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>
        <div xml:id="echoid-div244" type="section" level="1" n="81">
          <head xml:id="echoid-head114" xml:space="preserve">PROPOSITIO LX.</head>
          <p>
            <s xml:id="echoid-s2725" xml:space="preserve">D Einde perpendicularis egrediens ex
              <lb/>
              <figure xlink:label="fig-0100-01" xlink:href="fig-0100-01a" number="78">
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              <note position="right" xlink:label="note-0100-01" xlink:href="note-0100-01a" xml:space="preserve">a</note>
            C cadat ad centrum D ſectionis A B
              <lb/>
            hyperboles, & </s>
            <s xml:id="echoid-s2726" xml:space="preserve">ponamus C E ad E D, vt
              <lb/>
            proportio figuræ, & </s>
            <s xml:id="echoid-s2727" xml:space="preserve">producamus ex E ad
              <lb/>
            ſectionem rectã lineam E B, quæ parallela
              <lb/>
            ſit D E, producaturque C B, quæ occur-
              <lb/>
            rat axi in G. </s>
            <s xml:id="echoid-s2728" xml:space="preserve">Et quia C E ad E D, nempe
              <lb/>
              <note position="right" xlink:label="note-0100-02" xlink:href="note-0100-02a" xml:space="preserve">b</note>
            C B ad B G, nempe D H ad H G eſt, vt
              <lb/>
            proportio figuræ; </s>
            <s xml:id="echoid-s2729" xml:space="preserve">erit G B linea breuiſſima
              <lb/>
            (nona ex quinto) quod erat oſtenden-
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            dum.</s>
            <s xml:id="echoid-s2730" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div246" type="section" level="1" n="82">
          <head xml:id="echoid-head115" xml:space="preserve">PROPOSITIO LXI.</head>
          <p>
            <s xml:id="echoid-s2731" xml:space="preserve">S It poſtea punctum C, & </s>
            <s xml:id="echoid-s2732" xml:space="preserve">
              <lb/>
              <figure xlink:label="fig-0100-02" xlink:href="fig-0100-02a" number="79">
                <image file="0100-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0100-02"/>
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            perpendicularis C F, & </s>
            <s xml:id="echoid-s2733" xml:space="preserve">
              <lb/>
            F remotius à vertice ſectio-
              <lb/>
            nis, quàm ſit centrum, & </s>
            <s xml:id="echoid-s2734" xml:space="preserve">po-
              <lb/>
            namus C E ad E F, vt eſt
              <lb/>
            proportio figuræ, & </s>
            <s xml:id="echoid-s2735" xml:space="preserve">ſimiliter
              <lb/>
            D G ad G F, & </s>
            <s xml:id="echoid-s2736" xml:space="preserve">ex E pro-
              <lb/>
            ducamus E H, quæ ſit paral-
              <lb/>
            lela ipſi F A, & </s>
            <s xml:id="echoid-s2737" xml:space="preserve">ex G, D.
              <lb/>
            </s>
            <s xml:id="echoid-s2738" xml:space="preserve">ad illam G I, D K, quæ ſint
              <lb/>
            parallelæ ipſi C F; </s>
            <s xml:id="echoid-s2739" xml:space="preserve">& </s>
            <s xml:id="echoid-s2740" xml:space="preserve">duca-
              <lb/>
            mus ſectionem hyperbolen
              <lb/>
              <note position="left" xlink:label="note-0100-03" xlink:href="note-0100-03a" xml:space="preserve">4 lib. 2.</note>
            tranſeuntem per D, quam
              <lb/>
            contineant I H, I G, quæ occurret ſectioni A B ſimiliter in B; </s>
            <s xml:id="echoid-s2741" xml:space="preserve">Itaque
              <lb/>
              <note position="right" xlink:label="note-0100-04" xlink:href="note-0100-04a" xml:space="preserve">a</note>
            per B, C producamus lineam, quæ occurrat axi F A in L, & </s>
            <s xml:id="echoid-s2742" xml:space="preserve">ipſi E H
              <lb/>
            in M. </s>
            <s xml:id="echoid-s2743" xml:space="preserve">Dico, quod B L eſt linea breuiſſima. </s>
            <s xml:id="echoid-s2744" xml:space="preserve">quia ducta perpendiculari
              <lb/>
              <note position="right" xlink:label="note-0100-05" xlink:href="note-0100-05a" xml:space="preserve">b</note>
            H N, C E ad E F, ſeu ad K D, eſt vt D G ad G F, nempe vt K I ad
              <lb/>
            I E, & </s>
            <s xml:id="echoid-s2745" xml:space="preserve">propterea E C in E I erit æquale rectangulo D I ſubſequenti
              <lb/>
            (octaua ex ſecundo) nempe rectangulo B I conſequenti; </s>
            <s xml:id="echoid-s2746" xml:space="preserve">Ergo C E in
              <lb/>
              <note position="left" xlink:label="note-0100-06" xlink:href="note-0100-06a" xml:space="preserve">12. lib. 2.</note>
            E I eſt æquale B H in H I, & </s>
            <s xml:id="echoid-s2747" xml:space="preserve">propterea B H ad C E, nempe H M ad
              <lb/>
            M E eſt, vt E I ad I H; </s>
            <s xml:id="echoid-s2748" xml:space="preserve">ergo H I, nempe N G æqualis eſt E M, & </s>
            <s xml:id="echoid-s2749" xml:space="preserve">ideo
              <lb/>
            L F ad E M, nempe ad N G eſt, vt C F ad E C, nempe D F ad D G,
              <lb/>
            quia quælibet earum aſſignata eſt, vt proportio figuræ; </s>
            <s xml:id="echoid-s2750" xml:space="preserve">ergo L F ad N
              <lb/>
            G eſt, vt D F ad D G; </s>
            <s xml:id="echoid-s2751" xml:space="preserve">itaq; </s>
            <s xml:id="echoid-s2752" xml:space="preserve">comparando homologorum differentias L
              <lb/>
            D ad D N, vt D F ad D G; </s>
            <s xml:id="echoid-s2753" xml:space="preserve">& </s>
            <s xml:id="echoid-s2754" xml:space="preserve">per conuerſionem rationis, & </s>
            <s xml:id="echoid-s2755" xml:space="preserve">poſtea
              <lb/>
            diuidendo D N ad N L erit, vt D G, ad G F, quæ eſt vt propor-
              <lb/>
            tio figuræ; </s>
            <s xml:id="echoid-s2756" xml:space="preserve">Ergo B L eſt linea breuiſſima ( nona ex quinto ) & </s>
            <s xml:id="echoid-s2757" xml:space="preserve">hoc erat
              <lb/>
            oſtendendum.</s>
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