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-div135" type="section" level="1" n="59">
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            <s xml:id="echoid-s1713" xml:space="preserve">
              <pb o="31" file="0069" n="69" rhead="Conicor. Lib. V."/>
            C ſub intermedijs contentum æquale ei, quod
              <lb/>
              <figure xlink:label="fig-0069-01" xlink:href="fig-0069-01a" number="46">
                <image file="0069-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0069-01"/>
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            ſub extremis E, D quatuor proportionaliũ con-
              <lb/>
            tinetur ; </s>
            <s xml:id="echoid-s1714" xml:space="preserve">ergo rectangulum A D maius eſt re-
              <lb/>
            ctangulo B C. </s>
            <s xml:id="echoid-s1715" xml:space="preserve">Poſtea ſit rectangulũ A D ma-
              <lb/>
            ius rectangulo B C; </s>
            <s xml:id="echoid-s1716" xml:space="preserve">Dico A ad B maiorem pro-
              <lb/>
            portionem habere, quàm C ad D; </s>
            <s xml:id="echoid-s1717" xml:space="preserve">Si enim hoc
              <lb/>
            verum non eſt, habebit A ad B eandem, aut
              <lb/>
            minorem proportionem quàm C ad D, quare rectangulum A D æquale, aut mi-
              <lb/>
            nus erit rectangulo B C, quæ ſunt contra hypotheſim ; </s>
            <s xml:id="echoid-s1718" xml:space="preserve">igitur A ad B maiorem
              <lb/>
            proportionem babet, quàm C ad D.</s>
            <s xml:id="echoid-s1719" xml:space="preserve"/>
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        </div>
        <div xml:id="echoid-div137" type="section" level="1" n="60">
          <head xml:id="echoid-head91" xml:space="preserve">LEMMA. VI.</head>
          <p style="it">
            <s xml:id="echoid-s1720" xml:space="preserve">SIrectæ linea A B ſecetur bifariam in C, & </s>
            <s xml:id="echoid-s1721" xml:space="preserve">non bifariam in D: </s>
            <s xml:id="echoid-s1722" xml:space="preserve">Dico,
              <lb/>
            quod ſemiſsis C B ad alterum ſegmentorum inæqualium D B habet
              <lb/>
            maiorẽ proportionẽ, quàm reliquum inæqualiũ AD ad alter ã medietatẽ AC.</s>
            <s xml:id="echoid-s1723" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s1724" xml:space="preserve">Quoniam quadratum ſemiſſis C B, ſeu re-
              <lb/>
              <figure xlink:label="fig-0069-02" xlink:href="fig-0069-02a" number="47">
                <image file="0069-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0069-02"/>
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            ctangulum B C A maius eſt rectangulo A D B
              <lb/>
            ſub inæqualibus ſegmentis contento;</s>
            <s xml:id="echoid-s1725" xml:space="preserve">ergo ex præ-
              <lb/>
            cedenti lemmate C B ad D B maiorem propor-
              <lb/>
            tionem habet, quàm A D ad A C; </s>
            <s xml:id="echoid-s1726" xml:space="preserve">Aſſumitur
              <lb/>
            in ſequenti prop. </s>
            <s xml:id="echoid-s1727" xml:space="preserve">52. </s>
            <s xml:id="echoid-s1728" xml:space="preserve">problema antiquum in-
              <lb/>
            uentionis duarum mediarum continuè proportionalium inter duas rectas lineas
              <lb/>
              <note position="right" xlink:label="note-0069-01" xlink:href="note-0069-01a" xml:space="preserve">Cõm. lib.
                <lb/>
              2. Arch. de
                <lb/>
              Sphę a, &
                <lb/>
              Cylin.
                <lb/>
              Prop. 2.</note>
            datas, cuius conſtructio, & </s>
            <s xml:id="echoid-s1729" xml:space="preserve">demonſtratio ab Apollonio inuenta adhuc legitur apud
              <lb/>
            Eutocium, ſed organica quidem illa eſt, & </s>
            <s xml:id="echoid-s1730" xml:space="preserve">ad manuum operationes maximè ac-
              <lb/>
            comodata, non omnino diuerſa ab ea, quàm Hero, & </s>
            <s xml:id="echoid-s1731" xml:space="preserve">philo ediderunt. </s>
            <s xml:id="echoid-s1732" xml:space="preserve">At Par-
              <lb/>
            menion aliam eiuſdem problematis demonſtrationem Apollonio tribuit paulò di-
              <lb/>
            uerſam ab ea , quàm Eutocius recenſuit : </s>
            <s xml:id="echoid-s1733" xml:space="preserve">eam ſane nec percepit, nec rite expo-
              <lb/>
              <note position="right" xlink:label="note-0069-02" xlink:href="note-0069-02a" xml:space="preserve">In lib. 5.
