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-div146" type="section" level="1" n="64">
          <p>
            <s xml:id="echoid-s1842" xml:space="preserve">
              <pb o="35" file="0073" n="73" rhead="Conicor. Lib. V."/>
            egrediens ex puncto L cadit extra L S, quapropter duci non poteſt ex E
              <lb/>
            ad ſectionem L B A linea, aliqua cuius portio intercepta inter axim, & </s>
            <s xml:id="echoid-s1843" xml:space="preserve">
              <lb/>
            ſectionem, ſit linea breuiſſima.</s>
            <s xml:id="echoid-s1844" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1845" xml:space="preserve">Pariter demonſtrabitur, quemadmodum iam oſtenſum eſt, quod ſi E D
              <lb/>
            fuerit æqualis H, tunc GI æqualis erit D F, quæ eſt æqualis ipſi A C; </s>
            <s xml:id="echoid-s1846" xml:space="preserve">& </s>
            <s xml:id="echoid-s1847" xml:space="preserve">
              <lb/>
              <note position="left" xlink:label="note-0073-01" xlink:href="note-0073-01a" xml:space="preserve">g</note>
            ideo B I (8. </s>
            <s xml:id="echoid-s1848" xml:space="preserve">ex quinto) vna eſt ex breuiſſimis, non autem R K, quia de-
              <lb/>
            monſtrabitur, quod E D ad M K, nempe D R ad R M maiorem rationem
              <lb/>
            habet, quàm M F ad F D, & </s>
            <s xml:id="echoid-s1849" xml:space="preserve">propterea D F maior erit, quàm R M; </s>
            <s xml:id="echoid-s1850" xml:space="preserve">bre-
              <lb/>
            uiſſima ergo cadit extra R K. </s>
            <s xml:id="echoid-s1851" xml:space="preserve">(13. </s>
            <s xml:id="echoid-s1852" xml:space="preserve">ex quinto) Et S L quoque non eſt ex
              <lb/>
            breuiſſimis, quod ita demonſtrabimus; </s>
            <s xml:id="echoid-s1853" xml:space="preserve">Si N S minor eſt, quàm D F; </s>
            <s xml:id="echoid-s1854" xml:space="preserve">ergo
              <lb/>
            breuiſſima egrediens ex L cadit extra S L; </s>
            <s xml:id="echoid-s1855" xml:space="preserve">Non igitur ex E duci poteſt
              <lb/>
            ad ſectionem linea breuiſecans præter E B, & </s>
            <s xml:id="echoid-s1856" xml:space="preserve">hoc erat oſtendendum.</s>
            <s xml:id="echoid-s1857" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1858" xml:space="preserve">Tertio loco ſit E D minor quàm H, & </s>
            <s xml:id="echoid-s1859" xml:space="preserve">oſtendetur quod E D in D F
              <lb/>
            minus eſt, quàm B G in G F; </s>
            <s xml:id="echoid-s1860" xml:space="preserve">poſtea ponamus T G in G F æquale illi, & </s>
            <s xml:id="echoid-s1861" xml:space="preserve">
              <lb/>
              <note position="left" xlink:label="note-0073-02" xlink:href="note-0073-02a" xml:space="preserve">h</note>
            erigamus ſuper F perpendicularem F V, & </s>
            <s xml:id="echoid-s1862" xml:space="preserve">ducamus per T ſectionem
              <lb/>
              <note position="right" xlink:label="note-0073-03" xlink:href="note-0073-03a" xml:space="preserve">4. lib. 2.</note>
            hyperbolicam circa duas continentes A F, & </s>
            <s xml:id="echoid-s1863" xml:space="preserve">F V; </s>
            <s xml:id="echoid-s1864" xml:space="preserve">duæ ſectiones ſe mu-
              <lb/>
            tuò ſecabunt in duobus punctis, & </s>
            <s xml:id="echoid-s1865" xml:space="preserve">ſint K, L, & </s>
            <s xml:id="echoid-s1866" xml:space="preserve">educamus ex illis duas
              <lb/>
            L N, P K M perpendiculares ad A D. </s>
            <s xml:id="echoid-s1867" xml:space="preserve">Et quoniam perpendiculares K M,
              <lb/>
            T G, L N parallelæ ſunt continenti V F, erit K M in M F æquale L N in
              <lb/>
            N F (12. </s>
            <s xml:id="echoid-s1868" xml:space="preserve">ex ſecundo) & </s>
            <s xml:id="echoid-s1869" xml:space="preserve">quodlibet eorum æquale eſt T G in G F, quod fa-
              <lb/>
            ctum eſt æquale E D in D F; </s>
