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-div464" type="section" level="1" n="142">
          <pb o="126" file="0164" n="164" rhead="Apollonij Pergæi"/>
        </div>
        <div xml:id="echoid-div467" type="section" level="1" n="143">
          <head xml:id="echoid-head192" xml:space="preserve">PROPOSITIO XXXVI.</head>
          <p>
            <s xml:id="echoid-s4990" xml:space="preserve">IN ſectione elliptica quatuor lineæ
              <lb/>
              <figure xlink:label="fig-0164-01" xlink:href="fig-0164-01a" number="158">
                <image file="0164-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0164-01"/>
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            breuiſſimæ, vt B D, F I, G K,
              <lb/>
            H L, non conueniunt omnes in vno
              <lb/>
            puncto.</s>
            <s xml:id="echoid-s4991" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4992" xml:space="preserve">Alioquin ſit occurſus in E, & </s>
            <s xml:id="echoid-s4993" xml:space="preserve">prius ſit
              <lb/>
            B D perpendicularis ſuper A C, tranſi-
              <lb/>
            ens per D centrum ſectionis; </s>
            <s xml:id="echoid-s4994" xml:space="preserve">& </s>
            <s xml:id="echoid-s4995" xml:space="preserve">quia E
              <lb/>
            eſt occurſus duarum breuiſſimarum B D,
              <lb/>
              <note position="left" xlink:label="note-0164-01" xlink:href="note-0164-01a" xml:space="preserve">35. huius.</note>
            F I, & </s>
            <s xml:id="echoid-s4996" xml:space="preserve">B E tranſit per centrum; </s>
            <s xml:id="echoid-s4997" xml:space="preserve">igitur
              <lb/>
              <figure xlink:label="fig-0164-02" xlink:href="fig-0164-02a" number="159">
                <image file="0164-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0164-02"/>
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            G K non eſt linea breuiſſima, quod eſt
              <lb/>
            contra hypotheſim. </s>
            <s xml:id="echoid-s4998" xml:space="preserve">Si vero nullus eorũ
              <lb/>
            tranſit per centrum, educamus per cen-
              <lb/>
            trum D O perpendicularem ad A C; </s>
            <s xml:id="echoid-s4999" xml:space="preserve">qua-
              <lb/>
            re duæ breuiſſimæ F I, G K conueniunt
              <lb/>
            intra angulum A D O (34. </s>
            <s xml:id="echoid-s5000" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s5001" xml:space="preserve">ſimi-
              <lb/>
            liter H L, M N breuiſſimæ occurrunt in-
              <lb/>
            tra angulum C D O (34. </s>
            <s xml:id="echoid-s5002" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s5003" xml:space="preserve">ſed cõ-
              <lb/>
            ueniunt in E, quod eſt abſurdum; </s>
            <s xml:id="echoid-s5004" xml:space="preserve">igitur
              <lb/>
            quatuor lineæ breuiſſimæ non cõueniunt in vno puncto; </s>
            <s xml:id="echoid-s5005" xml:space="preserve">quod erat oſten-
              <lb/>
            dendum.</s>
            <s xml:id="echoid-s5006" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div470" type="section" level="1" n="144">
          <head xml:id="echoid-head193" xml:space="preserve">PROPOSITIO XXXVII. XLVI.</head>
          <p>
            <s xml:id="echoid-s5007" xml:space="preserve">IN coniſectione A B, cuius centrum D duci non poſſunt-duæ
              <lb/>
            lineæ maximæ in ellipſi, neque duæbreuiſſimæ in omnibus
              <lb/>
            ſectionibus, vt A E, A F ad vnum punctum A circumferentiæ
              <lb/>
            ſectionis terminatæ.</s>
            <s xml:id="echoid-s5008" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5009" xml:space="preserve">Educamus A G perpendicularem ad axim B E. </s>
            <s xml:id="echoid-s5010" xml:space="preserve">Si itaque ſectio fue-
              <lb/>
            rit parabole, fiet E G æqualis F G, quia quælibet earum eſt æqualis di-
              <lb/>
            midio erecti (13. </s>
            <s xml:id="echoid-s5011" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s5012" xml:space="preserve">ſi vero fuerit hyperbole, aut ellipſis, fiet D G
              <lb/>
            ad G E, vt D G ad G F; </s>
            <s xml:id="echoid-s5013" xml:space="preserve">quia quælibet earum eſt, vt proportio figuræ
              <lb/>
            (14. </s>
            <s xml:id="echoid-s5014" xml:space="preserve">15. </s>
            <s xml:id="echoid-s5015" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s5016" xml:space="preserve">igitur G F æqualis eſt G E, quod eſt abſurdum. </s>
            <s xml:id="echoid-s5017" xml:space="preserve">Simi-
              <lb/>
            liter ſi B G fuerit minor duarum axium ellipſis, & </s>
            <s xml:id="echoid-s5018" xml:space="preserve">fuerint A E, A F
              <lb/>
            rami maximi oſtendetur, quod G F æqualis ſit G E (23. </s>
            <s xml:id="echoid-s5019" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s5020" xml:space="preserve">Patet
              <lb/>
            igitur, vt dictum eſt, quod ex vno puncto ſectionis educi non poſſunt
              <lb/>
            ad axim illius duæ lineæ maximæ, neque breuiſſimæ, & </s>
            <s xml:id="echoid-s5021" xml:space="preserve">hoc erat oſten-
              <lb/>
            dendum.</s>
            <s xml:id="echoid-s5022" xml:space="preserve"/>
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