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-div458" type="section" level="1" n="140">
          <p>
            <s xml:id="echoid-s4940" xml:space="preserve">
              <pb o="125" file="0163" n="163" rhead="Conicor. Lib. V."/>
            P D; </s>
            <s xml:id="echoid-s4941" xml:space="preserve">ergo B E ſemiſsis erecti ad B D ſemiſsim tranſuerſi eſt, vt N P ad
              <lb/>
            P D, & </s>
            <s xml:id="echoid-s4942" xml:space="preserve">ideo N H eſt breuiſsima linearum egredientium ex N (10.
              <lb/>
            </s>
            <s xml:id="echoid-s4943" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s4944" xml:space="preserve">& </s>
            <s xml:id="echoid-s4945" xml:space="preserve">fic oſtendetur, quod ſi K I fuerit maximus, erit K M breuiſſima.</s>
            <s xml:id="echoid-s4946" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div461" type="section" level="1" n="141">
          <head xml:id="echoid-head190" xml:space="preserve">PROPOSITIO XXXIII. XXXIV.</head>
          <p>
            <s xml:id="echoid-s4947" xml:space="preserve">EContra oſtendetur, quod duæ breuiſſimæ, ſi producantur
              <lb/>
              <note position="left" xlink:label="note-0163-01" xlink:href="note-0163-01a" xml:space="preserve">a</note>
            ad partes ſuarum originum vſque ad axim minorem rectũ
              <lb/>
            ellipſis, fient duo maximi; </s>
            <s xml:id="echoid-s4948" xml:space="preserve">& </s>
            <s xml:id="echoid-s4949" xml:space="preserve">lineæ maximæ mutuò ſe ſecant in-
              <lb/>
            ter tranſuerſum, & </s>
            <s xml:id="echoid-s4950" xml:space="preserve">rectum in eadem parte, & </s>
            <s xml:id="echoid-s4951" xml:space="preserve">quod continent
              <lb/>
            cum menſura angulos, quorum proximior vertici ſectiouis ma-
              <lb/>
            ior eſt.</s>
            <s xml:id="echoid-s4952" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4953" xml:space="preserve">Quia D Q ad Q I eſt, vt D O ad O G, quia quælibet earum eſt, vt
              <lb/>
              <note position="left" xlink:label="note-0163-02" xlink:href="note-0163-02a" xml:space="preserve">b</note>
            D A ad A F (22. </s>
            <s xml:id="echoid-s4954" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s4955" xml:space="preserve">diuidendo, & </s>
            <s xml:id="echoid-s4956" xml:space="preserve">permutando, fiet D Q minor ad
              <lb/>
            D O maiorem, vt D I ad D G; </s>
            <s xml:id="echoid-s4957" xml:space="preserve">ergo D I minor eſt, quàm D G, & </s>
            <s xml:id="echoid-s4958" xml:space="preserve">K Q
              <lb/>
            maior, quàm H O; </s>
            <s xml:id="echoid-s4959" xml:space="preserve">quare angulus I maior eſt, quàm G; </s>
            <s xml:id="echoid-s4960" xml:space="preserve">igitur H G, K I,
              <lb/>
            ſe mutuo ſecantes, conueniunt in L.</s>
            <s xml:id="echoid-s4961" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4962" xml:space="preserve">Et conſtat, quod occurſus duarum breuiſsimarum (ſi producantur ver-
              <lb/>
            ſus ſuam originem) erit intra angulum contentum à duabus medietati-
              <lb/>
            bus axium ellipſis B D, D C ſupra vnum eorum, nempe punctum L ca-
              <lb/>
            dit intra angulum B D C. </s>
            <s xml:id="echoid-s4963" xml:space="preserve">Quoniam breuiſsimæ N H, M K ſe mutuò ſe-
              <lb/>
            cant, ſi producantur ad partes ſuæ originis (28. </s>
            <s xml:id="echoid-s4964" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s4965" xml:space="preserve">occurrent vtique
              <lb/>
            extra B D, & </s>
            <s xml:id="echoid-s4966" xml:space="preserve">intra A G (33. </s>
            <s xml:id="echoid-s4967" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s4968" xml:space="preserve">& </s>
            <s xml:id="echoid-s4969" xml:space="preserve">hoc erat oſtendendum.</s>
            <s xml:id="echoid-s4970" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div464" type="section" level="1" n="142">
          <head xml:id="echoid-head191" xml:space="preserve">PROPOSITIO XXXV.</head>
          <p>
            <s xml:id="echoid-s4971" xml:space="preserve">SI per centrũ ellipfis tranſierit vna
              <lb/>
              <note position="left" xlink:label="note-0163-03" xlink:href="note-0163-03a" xml:space="preserve">a</note>
              <figure xlink:label="fig-0163-01" xlink:href="fig-0163-01a" number="157">
                <image file="0163-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0163-01"/>
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            duarum breuiſſimarum, vtique
              <lb/>
            rami egrediẽtes ab eorum occurſu ad
              <lb/>
            ſectionis quadrantem alterius breuiſſi-
              <lb/>
            mæ habebunt proprietates expoſitas
              <lb/>
            in propoſitionibus 54. </s>
            <s xml:id="echoid-s4972" xml:space="preserve">& </s>
            <s xml:id="echoid-s4973" xml:space="preserve">55.</s>
            <s xml:id="echoid-s4974" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4975" xml:space="preserve">In ellipſi A B C ſit punctum E occur-
              <lb/>
            ſus duarum breuiſſimarum B D, C I, & </s>
            <s xml:id="echoid-s4976" xml:space="preserve">
              <lb/>
            centrum ſectionis D: </s>
            <s xml:id="echoid-s4977" xml:space="preserve">& </s>
            <s xml:id="echoid-s4978" xml:space="preserve">ex E educamus E F, quæ ſecet tranſuerſum a-
              <lb/>
            xim in H. </s>
            <s xml:id="echoid-s4979" xml:space="preserve">Dico, quod H F nõ eſt breuiſſima, & </s>
            <s xml:id="echoid-s4980" xml:space="preserve">quod breuiſſima egre-
              <lb/>
            diens ex F abſcindit ex ſagitta A C cum A lineam maiorem, quàm A
              <lb/>
            H. </s>
            <s xml:id="echoid-s4981" xml:space="preserve">Quoniam G I eſt breuiſſima; </s>
            <s xml:id="echoid-s4982" xml:space="preserve">igitur F H, ſi eſſet quoque breuiſſima,
              <lb/>
              <note position="right" xlink:label="note-0163-04" xlink:href="note-0163-04a" xml:space="preserve">34. Huius.</note>
            occurreret ipſi G I intra angulum A D E: </s>
            <s xml:id="echoid-s4983" xml:space="preserve">ſed non occurrit ei, niſi in E,
              <lb/>
            ergo F H non eſt breuiſſima; </s>
            <s xml:id="echoid-s4984" xml:space="preserve">& </s>
            <s xml:id="echoid-s4985" xml:space="preserve">quia F E non cadit inter duas breuiſe-
              <lb/>
            cantes E B, E G; </s>
            <s xml:id="echoid-s4986" xml:space="preserve">ergo breuiſſima, egrediens ex F, abſcindit ex ſagitta
              <lb/>
            lineam maiorem, quàm A H (54. </s>
            <s xml:id="echoid-s4987" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s4988" xml:space="preserve">quod erat oſtenden@um.</s>
            <s xml:id="echoid-s4989" xml:space="preserve"/>
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