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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          <pb o="56" file="0094" n="94" rhead="Apollonij Pergæi"/>
          <p style="it">
            <s xml:id="echoid-s2543" xml:space="preserve">Sit ſectio ellipſis
              <lb/>
              <note position="left" xlink:label="note-0094-01" xlink:href="note-0094-01a" xml:space="preserve">b</note>
            A C B tranſuerſa A
              <lb/>
              <figure xlink:label="fig-0094-01" xlink:href="fig-0094-01a" number="70">
                <image file="0094-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0094-01"/>
              </figure>
            B, &</s>
            <s xml:id="echoid-s2544" xml:space="preserve">c. </s>
            <s xml:id="echoid-s2545" xml:space="preserve">Lego; </s>
            <s xml:id="echoid-s2546" xml:space="preserve">Sit ſe-
              <lb/>
            ctio ellipſis A C B, & </s>
            <s xml:id="echoid-s2547" xml:space="preserve">
              <lb/>
            axis maior A B, cen-
              <lb/>
            trum D, & </s>
            <s xml:id="echoid-s2548" xml:space="preserve">perpendi-
              <lb/>
            cularis E F ſecans a-
              <lb/>
            xim in F inter cen-
              <lb/>
            trũ ellipſis D, & </s>
            <s xml:id="echoid-s2549" xml:space="preserve">ver-
              <lb/>
            ticem A.</s>
            <s xml:id="echoid-s2550" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2551" xml:space="preserve">Et ducamus per
              <lb/>
              <note position="left" xlink:label="note-0094-02" xlink:href="note-0094-02a" xml:space="preserve">c</note>
            punctum E ſectionẽ
              <lb/>
            hyperbolicam E M
              <lb/>
            C circa duas eius continentes, &</s>
            <s xml:id="echoid-s2552" xml:space="preserve">c. </s>
            <s xml:id="echoid-s2553" xml:space="preserve">Ideſt circa duas asymptotos I L, I H per
              <lb/>
            E deſcribatur hyperbole E M C, quæ ſecet axim A B æquidiſtantem alteri asym-
              <lb/>
              <note position="left" xlink:label="note-0094-03" xlink:href="note-0094-03a" xml:space="preserve">12. & 13.
                <lb/>
              lib. 2.</note>
            ptoton in aliquo puncto vt in M; </s>
            <s xml:id="echoid-s2554" xml:space="preserve">oſtendetur punctum M ſuper ellipſis centrum
              <lb/>
            D cadere.</s>
            <s xml:id="echoid-s2555" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2556" xml:space="preserve">Ergo E H prima in proportione in IH ſubſequentem, nempe G F ſub-
              <lb/>
              <note position="left" xlink:label="note-0094-04" xlink:href="note-0094-04a" xml:space="preserve">d</note>
            ſequens ipſam M G quartam, æquale eſt ſubſequenti D G ſecundæ in,
              <lb/>
            I G nempe F H tertiam. </s>
            <s xml:id="echoid-s2557" xml:space="preserve">Ergo punctum N, &</s>
            <s xml:id="echoid-s2558" xml:space="preserve">c. </s>
            <s xml:id="echoid-s2559" xml:space="preserve">Textus corruptus ſic reſti-
              <lb/>
            tui poſſe cenſeo; </s>
            <s xml:id="echoid-s2560" xml:space="preserve">Ergo E H prima proportionalium in H I, nempe G F quartam
              <lb/>
            æquale eſt D G ſecundæ in I G, nempe F H tertiam, &</s>
            <s xml:id="echoid-s2561" xml:space="preserve">c. </s>
            <s xml:id="echoid-s2562" xml:space="preserve">Propterea quod E H ad
              <lb/>
            F H, atque D G ad G F poſitæ fuerunt, vt latus tranſuerſum ad rectum; </s>
            <s xml:id="echoid-s2563" xml:space="preserve">ergo re-
              <lb/>
            ctangulum ſub D G, & </s>
            <s xml:id="echoid-s2564" xml:space="preserve">H F, ſeu I G, extremis quatuor proportionalium, æqua-
              <lb/>
            le eſt rectangulo ſub intermedijs E H, & </s>
            <s xml:id="echoid-s2565" xml:space="preserve">F G, ſeu H I, eſt que punctum E in,
              <lb/>
            hyperbola E M C cuius aſymptoti K I, L I; </s>
            <s xml:id="echoid-s2566" xml:space="preserve">ergo punctum D in eadem hyperbola
              <lb/>
            exiſtit; </s>
            <s xml:id="echoid-s2567" xml:space="preserve">ſed erat prius in ellipſis diametro A B, ſcilicet in centro; </s>
            <s xml:id="echoid-s2568" xml:space="preserve">quare in eorum
