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-div528" type="section" level="1" n="173">
          <p>
            <s xml:id="echoid-s5557" xml:space="preserve">
              <pb o="142" file="0180" n="180" rhead="Apollonij Pergæi"/>
            niam, ſuperpoſita axi C H ſuper axim A G,
              <lb/>
            &</s>
            <s xml:id="echoid-s5558" xml:space="preserve">c. </s>
            <s xml:id="echoid-s5559" xml:space="preserve">vt in textu habetur. </s>
            <s xml:id="echoid-s5560" xml:space="preserve">Si enim axis C H
              <lb/>
              <figure xlink:label="fig-0180-01" xlink:href="fig-0180-01a" number="183">
                <image file="0180-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0180-01"/>
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            ſuper axim A G applicatur, ita vt vertices A,
              <lb/>
            C coincidant, neceſſariò ſectio C D cadet ſu-
              <lb/>
            per ſectionem A B alias aſſignari poſſet pun-
              <lb/>
            ctum eius D, extra ſectionem A B cadens.</s>
            <s xml:id="echoid-s5561" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s5562" xml:space="preserve">Præterea ponamus duas ſectiones æqua-
              <lb/>
              <note position="right" xlink:label="note-0180-01" xlink:href="note-0180-01a" xml:space="preserve">c</note>
            les, & </s>
            <s xml:id="echoid-s5563" xml:space="preserve">C F æqualis A E, &</s>
            <s xml:id="echoid-s5564" xml:space="preserve">c. </s>
            <s xml:id="echoid-s5565" xml:space="preserve">Textum cor-
              <lb/>
            ruptum ſic reſtituendum cenſeo. </s>
            <s xml:id="echoid-s5566" xml:space="preserve">Præterea ſup-
              <lb/>
            ponamus, duas illas ſectiones æquales eſſe in-
              <lb/>
            ter ſe, & </s>
            <s xml:id="echoid-s5567" xml:space="preserve">fiat C F æqualis A E, educamus ad
              <lb/>
            axes perpendiculares B E, D F, &</s>
            <s xml:id="echoid-s5568" xml:space="preserve">c. </s>
            <s xml:id="echoid-s5569" xml:space="preserve">Sic enim
              <lb/>
            diſtinguitur hypotheſis propoſitionis à conſtru-
              <lb/>
            ctione eius.</s>
            <s xml:id="echoid-s5570" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s5571" xml:space="preserve">Ergo ſectio A B cadit ſuper ſectionem.
              <lb/>
            </s>
            <s xml:id="echoid-s5572" xml:space="preserve">
              <note position="right" xlink:label="note-0180-02" xlink:href="note-0180-02a" xml:space="preserve">d</note>
            C D, & </s>
            <s xml:id="echoid-s5573" xml:space="preserve">A E ſuper C F: </s>
            <s xml:id="echoid-s5574" xml:space="preserve">alioqui eſſent ſe-
              <lb/>
            ctioni parabolicæ duo axes; </s>
            <s xml:id="echoid-s5575" xml:space="preserve">ergo F cadit
              <lb/>
            ſuper E, &</s>
            <s xml:id="echoid-s5576" xml:space="preserve">c. </s>
            <s xml:id="echoid-s5577" xml:space="preserve">Quoniam (ex hypotheſi) ſectio-
              <lb/>
            nes A B, & </s>
            <s xml:id="echoid-s5578" xml:space="preserve">C D æquales ſunt, facta intellectuali conuenienti ſuperpoſitione, ſi-
              <lb/>
            bi mutuò congruent, & </s>
            <s xml:id="echoid-s5579" xml:space="preserve">vertex A cadet ſuper verticcm C. </s>
            <s xml:id="echoid-s5580" xml:space="preserve">Dico iam, axim A
              <lb/>
            E cadere ſuper axim C F: </s>
            <s xml:id="echoid-s5581" xml:space="preserve">alioquin in eadem parabola, ſcilicet in duabus pa-
              <lb/>
            rabolis ſibi congruentibus à communi vertice C, vel A, duo axes A E, & </s>
            <s xml:id="echoid-s5582" xml:space="preserve">C F
              <lb/>
            ducerentur: </s>
            <s xml:id="echoid-s5583" xml:space="preserve">quod eſt impoſſibile. </s>
            <s xml:id="echoid-s5584" xml:space="preserve">Quare axis A E cadit ſuper axim C F.</s>
            <s xml:id="echoid-s5585" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div533" type="section" level="1" n="174">
