-
              <lb/>
            rentia ipſarum G a, & </s>
            <s xml:id="echoid-s12143" xml:space="preserve">G O
              <lb/>
            in H O, ſeu remanet duplum rectanguli a O H æquale quaàrato H G minus
              <lb/>
            duplo rectanguli G O H: </s>
            <s xml:id="echoid-s12144" xml:space="preserve">huic verò differentiæ æqualia ſunt duo quadrata ex
              <lb/>
            G O, & </s>
            <s xml:id="echoid-s12145" xml:space="preserve">ex H O, ergo duplum rectanguli a O H æquale eſt ſummæ quadrato-
              <lb/>
            rum ex G O, & </s>
            <s xml:id="echoid-s12146" xml:space="preserve">ex O H.</s>
            <s xml:id="echoid-s12147" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s12148" xml:space="preserve">Secundo, quia duplum O H ad H G maiorem proportionem habet, quàm
              <lb/>
            G H ad H E, ergo duplum rectanguli O H E maius erit quadrato G H, & </s>
            <s xml:id="echoid-s12149" xml:space="preserve">
              <lb/>
            ablato communiter duplo rectanguli G O H erit duplum rectanguli ex differen-
              <lb/>
            tia ipſarum E H, & </s>
            <s xml:id="echoid-s12150" xml:space="preserve">G O in H O maius, quàm ſumma quadratorum ex G O,
              <lb/>
            & </s>
            <s xml:id="echoid-s12151" xml:space="preserve">ex H O.</s>
            <s xml:id="echoid-s12152" xml:space="preserve"/>
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