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-div759" type="section" level="1" n="236">
          <pb o="238" file="0276" n="276" rhead="Apollonij Pergæi"/>
          <p>
            <s xml:id="echoid-s8914" xml:space="preserve">Vt quadratum A C ad C B in BA,
              <lb/>
              <figure xlink:label="fig-0276-01" xlink:href="fig-0276-01a" number="321">
                <image file="0276-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0276-01"/>
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            ita ponatur E G ad B H: </s>
            <s xml:id="echoid-s8915" xml:space="preserve">& </s>
            <s xml:id="echoid-s8916" xml:space="preserve">educa-
              <lb/>
            mus H I parallelam B C, & </s>
            <s xml:id="echoid-s8917" xml:space="preserve">exten-
              <lb/>
            datur per H I planum eleuatum ſuper
              <lb/>
            triangulum A B C ad angulos rectos
              <lb/>
            efficiens in cono ſectionem K H L.
              <lb/>
            </s>
            <s xml:id="echoid-s8918" xml:space="preserve">Dico eam æqualem eſſe ſectioni D E. </s>
            <s xml:id="echoid-s8919" xml:space="preserve">
              <lb/>
            Quia quadratum A C ad C B in B
              <lb/>
            A eſt, vt E G ad B H; </s>
            <s xml:id="echoid-s8920" xml:space="preserve">ergo poten-
              <lb/>
            tes eductæ ad axim H I in ſectione
              <lb/>
              <note position="right" xlink:label="note-0276-01" xlink:href="note-0276-01a" xml:space="preserve">a</note>
            K H L poſſunt applicata contenta ab
              <lb/>
            abſciſſis illarum potentium, & </s>
            <s xml:id="echoid-s8921" xml:space="preserve">ab E
              <lb/>
            G; </s>
            <s xml:id="echoid-s8922" xml:space="preserve">quare E G erit erectum ſectionis
              <lb/>
            K H, & </s>
            <s xml:id="echoid-s8923" xml:space="preserve">idem etiam eſt erectum ſectionis D E; </s>
            <s xml:id="echoid-s8924" xml:space="preserve">ergo duo erecta duarum
              <lb/>
            ſectionum ſunt æqualia, & </s>
            <s xml:id="echoid-s8925" xml:space="preserve">propterea ſectiones æquales ſunt (1. </s>
            <s xml:id="echoid-s8926" xml:space="preserve">ex 6.)</s>
            <s xml:id="echoid-s8927" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s8928" xml:space="preserve">Et dico, quod in cono A B C reperiri non poteſt ſectio alia parabo-
              <lb/>
              <note position="right" xlink:label="note-0276-02" xlink:href="note-0276-02a" xml:space="preserve">b</note>
            lica, cuius vertex ſit ſuper A B, quæ eidem D E ſit æqualis. </s>
            <s xml:id="echoid-s8929" xml:space="preserve">Si enim
              <lb/>
            hoc eſt poſſibile, ſit axis illius ſectionis M N, qui quidem cadet in trian-
              <lb/>
            gulo A B C; </s>
            <s xml:id="echoid-s8930" xml:space="preserve">quia conus eſt rectus, & </s>
            <s xml:id="echoid-s8931" xml:space="preserve">erectum illius ſit M O; </s>
            <s xml:id="echoid-s8932" xml:space="preserve">atq; </s>
            <s xml:id="echoid-s8933" xml:space="preserve">M O
              <lb/>
            ad M B erit, vt G E ad B H; </s>
            <s xml:id="echoid-s8934" xml:space="preserve">eſtque B H maior, quàm B M; </s>
            <s xml:id="echoid-s8935" xml:space="preserve">ergo M O
              <lb/>
              <note position="left" xlink:label="note-0276-03" xlink:href="note-0276-03a" xml:space="preserve">ex conu.
                <lb/>
              Prop. 1.
                <lb/>
              huius.</note>
            minor eſt, quàm G E; </s>
            <s xml:id="echoid-s8936" xml:space="preserve">quare ſectio, cuius axis eſt M N non eſt æqualis
              <lb/>
            ſectioni D E; </s>
            <s xml:id="echoid-s8937" xml:space="preserve">& </s>
            <s xml:id="echoid-s8938" xml:space="preserve">tamen ſuppoſita fuit æqualis illi, quod eſt abſurdum.
