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-div796" type="section" level="1" n="245">
          <pb o="253" file="0291" n="291" rhead="Conicor. Lib. VI."/>
        </div>
        <div xml:id="echoid-div797" type="section" level="1" n="246">
          <head xml:id="echoid-head309" xml:space="preserve">Notæ in Propoſit. XXIX.</head>
          <p>
            <s xml:id="echoid-s9456" xml:space="preserve">ET faciamus ſuper E K triangulum ſimile triangulo A B C, &</s>
            <s xml:id="echoid-s9457" xml:space="preserve">c. </s>
            <s xml:id="echoid-s9458" xml:space="preserve">
              <note position="left" xlink:label="note-0291-01" xlink:href="note-0291-01a" xml:space="preserve">a</note>
            mirum, fiat angulus K E L æqualis angulo A, & </s>
            <s xml:id="echoid-s9459" xml:space="preserve">angulus L fiat æqualis angulo B.</s>
            <s xml:id="echoid-s9460" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9461" xml:space="preserve">Ergo L K, quæ eſt latus trianguli tranſeuntis per axim E G para llelũ
              <lb/>
              <note position="left" xlink:label="note-0291-02" xlink:href="note-0291-02a" xml:space="preserve">b</note>
            eſt E G, &</s>
            <s xml:id="echoid-s9462" xml:space="preserve">c. </s>
            <s xml:id="echoid-s9463" xml:space="preserve">Legi debet, vt in textu videre eſt. </s>
            <s xml:id="echoid-s9464" xml:space="preserve">Hoc conſtat ex conſtructio-
              <lb/>
            ne; </s>
            <s xml:id="echoid-s9465" xml:space="preserve">nam duo anguli alterni G E K,, & </s>
            <s xml:id="echoid-s9466" xml:space="preserve">L K E æquales ſunt eidem angulo C.</s>
            <s xml:id="echoid-s9467" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9468" xml:space="preserve">Et propterea planum, in quo eſt ſectio D E
              <lb/>
              <note position="left" xlink:label="note-0291-03" xlink:href="note-0291-03a" xml:space="preserve">C</note>
              <figure xlink:label="fig-0291-01" xlink:href="fig-0291-01a" number="339">
                <image file="0291-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0291-01"/>
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            F producit in cono ſectionem parabolicam, &</s>
            <s xml:id="echoid-s9469" xml:space="preserve">c.
              <lb/>
            </s>
            <s xml:id="echoid-s9470" xml:space="preserve">Quoniam planum circuli, cuius diameter E K
              <lb/>
            perpendiculare eſt ad planum trianguli L E K: </s>
            <s xml:id="echoid-s9471" xml:space="preserve">igi-
              <lb/>
            tur ſi ducatur planum N F O æquidiſtans circulo E
              <lb/>
            K ſecans planum D E F in recta linea D G F, erit
              <lb/>
            quoque circulus, & </s>
            <s xml:id="echoid-s9472" xml:space="preserve">perpendicularis ad planum triã-
              <lb/>
            guli per axim L E K: </s>
            <s xml:id="echoid-s9473" xml:space="preserve">ſed ex conſtructione planum
              <lb/>
            D E F perpendiculare quoque erat ad idem trian-
              <lb/>
            gulum per axim E L K; </s>
            <s xml:id="echoid-s9474" xml:space="preserve">igitur D F communis ſectio
              <lb/>
            eorundem planorum perpendicularis quoque erit ad
              <lb/>
            idem planum L N O, & </s>
            <s xml:id="echoid-s9475" xml:space="preserve">efficiet angulos rectos cum
              <lb/>
            diametro circuli N O, & </s>
            <s xml:id="echoid-s9476" xml:space="preserve">cum E G, quæ in eodẽ pla-
              <lb/>
            no exiſtunt, & </s>
            <s xml:id="echoid-s9477" xml:space="preserve">cũ illo conueniunt in puncto G; </s>
            <s xml:id="echoid-s9478" xml:space="preserve">ſuntq; </s>
            <s xml:id="echoid-s9479" xml:space="preserve">E G, & </s>
            <s xml:id="echoid-s9480" xml:space="preserve">L O parallelæ: </s>
            <s xml:id="echoid-s9481" xml:space="preserve">igitur
              <lb/>
              <note position="right" xlink:label="note-0291-04" xlink:href="note-0291-04a" xml:space="preserve">11. lib. 1.</note>
            planum ſectionis D E F producit neceſſariò in cono L N O producto parabolam.</s>
            <s xml:id="echoid-s9482" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9483" xml:space="preserve">Igitur H E ad E L, quæ eſt æqualis ipſi L K eamdem proportionem,