              <lb/>
            communi ſectione exiſtet: </s>
            <s xml:id="echoid-s2569" xml:space="preserve">erat autem punctum M communis ſectio hyperboles
              <lb/>
            E C, & </s>
            <s xml:id="echoid-s2570" xml:space="preserve">axis ellipſis A B; </s>
            <s xml:id="echoid-s2571" xml:space="preserve">igitur puncta M, & </s>
            <s xml:id="echoid-s2572" xml:space="preserve">D coincidunt, & </s>
            <s xml:id="echoid-s2573" xml:space="preserve">hyperbole E D C
              <lb/>
            tranſit per centrũ ſectionis ellipticæ A C B, & </s>
            <s xml:id="echoid-s2574" xml:space="preserve">ideo hyperbole E D C, quæ in infinitũ
              <lb/>
              <note position="left" xlink:label="note-0094-05" xlink:href="note-0094-05a" xml:space="preserve">8. lib. I.</note>
            extendi, & </s>
            <s xml:id="echoid-s2575" xml:space="preserve">dilatari poteſt neceſſario ſecabit finitam ellipſim alicubi, vt in C.</s>
            <s xml:id="echoid-s2576" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2577" xml:space="preserve">Et producamus per E C lineam, &</s>
            <s xml:id="echoid-s2578" xml:space="preserve">c. </s>
            <s xml:id="echoid-s2579" xml:space="preserve">Et producamus per E C rectam li-
              <lb/>
              <note position="right" xlink:label="note-0094-06" xlink:href="note-0094-06a" xml:space="preserve">e</note>
            neam, quæ occurrat continentibus in L, K, & </s>
            <s xml:id="echoid-s2580" xml:space="preserve">ſecet axim ellipſis in P.</s>
            <s xml:id="echoid-s2581" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2582" xml:space="preserve">Erit G F æqualis O N, quare F O, &</s>
            <s xml:id="echoid-s2583" xml:space="preserve">c. </s>
            <s xml:id="echoid-s2584" xml:space="preserve">Quia duæ rectæ lineæ A O, L K
              <lb/>
            ſecantur à parallelis I L, F E, C N, K O proportionaliter, & </s>
            <s xml:id="echoid-s2585" xml:space="preserve">ſunt K C, L E
              <lb/>
            æquales, ergo O N, F G inter ſe æquales erunt, & </s>
            <s xml:id="echoid-s2586" xml:space="preserve">addita communiter N F erit
              <lb/>
              <note position="left" xlink:label="note-0094-07" xlink:href="note-0094-07a" xml:space="preserve">8. lib. 2.</note>
            F O æqualis N G; </s>
            <s xml:id="echoid-s2587" xml:space="preserve">Et quoniam E H ad H F eſt vt E K ad K P (propter pa-
              <lb/>
            rallelas K I, O A) nempe vt F O, ſeu ei æqualis G N ad O P (propter paral-
              <lb/>
            lelas E F, O K) ſed eandem proportionẽ habet D G ad G F, quàm E H ad H F;
              <lb/>
            </s>
            <s xml:id="echoid-s2588" xml:space="preserve">ergo G N ad O P eandem proportionem habet quàm D G ad G F, & </s>
            <s xml:id="echoid-s2589" xml:space="preserve">compa-
              <lb/>
            rando homologorum differentias D N ad N P erit vt D G ad G F, ſeu vt latus
              <lb/>
              <note position="left" xlink:label="note-0094-08" xlink:href="note-0094-08a" xml:space="preserve">Lem. 3.
                <lb/>
              10. huius.</note>
            tranſuerſum ad rectum; </s>
            <s xml:id="echoid-s2590" xml:space="preserve">& </s>
            <s xml:id="echoid-s2591" xml:space="preserve">ideo C P eſt breuiſsima.</s>
            <s xml:id="echoid-s2592" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2593" xml:space="preserve">Quia in ſequenti propoſitione 57; </s>
            <s xml:id="echoid-s2594" xml:space="preserve">& </s>
            <s xml:id="echoid-s2595" xml:space="preserve">in alijs adhibetur propoſitio non adhuc
              <lb/>
            demonſtrata; </s>
            <s xml:id="echoid-s2596" xml:space="preserve">nimirum poſita C P linea breuiſsima, pariter que I D ſemiſsi axis
              <lb/>
            recti minoris etiam breuiſsima (ex II. </s>
            <s xml:id="echoid-s2597" xml:space="preserve">huius) quæ occurrant vltra axim in,
              <lb/>
            M deducuntur ea omnia, quæ in propoſitionibus 51. </s>
            <s xml:id="echoid-s2598" xml:space="preserve">& </s>
            <s xml:id="echoid-s2599" xml:space="preserve">52. </s>
            <s xml:id="echoid-s2600" xml:space="preserve">ex hypotheſi </s>
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