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-div88" type="section" level="1" n="49">
          <p style="it">
            <s xml:id="echoid-s1423" xml:space="preserve">
              <pb o="23" file="0061" n="61" rhead="Conicor. Lib. V."/>
            H C; </s>
            <s xml:id="echoid-s1424" xml:space="preserve">igitur prædisti exceſſus tam in parabola, quàm in reliquis ſectioni-
              <lb/>
            bus æquales ſunt inter ſe, & </s>
            <s xml:id="echoid-s1425" xml:space="preserve">ideò quadrata ramorum I O, 10, I C, & </s>
            <s xml:id="echoid-s1426" xml:space="preserve">rami ipſi
              <lb/>
            æquales erunt: </s>
            <s xml:id="echoid-s1427" xml:space="preserve">cumque quilibet alius ramus ſupra, vel infra ramum I O maior,
              <lb/>
            vel minor ſit illo, non crunt plures, quam tres rami inter ſe æquales.</s>
            <s xml:id="echoid-s1428" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s1429" xml:space="preserve">Secundò H D differentia abſciſſarum rami I A, & </s>
            <s xml:id="echoid-s1430" xml:space="preserve">breniſsimi I N ſupponatur
              <lb/>
            maior, quàm H C quæ eſt abſciſſa breuiſsimi rami I N; </s>
            <s xml:id="echoid-s1431" xml:space="preserve">& </s>
            <s xml:id="echoid-s1432" xml:space="preserve">producta ſimiliter
              <lb/>
            ordinata D A vltra axim ad ſectionem in a, & </s>
            <s xml:id="echoid-s1433" xml:space="preserve">coniuncta I a; </s>
            <s xml:id="echoid-s1434" xml:space="preserve">Dico, quod duo
              <lb/>
            rami tantummodo I A, & </s>
            <s xml:id="echoid-s1435" xml:space="preserve">I a inter ſe æquales ſunt: </s>
            <s xml:id="echoid-s1436" xml:space="preserve">Quia H D maior eſt, quàm
              <lb/>
            H C, erit quadratum ex H D maius quadrato H C; </s>
            <s xml:id="echoid-s1437" xml:space="preserve">pariterque exemplar appli-
              <lb/>
            catum ad H D maius erit exemplari ei ſimili applicato ad H C, & </s>
            <s xml:id="echoid-s1438" xml:space="preserve">ideo tam.
              <lb/>
            </s>
            <s xml:id="echoid-s1439" xml:space="preserve">quadratum I A, quàm I a maius erit quadrato I C, cum quodlibet illorum ma-
              <lb/>
            iori exceſſu ſuperet quadratum breuiſsimi rami I N quam quadratqm I C, qua-
              <lb/>
            re tam ramus I A, quàm I a (qui æquales ſunt) maiores erunt, quàm I C, & </s>
            <s xml:id="echoid-s1440" xml:space="preserve">
              <lb/>
            ideo maiores quàm intercepti inter I C, & </s>
            <s xml:id="echoid-s1441" xml:space="preserve">I N, pariterque maiores, quàm in-
              <lb/>
            terpoſiti inter I N, & </s>
            <s xml:id="echoid-s1442" xml:space="preserve">I A, & </s>
            <s xml:id="echoid-s1443" xml:space="preserve">minores omnibus alijs, qui infra ipſos cadunt. </s>
            <s xml:id="echoid-s1444" xml:space="preserve">
              <lb/>
            Quapropter duo tantùm rami I A, I a ab origine ad ſectionem duci poſſunt in-
              <lb/>
            ter ſe æquales.</s>
            <s xml:id="echoid-s1445" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s1446" xml:space="preserve">Tertiò ſint duæ abſciſſarum differentiæ H P, & </s>
            <s xml:id="echoid-s1447" xml:space="preserve">H I æquales inter ſe, & </s>
            <s xml:id="echoid-s1448" xml:space="preserve">quæ-
              <lb/>
            libet earum minor H C abſciſſa rami breuiſsimi, & </s>
            <s xml:id="echoid-s1449" xml:space="preserve">producantur perpendicula-
              <lb/>
            res ad axim L P, B I, donec conueniant ex altera parte cum ſectione in l, & </s>
            <s xml:id="echoid-s1450" xml:space="preserve">b,