            <s xml:id="echoid-s1870" xml:space="preserve">igitur E D ad K M, nempe D R ad R M eſt
              <lb/>
            vt M F ad F D, & </s>
            <s xml:id="echoid-s1871" xml:space="preserve">componendo patet, quod D F eſt æqualis R M, & </s>
            <s xml:id="echoid-s1872" xml:space="preserve">pro-
              <lb/>
              <note position="left" xlink:label="note-0073-04" xlink:href="note-0073-04a" xml:space="preserve">i</note>
            pterea K R eſt linea breuiſſima (8. </s>
            <s xml:id="echoid-s1873" xml:space="preserve">ex quinto.)</s>
            <s xml:id="echoid-s1874" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1875" xml:space="preserve">Et ſimiliter patebit, quod L S ſit breuiſſima.</s>
            <s xml:id="echoid-s1876" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">k</note>
          <p>
            <s xml:id="echoid-s1877" xml:space="preserve">Et cum B I intercipiatur inter illas patet etiam, quod B G in G F ma-
              <lb/>
              <note position="left" xlink:label="note-0073-06" xlink:href="note-0073-06a" xml:space="preserve">l</note>
            ius ſit, quàm E D in D F, oſtendetur vt dictum eſt, quod I G maior ſit,
              <lb/>
            quàm D F; </s>
            <s xml:id="echoid-s1878" xml:space="preserve">breuiſſima ergo ducta ex B cadit inter I, & </s>
            <s xml:id="echoid-s1879" xml:space="preserve">A.</s>
            <s xml:id="echoid-s1880" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1881" xml:space="preserve">Deindè ex concurſu E ad ſectionem parobolicam A B Z educamus E X,
              <lb/>
              <note position="left" xlink:label="note-0073-07" xlink:href="note-0073-07a" xml:space="preserve">m</note>
            E Z; </s>
            <s xml:id="echoid-s1882" xml:space="preserve">quas interſecant l Z, X Y perpendiculares ad A D, quæ parallelæ
              <lb/>
            ſunt continenti F V ſecantes K T L hyperbolen, ergo a Y in Y F æquale
              <lb/>
            eſt G T in G F, quod factum eſt æquale E D in D F, itaque E D in D F
              <lb/>
            maius eſt, quàm X Y in Y F; </s>
            <s xml:id="echoid-s1883" xml:space="preserve">igitur E D ad X Y, quæ eſt vt D b ad b Y
              <lb/>
            maiorem rationem habet, quàm Y F ad F D, & </s>
            <s xml:id="echoid-s1884" xml:space="preserve">componendo patet, quod
              <lb/>
            F D maior eſt quàm b Y; </s>
            <s xml:id="echoid-s1885" xml:space="preserve">itaque breuiſſima egrediens ex X abſcindit ex
              <lb/>
            A D lineam maiorem, quàm b A; </s>
            <s xml:id="echoid-s1886" xml:space="preserve">Simili modo demonſtrabitur, quod Z c
              <lb/>
            non ſit breuiſſima, & </s>
            <s xml:id="echoid-s1887" xml:space="preserve">quod breuiſſima egrediens ex Z abſcindit ex A D
              <lb/>
              <note position="left" xlink:label="note-0073-08" xlink:href="note-0073-08a" xml:space="preserve">n</note>
            lineam maiorem, quàm A c, & </s>
            <s xml:id="echoid-s1888" xml:space="preserve">hoc erat propoſitum.</s>
            <s xml:id="echoid-s1889" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div153" type="section" level="1" n="65">
          <head xml:id="echoid-head97" xml:space="preserve">PROPOSITIO LII. LIII.</head>
          <p>
            <s xml:id="echoid-s1890" xml:space="preserve">Deindè ſit ſectio hyperbole, aut ellipſis A B, & </s>
            <s xml:id="echoid-s1891" xml:space="preserve">axis illius C
              <lb/>
            A D, centrum C, & </s>
            <s xml:id="echoid-s1892" xml:space="preserve">D A menſura, quæ ſit maior dimidio ere-
              <lb/>
            cti, & </s>
            <s xml:id="echoid-s1893" xml:space="preserve">perpendicularis E D. </s>
            <s xml:id="echoid-s1894" xml:space="preserve">Dico, quod rami egredientes ex E
              <lb/>
            habent ſuperiùs expoſitas proprietates.</s>
            <s xml:id="echoid-s1895" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">a</note>
        </div>
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