          <head xml:id="echoid-head228" xml:space="preserve">Notæ in Propoſit. II.</head>
          <p>
            <s xml:id="echoid-s5586" xml:space="preserve">SI fuerint figuræ duarum ſectionem hyperbolicarum, aut duarum elli-
              <lb/>
              <note position="right" xlink:label="note-0180-03" xlink:href="note-0180-03a" xml:space="preserve">a</note>
            pſium, vt duo plana G I, H K in A B, D E ſimiles, & </s>
            <s xml:id="echoid-s5587" xml:space="preserve">æquales;
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            </s>
            <s xml:id="echoid-s5588" xml:space="preserve">vtique duæ ſectiones æquales erunt: </s>
            <s xml:id="echoid-s5589" xml:space="preserve">ſi vero duæ ſectiones ſint æquales
              <lb/>
            earum figuræ erunt æquales, ſimiles, &</s>
            <s xml:id="echoid-s5590" xml:space="preserve">c. </s>
            <s xml:id="echoid-s5591" xml:space="preserve">In duabus ſectionibus A B, & </s>
            <s xml:id="echoid-s5592" xml:space="preserve">
              <lb/>
            D E ſumi debent figuræ G I, & </s>
            <s xml:id="echoid-s5593" xml:space="preserve">H K, non qualeſcunque, ſed illæ, quæ ad axes
              <lb/>
            fiunt, nimirum debent eſſe G A, & </s>
            <s xml:id="echoid-s5594" xml:space="preserve">H D axes inclinati, ſeu tranſuerſi, & </s>
            <s xml:id="echoid-s5595" xml:space="preserve">A
              <lb/>
            I, atque D K eorum latera recta; </s>
            <s xml:id="echoid-s5596" xml:space="preserve">tunc quidem, ſi figuræ axium G I, H K fue-
              <lb/>
            rint ſimiles, & </s>
            <s xml:id="echoid-s5597" xml:space="preserve">æquales, conicæ ſectiones B A, D E æquales quoque oſtenduntur
              <lb/>
            in propoſitione. </s>
            <s xml:id="echoid-s5598" xml:space="preserve">Quod verò particula illa (axium) deſideretur in textu propo-
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            ſitionis, conſtat ex primis verbis immediatè ſequentis conſtructionis. </s>
            <s xml:id="echoid-s5599" xml:space="preserve">Inquit
              <lb/>
            enim. </s>
            <s xml:id="echoid-s5600" xml:space="preserve">Quoniam ſi ponamus axim A M ſuper axim D O, &</s>
            <s xml:id="echoid-s5601" xml:space="preserve">c.</s>
            <s xml:id="echoid-s5602" xml:space="preserve"/>
          </p>
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            <s xml:id="echoid-s5603" xml:space="preserve">Cumque G I, H K ſint duæ figuræ ſimiles, & </s>
            <s xml:id="echoid-s5604" xml:space="preserve">æquales, pariterque
              <lb/>
              <note position="right" xlink:label="note-0180-04" xlink:href="note-0180-04a" xml:space="preserve">b</note>
            I P, K R; </s>
            <s xml:id="echoid-s5605" xml:space="preserve">ergo duo plana A P, D R ſunt æqualia, &</s>
            <s xml:id="echoid-s5606" xml:space="preserve">c. </s>
            <s xml:id="echoid-s5607" xml:space="preserve">Quia rectangula
              <lb/>
            I P, G I circa communcm diametrum G I P conſiſtunt, erunt inter ſe ſimilia:
              <lb/>
            </s>
            <s xml:id="echoid-s5608" xml:space="preserve">pariterque K R ſimile erit rectangulo K H: </s>
            <s xml:id="echoid-s5609" xml:space="preserve">quare duo rectangula I P, & </s>
            <s xml:id="echoid-s5610" xml:space="preserve">K R
              <lb/>
            ſimilia ſunt duobus rectangulis G I, H K inter ſe ſimilibus; </s>
            <s xml:id="echoid-s5611" xml:space="preserve">& </s>
            <s xml:id="echoid-s5612" xml:space="preserve">ideo illa inter
              <lb/>
            ſe quoque ſimilia erunt, & </s>
            <s xml:id="echoid-s5613" xml:space="preserve">habent latera homologa æqualia, illa nimirum, quæ
              <lb/>
            opponuntur æqualibus abciſsis A L, & </s>
            <s xml:id="echoid-s5614" xml:space="preserve">D N, igitur rectangula P I, & </s>
            <s xml:id="echoid-s5615" xml:space="preserve">R </s>
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