              <lb/>
            coniunganturque rami ad l, b. </s>
            <s xml:id="echoid-s1451" xml:space="preserve">Dico, quatuor ramos I B, I L, I l, I b æquales
              <lb/>
            inter ſe tantummodo duci poſſe; </s>
            <s xml:id="echoid-s1452" xml:space="preserve">quia, vt dictum eſt, quilibet eorum ſuperat ra-
              <lb/>
            mum breuiſsimum I N potentia eodem exceſſu, erunt radij ipſi I B, I L, I l, I b
              <lb/>
            æquales inter ſe, reliqui verò ſupra, & </s>
            <s xml:id="echoid-s1453" xml:space="preserve">infra ipſos maiores, aut minores erunt,
              <lb/>
            & </s>
            <s xml:id="echoid-s1454" xml:space="preserve">ideo non poſſunt duci plures, quàm quatuor rami iam dicti æquales. </s>
            <s xml:id="echoid-s1455" xml:space="preserve">Quod
              <lb/>
            erat oſtendendum.</s>
            <s xml:id="echoid-s1456" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s1457" xml:space="preserve">Et inſuper quadratum rami
              <lb/>
              <note position="right" xlink:label="note-0061-01" xlink:href="note-0061-01a" xml:space="preserve">PROP.
                <lb/>
              IV. Add.</note>
            à breuiſsimo remotioris ſuper at
              <lb/>
            quadratum rami propinquioris,
              <lb/>
              <figure xlink:label="fig-0061-01" xlink:href="fig-0061-01a" number="34">
                <image file="0061-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0061-01"/>
              </figure>
            in parabola quidem rectangulo
              <lb/>
            ſub exceſſu, & </s>
            <s xml:id="echoid-s1458" xml:space="preserve">ſub aggregato
              <lb/>
            differẽtiali ſuarum abſciſſarum
              <lb/>
            ab abſciſſa rami breuiſsimi, in
              <lb/>
            reliquis verò ſectionibus rectã-
              <lb/>
            gulo ſub codem exceſſu differen-
              <lb/>
            tiali, & </s>
            <s xml:id="echoid-s1459" xml:space="preserve">ſub recta linea, ad quam
              <lb/>
            ſumma differentialis eandem
              <lb/>
            proportionem habet, quam latus
              <lb/>
            tranſuer ſum ad ſummam in hy-
              <lb/>
            perbola, & </s>
            <s xml:id="echoid-s1460" xml:space="preserve">ad differentiam in ellipſi laterum recti, & </s>
            <s xml:id="echoid-s1461" xml:space="preserve">tranſuerſi.</s>
            <s xml:id="echoid-s1462" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s1463" xml:space="preserve">Quoniam in parabola quadratum I L ſuperat quadratum I M eodem exceſſu,
              <lb/>
            quo quadratum H P ſuperat quadratum H Q (cum quadratum H P, atque qua-
              <lb/>
              <note position="right" xlink:label="note-0061-02" xlink:href="note-0061-02a" xml:space="preserve">Ex 8. hu.</note>
            dratum I N ſimul ſumpta æqualia ſint quadrato L I, & </s>
            <s xml:id="echoid-s1464" xml:space="preserve">quadrata ex H Q, & </s>
            <s xml:id="echoid-s1465" xml:space="preserve">
              <lb/>
            ex I N æqualia ſint quadrato I M) ſed exceſſus quadrati H P ſupra quadratum
              <lb/>
            H Q æqualis eſt rectangulo ſub P Q differentia, & </s>
            <s xml:id="echoid-s1466" xml:space="preserve">P H, H Q, ſumma laterum
              <lb/>
            eorundem quadratorum contento; </s>
            <s xml:id="echoid-s1467" xml:space="preserve">igitur quadratum I L ſuperat quadratum ra-
              <lb/>
            mi I M propinquioris breuiſsimo I N rectangulo ſub P Q exceſſu, & </s>
            <s xml:id="echoid-s1468" xml:space="preserve">P H </s>
          </p>
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    </echo>
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