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-div223" type="section" level="1" n="75">
          <p style="it">
            <s xml:id="echoid-s2600" xml:space="preserve">
              <pb o="57" file="0095" n="95" rhead="Conicor. Lib. V."/>
            no diuerſa eliciebantur; </s>
            <s xml:id="echoid-s2601" xml:space="preserve">nam in dictis propoſitionibus perpendicularis ex concur-
              <lb/>
            ſu ad axim ducta efficiebat in ellipſi menſuram (iuxta deſinitionem 15. </s>
            <s xml:id="echoid-s2602" xml:space="preserve">huius
              <lb/>
            libri) minorem medietate axis tranſuerſi, ideſt perpendicularis ex concurſu ca-
              <lb/>
            debat inter centrum ſectionis, & </s>
            <s xml:id="echoid-s2603" xml:space="preserve">proximiorem verticem: </s>
            <s xml:id="echoid-s2604" xml:space="preserve">hic vero perpendicu-
              <lb/>
            laris ex concurſu M per centrum D ellipſis tranſit.</s>
            <s xml:id="echoid-s2605" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2606" xml:space="preserve">Animaduertendum eſt hoc theorema demonſtratum fuiſſe ab Apollonio Propoſ.
              <lb/>
            </s>
            <s xml:id="echoid-s2607" xml:space="preserve">35. </s>
            <s xml:id="echoid-s2608" xml:space="preserve">huius libri, quod tamen paraphraſtes neſcio an iure in fine huius voluminis
              <lb/>
            tranſpoſuit; </s>
            <s xml:id="echoid-s2609" xml:space="preserve">Sed quia predicta propoſitio 35. </s>
            <s xml:id="echoid-s2610" xml:space="preserve">omnino hic eſt neceſſaria, & </s>
            <s xml:id="echoid-s2611" xml:space="preserve">pendet
              <lb/>
            ex alijs præcedentibus, libuit potius aliam independentem demonſtrationem af-
              <lb/>
            ferre quam ordinem propoſitionum ſatis alter atum denuo perturbare.</s>
            <s xml:id="echoid-s2612" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div230" type="section" level="1" n="76">
          <head xml:id="echoid-head109" xml:space="preserve">LEMMA VIII.</head>
          <p style="it">
            <s xml:id="echoid-s2613" xml:space="preserve">IN ellipſi ABC linea breuiſsima F G, & </s>
            <s xml:id="echoid-s2614" xml:space="preserve">ſemiaxis minor rectus B
              <lb/>
            D conueniant in E, erunt E F, & </s>
            <s xml:id="echoid-s2615" xml:space="preserve">E B duæ breuiſecantes, duca-
              <lb/>
            tur quilibet ramus E H inter eos: </s>
            <s xml:id="echoid-s2616" xml:space="preserve">Dico E H non eſſe breuiſecantem, & </s>
            <s xml:id="echoid-s2617" xml:space="preserve">
              <lb/>
            cadere infra lineam breuiſsimam ductam ex puncto H ad axim.</s>
            <s xml:id="echoid-s2618" xml:space="preserve"/>
          </p>
          <figure number="71">
            <image file="0095-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0095-01"/>
          </figure>
          <p style="it">
            <s xml:id="echoid-s2619" xml:space="preserve">Ducantur ex F, & </s>
            <s xml:id="echoid-s2620" xml:space="preserve">H rectæ F K, H L perpendiculares aa axim rectum B
              <lb/>
            D eum ſecantes in K, & </s>
            <s xml:id="echoid-s2621" xml:space="preserve">L, pariterque ducantur F M, H N perpendiculares ad
              <lb/>
            axim tranſuerſum A D eum ſecantes in M, N. </s>
            <s xml:id="echoid-s2622" xml:space="preserve">Et quia F G eſt breuiſsima, ergo
              <lb/>
            D M ad M G eandem proportionem habet, quàm latus tranſuerſum C A ad eius
              <lb/>
              <note position="right" xlink:label="note-0095-01" xlink:href="note-0095-01a" xml:space="preserve">15. huius.</note>
            latus rectum; </s>
            <s xml:id="echoid-s2623" xml:space="preserve">ſed propter parallelas D E, M F, eſt D M ad M G, vt E F ad F
              <lb/>
            G, ſeu E K ad K D (propter parallelas G D, F K) quare E K ad K D eandem
              <lb/>
            proportionem habet, quàm latus tranſuerſum ad rectum, & </s>
            <s xml:id="echoid-s2624" xml:space="preserve">diuidendo E D ad
              <lb/>
            D K eandem proportionem habebit, quàm differentia lateris tranuerſi, & </s>
            <s xml:id="echoid-s2625" xml:space="preserve">recti
              <lb/>
            ad latus rectum, eſt vero D L maior, quàm D K (cum H L parallela ipſi F K
              <lb/>
            cadat inter punctum K, & </s>
            <s xml:id="echoid-s2626" xml:space="preserve">B) igitur E D ad maiorem D L minorem proportio-
              <lb/>
            nem habet, quàm ad D K, & </s>
            <s xml:id="echoid-s2627" xml:space="preserve">propterea componendo E L ad L D minorem pro-
              <lb/>
            portionem habebit, quàm latus tranſuerſum ad rectum: </s>
            <s xml:id="echoid-s2628" xml:space="preserve">eſt vero E H ad H </s>
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