                <lb/>
              Poſt Ana-
                <lb/>
              lit. comm.
                <lb/>
              36.</note>
            ſuit, Philoponus, quàm enim petitionem non demonſtratam ipſe vocat conſequẽ-
              <lb/>
            tia eſt neceſſaria ex deſcriptione hyperboles, quæ omnino ſubintelligi, & </s>
            <s xml:id="echoid-s1734" xml:space="preserve">adiun-
              <lb/>
            gi debet, vt colligitur ex Pappi verbis : </s>
            <s xml:id="echoid-s1735" xml:space="preserve">hi enim (ſcilicet Hero, & </s>
            <s xml:id="echoid-s1736" xml:space="preserve">Philo)
              <lb/>
              <note position="right" xlink:label="note-0069-03" xlink:href="note-0069-03a" xml:space="preserve">Coll. lib. 3.
                <lb/>
              Prop. 4.</note>
            aßerentes problema ſolidum eße, ipſius conſtructionem inſtrumentis tantum per-
              <lb/>
            fecerunt congruenter Apollonio Pergæo, qui reſolutionem eius fecit per coniſe-
              <lb/>
            ctiones. </s>
            <s xml:id="echoid-s1737" xml:space="preserve">Erit igitur Apollonij propoſitio huiuſmodi.</s>
            <s xml:id="echoid-s1738" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div139" type="section" level="1" n="61">
          <head xml:id="echoid-head92" xml:space="preserve">LEMMA VII.</head>
          <p style="it">
            <s xml:id="echoid-s1739" xml:space="preserve">INter rectam lineam A C maiorem , & </s>
            <s xml:id="echoid-s1740" xml:space="preserve">B C minorem duas medias
              <lb/>
            proportionales reperire.</s>
            <s xml:id="echoid-s1741" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s1742" xml:space="preserve">Conueniant illæ ad angulos rectos in A , & </s>
            <s xml:id="echoid-s1743" xml:space="preserve">compleatur Parallelogrammum
              <lb/>
              <note position="right" xlink:label="note-0069-04" xlink:href="note-0069-04a" xml:space="preserve">Prop. 4.
                <lb/>
              lib. 2.</note>
            A B D C, cui circumſcribatur circulus diametro D A, & </s>
            <s xml:id="echoid-s1744" xml:space="preserve">per punctum D circa
              <lb/>
            aſymptotos C A B deſcribatur hyperbole D F, & </s>
            <s xml:id="echoid-s1745" xml:space="preserve">ducatur recta D M circulum
              <lb/>
              <note position="right" xlink:label="note-0069-05" xlink:href="note-0069-05a" xml:space="preserve">Prop. 34.
                <lb/>
              lib. 1.</note>
            tangens in D, & </s>
            <s xml:id="echoid-s1746" xml:space="preserve">recta I D K ſectionem ibidem contingens , occurrens aſym-
              <lb/>
            ptotis in I , & </s>
            <s xml:id="echoid-s1747" xml:space="preserve">K, erunt quidem I D, & </s>
            <s xml:id="echoid-s1748" xml:space="preserve">I K æquales inter ſe, & </s>
            <s xml:id="echoid-s1749" xml:space="preserve">D C paral-
              <lb/>
              <note position="right" xlink:label="note-0069-06" xlink:href="note-0069-06a" xml:space="preserve">3. lib. 1.</note>
            lela eſt A K , ergo I C æqualis eſt C A : </s>
            <s xml:id="echoid-s1750" xml:space="preserve">pari ratione K B æqualis erit B A,
              <lb/>
            ſed poſita fuit C A maior quàm A B, ergo in triangulis I A D, & </s>
            <s xml:id="echoid-s1751" xml:space="preserve">K D A baſis
              <lb/>
            I A maior erit, quàm A K, & </s>
            <s xml:id="echoid-s1752" xml:space="preserve">latera I D, D K æqualia ſunt, & </s>
            <s xml:id="echoid-s1753" xml:space="preserve">D A eſt commune,
              <lb/>
            igitur angulus A D I maior erit angulo A D K, & </s>
            <s xml:id="echoid-s1754" xml:space="preserve">propterearecta line a I K </s>
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