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-div313" type="section" level="1" n="104">
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              <pb o="79" file="0117" n="117" rhead="Conicor. Lib. V."/>
            (qui eſt D B) cadit ſupra breuiſsimam ex puncto I ad axim ductam, quæ per-
              <lb/>
              <note position="right" xlink:label="note-0117-01" xlink:href="note-0117-01a" xml:space="preserve">51. 52.
                <lb/>
              huius.</note>
            pendicularis eſt ad tangentem L I H, & </s>
            <s xml:id="echoid-s3300" xml:space="preserve">propterea angulus D I L, verticem
              <lb/>
              <note position="right" xlink:label="note-0117-02" xlink:href="note-0117-02a" xml:space="preserve">29. 30.
                <lb/>
              huius.</note>
            A reſpiciens erit acutus, & </s>
            <s xml:id="echoid-s3301" xml:space="preserve">conſequens angulus D I H obtuſus.</s>
            <s xml:id="echoid-s3302" xml:space="preserve"/>
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        <div xml:id="echoid-div315" type="section" level="1" n="105">
          <head xml:id="echoid-head145" xml:space="preserve">LEMMA XI.</head>
          <p style="it">
            <s xml:id="echoid-s3303" xml:space="preserve">Iiſdem poſitis, ſi à concurſu D duo breuiſecantes D C, D B ad ſe-
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            ctionem A B duci poſſunt; </s>
            <s xml:id="echoid-s3304" xml:space="preserve">Dico, quod quilibet ramus ſecans D I poſi-
              <lb/>
            tus ſupra breuiſecantem D B vertici proximiorem, vel infra infimum
              <lb/>
            breuiſecantem D C, efficit cum recta L I H tangente ſectionem in I an-
              <lb/>
            gulum D I L, reſpicientem verticem A, acutum, & </s>
            <s xml:id="echoid-s3305" xml:space="preserve">conſequentem D
              <lb/>
            I H obtuſum, & </s>
            <s xml:id="echoid-s3306" xml:space="preserve">quilibet ramus D O inter breuiſecantes poſitus efficit
              <lb/>
            cum recta G O N ſectionem tangente in O angulum D O G verticem
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            reſpicientem obtuſum, conſequentem vero D O N acutum.</s>
            <s xml:id="echoid-s3307" xml:space="preserve"/>
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            <s xml:id="echoid-s3308" xml:space="preserve">Quia (ex conuerſa propoſitione 51. </s>
            <s xml:id="echoid-s3309" xml:space="preserve">& </s>
            <s xml:id="echoid-s3310" xml:space="preserve">52. </s>
            <s xml:id="echoid-s3311" xml:space="preserve">huius) perpendicularis D E mi-
              <lb/>
              <note position="right" xlink:label="note-0117-03" xlink:href="note-0117-03a" xml:space="preserve">51. 52.
                <lb/>
              huius.</note>
            nor eſſe debet Trutina F, & </s>
            <s xml:id="echoid-s3312" xml:space="preserve">propterea quilibet ramus D I ſupra breuiſecantem
              <lb/>
            D B, vel infra breuiſecãtem D C cadit ſupra breuiſsimam ex puncto I ad axim
              <lb/>
              <note position="right" xlink:label="note-0117-04" xlink:href="note-0117-04a" xml:space="preserve">29. 30.
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              huius.</note>
            ductam, cum qua contingens L I angulum rectum conſtituit; </s>
            <s xml:id="echoid-s3313" xml:space="preserve">ergo angulus D I
              <lb/>
            L verticem reſpiciens, eſt acutus, & </s>
            <s xml:id="echoid-s3314" xml:space="preserve">conſequens D I H obtuſus; </s>
            <s xml:id="echoid-s3315" xml:space="preserve">Similiter qui-
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            libet ramus D O inter breuiſecantes poſitus cadit infra breuiſsimam ex puncto
              <lb/>
            O ad axim ductam, & </s>
            <s xml:id="echoid-s3316" xml:space="preserve">cum illa ſectionem contingens G O efſicit angulos rectos,
              <lb/>
              <note position="right" xlink:label="note-0117-05" xlink:href="note-0117-05a" xml:space="preserve">Ibidem.</note>
            igitur angulus D O G verticem reſpiciens, eſt obtuſus, & </s>
            <s xml:id="echoid-s3317" xml:space="preserve">conſequens D O N
              <lb/>
            acutus.</s>
            <s xml:id="echoid-s3318" xml:space="preserve"/>
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        <div xml:id="echoid-div317" type="section" level="1" n="106">
          <head xml:id="echoid-head146" xml:space="preserve">Notæ in Propoſ. LXIV.
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          & LXV.</head>
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            <s xml:id="echoid-s3319" xml:space="preserve">ANtea Apollonius docuit qui nam rami ab origine ad coniſectionem ducti
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            eſſent minimi, & </s>
            <s xml:id="echoid-s3320" xml:space="preserve">quo ordine reliqui rami ſe ſe excederent, modo agit
              <lb/>
            de ramis axim ſecantibus à concurſu ductis, & </s>
            <s xml:id="echoid-s3321" xml:space="preserve">quærit qui minimus, & </s>
            <s xml:id="echoid-s3322" xml:space="preserve">qui
              <lb/>
            maximus ſit, & </s>
            <s xml:id="echoid-s3323" xml:space="preserve">quo ordine diſponantur.</s>
            <s xml:id="echoid-s3324" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s3325" xml:space="preserve">Producamus perpendicularem D E ſuper axim, &</s>
            <s xml:id="echoid-s3326" xml:space="preserve">c. </s>
            <s xml:id="echoid-s3327" xml:space="preserve">Si nullus ramus
              <lb/>
              <note position="right" xlink:label="note-0117-06" xlink:href="note-0117-06a" xml:space="preserve">a</note>
            breuiſecans à concurſu D ad ſectionem A C duci poteſt; </s>
            <s xml:id="echoid-s3328" xml:space="preserve">Dico, quod ramus ter-
              <lb/>
            minatus D A eſt minimus omnium ramorum ſecantium D B, D C, & </s>
            <s xml:id="echoid-s3329" xml:space="preserve">propin-
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            quiores vertici A minores ſunt remotioribus; </s>
            <s xml:id="echoid-s3330" xml:space="preserve">ducatur D E perpendicularis ad
              <lb/>
            axim eum ſecans in E, & </s>
            <s xml:id="echoid-s3331" xml:space="preserve">reperiatur Trutina F. </s>
            <s xml:id="echoid-s3332" xml:space="preserve">Et ſiquidem D A non eſt
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            minor quolibet alio ramo ſecante D B infra ipſum poſito erit æqualis, aut maior
              <lb/>
            illo; </s>
            <s xml:id="echoid-s3333" xml:space="preserve">ſitque prius D A æqualis D B, ſi fieri poteſt, & </s>
            <s xml:id="echoid-s3334" xml:space="preserve">ex puncto A verticis du-
              <lb/>
            catur A G perpendicularis ad axim A E, quæ continget ſectionem in A, pari-
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              <note position="right" xlink:label="note-0117-07" xlink:href="note-0117-07a" xml:space="preserve">17. lib. 1.
                <lb/>
              32. pr.</note>
            terque ducatur recta A H perpendicularis ad ramum A D inclinatum ad axim;</s>
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