              <lb/>
              <note position="left" xlink:label="note-0291-05" xlink:href="note-0291-05a" xml:space="preserve">d</note>
            habet, quàm quadratum E K ad quadratum K L, &</s>
            <s xml:id="echoid-s9484" xml:space="preserve">c. </s>
            <s xml:id="echoid-s9485" xml:space="preserve">Quoniam conus
              <lb/>
            L E K ſimilis eſt cono recto A B C erit quoque rectus: </s>
            <s xml:id="echoid-s9486" xml:space="preserve">& </s>
            <s xml:id="echoid-s9487" xml:space="preserve">propterea duo latera
              <lb/>
            trianguli per axim E L, & </s>
            <s xml:id="echoid-s9488" xml:space="preserve">L K æqualia erunt inter ſe, & </s>
            <s xml:id="echoid-s9489" xml:space="preserve">ideo E K ad K L,
              <lb/>
            atque ad E L eandem proportionem habebit, &</s>
            <s xml:id="echoid-s9490" xml:space="preserve">c.</s>
            <s xml:id="echoid-s9491" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9492" xml:space="preserve">Et dico, quod ſectio D E F non reperitur in alio cono ſimili cono A
              <lb/>
              <note position="left" xlink:label="note-0291-06" xlink:href="note-0291-06a" xml:space="preserve">e</note>
            B C, cuius vertex ſit ex parte plani ſectionis præter hunc conum, &</s>
            <s xml:id="echoid-s9493" xml:space="preserve">c.
              <lb/>
            </s>
            <s xml:id="echoid-s9494" xml:space="preserve">Ideſt. </s>
            <s xml:id="echoid-s9495" xml:space="preserve">Nullus alius conus rectus continebit eandem parabolam D E F, qui ſit
              <lb/>
            ſinilis cono A B C, & </s>
            <s xml:id="echoid-s9496" xml:space="preserve">vertex E parabole magis, aut minus recedat à vertice
              <lb/>
            coni, quàm E L.</s>
            <s xml:id="echoid-s9497" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9498" xml:space="preserve">Ergo E M eſt indirectum ipſi E L, &</s>
            <s xml:id="echoid-s9499" xml:space="preserve">c. </s>
            <s xml:id="echoid-s9500" xml:space="preserve">Quia D G baſis ſectionis conicæ
              <lb/>
              <note position="left" xlink:label="note-0291-07" xlink:href="note-0291-07a" xml:space="preserve">f</note>
            perpendicularis eße debet ad G O, & </s>
            <s xml:id="echoid-s9501" xml:space="preserve">ad G E, & </s>
            <s xml:id="echoid-s9502" xml:space="preserve">ideo ad triangulum per axim
              <lb/>
            vtriuſque coni recti L E K, & </s>
            <s xml:id="echoid-s9503" xml:space="preserve">M E I; </s>
            <s xml:id="echoid-s9504" xml:space="preserve">& </s>
            <s xml:id="echoid-s9505" xml:space="preserve">conueniunt plana eorundem trian-
              <lb/>
            gulorum in E G axi conicæ ſectionis geniti ab eis; </s>
            <s xml:id="echoid-s9506" xml:space="preserve">ergo dicta triangula in eo-
              <lb/>
            dem plano exiſtunt per rectas E G, & </s>
            <s xml:id="echoid-s9507" xml:space="preserve">G O ducto; </s>
            <s xml:id="echoid-s9508" xml:space="preserve">& </s>
            <s xml:id="echoid-s9509" xml:space="preserve">in vtroquè cono triangu-
              <lb/>
            lorum per axes latera L K, & </s>
            <s xml:id="echoid-s9510" xml:space="preserve">M I parallela ſunt eidem axi E G paraboles:
              <lb/>
            </s>
            <s xml:id="echoid-s9511" xml:space="preserve">ergo L K, M I parallelæ ſunt inter ſe, & </s>
            <s xml:id="echoid-s9512" xml:space="preserve">anguli L, & </s>
            <s xml:id="echoid-s9513" xml:space="preserve">M æquales ſunt pro-
              <lb/>
            pter ſimilitudinem triangulorum per axes in conis ſimilibus: </s>
            <s xml:id="echoid-s9514" xml:space="preserve">igitur L E, & </s>
            <s xml:id="echoid-s9515" xml:space="preserve">M
              <lb/>
            E ſunt quoq; </s>
            <s xml:id="echoid-s9516" xml:space="preserve">parallelæ, & </s>
            <s xml:id="echoid-s9517" xml:space="preserve">conueniunt in E vertice paraboles; </s>
            <s xml:id="echoid-s9518" xml:space="preserve">ergo in directum
              <lb/>
            ſunt conſtitutæ.</s>
            <s xml:id="echoid-s9519" xml:space="preserve"/>
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