              <lb/>
            </s>
            <s xml:id="echoid-s8939" xml:space="preserve">Quare patet propoſitum.</s>
            <s xml:id="echoid-s8940" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div762" type="section" level="1" n="237">
          <head xml:id="echoid-head298" xml:space="preserve">PROPOSITIO XXVII.</head>
          <p>
            <s xml:id="echoid-s8941" xml:space="preserve">SIt deinde hyperbole A B, cuius axis C D, inclinatus B
              <lb/>
              <note position="left" xlink:label="note-0276-04" xlink:href="note-0276-04a" xml:space="preserve">a</note>
            D, & </s>
            <s xml:id="echoid-s8942" xml:space="preserve">erectus B E; </s>
            <s xml:id="echoid-s8943" xml:space="preserve">atque quadratum axis F G dati coni
              <lb/>
            recti F H I ad quadratum G H ſemidiametri baſis eius, non
              <lb/>
            habeat maiorem proportionem, quàm habet figura, ſcilicet
              <lb/>
            quàm habet D B ad B E.</s>
            <s xml:id="echoid-s8944" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8945" xml:space="preserve">Sit prius proportio eadem, & </s>
            <s xml:id="echoid-s8946" xml:space="preserve">producamus I F ad K; </s>
            <s xml:id="echoid-s8947" xml:space="preserve">& </s>
            <s xml:id="echoid-s8948" xml:space="preserve">ducamus K
              <lb/>
            L ſubtendentem angulum H F K, quæ parallela ſit ipſi F G, & </s>
            <s xml:id="echoid-s8949" xml:space="preserve">æqualis
              <lb/>
            exiſtat ipſi D B; </s>
            <s xml:id="echoid-s8950" xml:space="preserve">& </s>
            <s xml:id="echoid-s8951" xml:space="preserve">per K L planum extendatur eleuatum ad angulos re-
              <lb/>
            ctos ſuper planum trianguli H F I, quod efficiet in ſuperſicie conica ſe-
              <lb/>
            ctionem hyperbolicam, cuius axis erit L M, & </s>
            <s xml:id="echoid-s8952" xml:space="preserve">inclinatus K L. </s>
            <s xml:id="echoid-s8953" xml:space="preserve">Et quia
              <lb/>
            F G parallela eſt K L, erit quadratum F G ad G I in G H, vt K L in-
              <lb/>
              <note position="left" xlink:label="note-0276-05" xlink:href="note-0276-05a" xml:space="preserve">12. lib. 1.</note>
            clinatus ad illius erectum, ſiue vt D B ad B E; </s>
            <s xml:id="echoid-s8954" xml:space="preserve">facta autem fuit K L æ-
              <lb/>
            qualis D B; </s>
            <s xml:id="echoid-s8955" xml:space="preserve">ergo erectus inclinati K L æqualis eſt B E; </s>
            <s xml:id="echoid-s8956" xml:space="preserve">& </s>
            <s xml:id="echoid-s8957" xml:space="preserve">propterea ſe-
              <lb/>
              <note position="left" xlink:label="note-0276-06" xlink:href="note-0276-06a" xml:space="preserve">2. huius.</note>
              <note position="right" xlink:label="note-0276-07" xlink:href="note-0276-07a" xml:space="preserve">b</note>
            ctio, cuius axis eſt L M æqualis eſt ſectioni A B. </s>
            <s xml:id="echoid-s8958" xml:space="preserve">Nec reperiri poterit
              <lb/>
            in cono H F I alia ſectio hyperbolica, cuius vertex ſit ſuper H F, quæ
              <lb/>
            æqualis ſit A B; </s>
            <s xml:id="echoid-s8959" xml:space="preserve">quia, ſi reperiri poſſet eſſet illius axis in plano trianguli
              <lb/>
            H F I, & </s>
            <s xml:id="echoid-s8960" xml:space="preserve">eius inclinatus, ſubtendens angulum H F K æqualis eſſet D B,
              <lb/>
            nec tamen eſſet K L, nequè ipſi æquidiſtans (eo quod, ſi </s>
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