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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            <s xml:id="echoid-s3861" xml:space="preserve">
              <pb o="95" file="0133" n="133" rhead="Conicor. Lib. V."/>
            occurrentes in M, & </s>
            <s xml:id="echoid-s3862" xml:space="preserve">quia angulus D B M rectus eſt contentus ab axe, & </s>
            <s xml:id="echoid-s3863" xml:space="preserve">tangen-
              <lb/>
              <note position="right" xlink:label="note-0133-01" xlink:href="note-0133-01a" xml:space="preserve">Conue ſ.
                <lb/>
              32. lib. 1.</note>
            te, & </s>
            <s xml:id="echoid-s3864" xml:space="preserve">cadit B E inter C, & </s>
            <s xml:id="echoid-s3865" xml:space="preserve">D ergo angulus E B M eſt obtuſus; </s>
            <s xml:id="echoid-s3866" xml:space="preserve">poſtea, quia E
              <lb/>
            K cadit infra breuiſsimam E G, & </s>
            <s xml:id="echoid-s3867" xml:space="preserve">ſupra minorem axim B D, ergo angulus
              <lb/>
              <note position="right" xlink:label="note-0133-02" xlink:href="note-0133-02a" xml:space="preserve">Lem. 12.</note>
            E K M reſpiciens verticem C propinquiorem concurſui, erit acutus, & </s>
            <s xml:id="echoid-s3868" xml:space="preserve">iuncta
              <lb/>
            M E erunt duo quadrata E B, B M minora quadrato E M, eſtque quadratum
              <lb/>
            E M minus duobus quadratis E K, K M circa acutum angulum (cum prior a
              <lb/>
            angulum obtuſum compræhendant,) Igitur duo quadrata E B, B M ſimul ſum-
              <lb/>
            pta minora ſunt duobus quadratis E K, K M: </s>
            <s xml:id="echoid-s3869" xml:space="preserve">eſtque quadratum M B maius
              <lb/>
            quadrato M K, cum contingens M K, proximior vertici A axis maioris minor
              <lb/>
              <note position="right" xlink:label="note-0133-03" xlink:href="note-0133-03a" xml:space="preserve">70. huius.</note>
            ſit remotiore B M; </s>
            <s xml:id="echoid-s3870" xml:space="preserve">igitur quadratum E B, ſcilicet reſiduum minoris ſummæ mi-
              <lb/>
            nus erit quadrato E K, & </s>
            <s xml:id="echoid-s3871" xml:space="preserve">propterea ramus E B minor erit, quàm E K.</s>
            <s xml:id="echoid-s3872" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3873" xml:space="preserve">Et educamus ex E ad ſectionem A G, E A, E O, & </s>
            <s xml:id="echoid-s3874" xml:space="preserve">patebit, quod E
              <lb/>
              <note position="left" xlink:label="note-0133-04" xlink:href="note-0133-04a" xml:space="preserve">e</note>
            G maior fit, quàm E O, & </s>
            <s xml:id="echoid-s3875" xml:space="preserve">E O, quàm E A: </s>
            <s xml:id="echoid-s3876" xml:space="preserve">erigamus itaque ad A C
              <lb/>
            perpendicularem A P; </s>
            <s xml:id="echoid-s3877" xml:space="preserve">ergo E A P eſt obtuſus: </s>
            <s xml:id="echoid-s3878" xml:space="preserve">& </s>
            <s xml:id="echoid-s3879" xml:space="preserve">ducamus P O Q tan-
              <lb/>
            gentem; </s>
            <s xml:id="echoid-s3880" xml:space="preserve">ergo P O E eſt acutus, quia linea breuiſsima egrediens ex O ab-
              <lb/>
            fcindit cum A lineam maiorem, & </s>
            <s xml:id="echoid-s3881" xml:space="preserve">P O eſt maior, quàm P A; </s>
            <s xml:id="echoid-s3882" xml:space="preserve">ergo E O
              <lb/>
            maior eſt quàm E A, atque ſic patet, quod E G maior ſit, quàm E O, &</s>
            <s xml:id="echoid-s3883" xml:space="preserve">c.
              <lb/>
            </s>
            <s xml:id="echoid-s3884" xml:space="preserve">Demonſtratio poſtremæ partis huius propoſitionis neglecta ab Apollonio ob ſui fa-
              <lb/>
            cilitatem occaſionem errandi alicui præbere poſſet, propter verba illa poſtrema
              <lb/>
            textui ſuperaddita; </s>
            <s xml:id="echoid-s3885" xml:space="preserve">non enim ex maiori ſumma duorum laterum P O, O E ſi au-
              <lb/>
            feratur maior O P, & </s>
            <s xml:id="echoid-s3886" xml:space="preserve">ex minori ſumma P A, A E auferatur minor P A, neceſſa-
              <lb/>
            rio reſiduum maioris, ideſt E O maior erit quam E A reſiduum minoris; </s>
            <s xml:id="echoid-s3887" xml:space="preserve">itaque
              <lb/>
            ſenſus huius contextus talis erit.</s>
            <s xml:id="echoid-s3888" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s3889" xml:space="preserve">Ex concurſu E ad ſectionem A G ducantur rami E A, & </s>
            <s xml:id="echoid-s3890" xml:space="preserve">quilibet alius E O;
              <lb/>
            </s>
            <s xml:id="echoid-s3891" xml:space="preserve">oſtendendum eſt, E G maiorem eſſe, quàm E O, & </s>
            <s xml:id="echoid-s3892" xml:space="preserve">E O maiorem, quàm E A: </s>
            <s xml:id="echoid-s3893" xml:space="preserve">du-
              <lb/>
            cantur A P, Q O tangentes ſectionem in A, & </s>
            <s xml:id="echoid-s3894" xml:space="preserve">O conuenientes in P, & </s>
            <s xml:id="echoid-s3895" xml:space="preserve">tangenti
              <lb/>
              <note position="right" xlink:label="note-0133-05" xlink:href="note-0133-05a" xml:space="preserve">Conuerſ.
                <lb/>
              32. lib. 1.</note>
            G Q in Q. </s>
            <s xml:id="echoid-s3896" xml:space="preserve">manifectum eſt angulum E A P obtuſum eſſe, cum angulus C A P ſit
              <lb/>
            rectus pariterque quilibet ramus E O inter breuiſecantem E G, & </s>
            <s xml:id="echoid-s3897" xml:space="preserve">verticem A
              <lb/>
              <note position="right" xlink:label="note-0133-06" xlink:href="note-0133-06a" xml:space="preserve">Lem. 12.</note>
            remotiorem interceptus efficit angulum E O P, verticem A reſpicientem acutum,
              <lb/>
            & </s>
            <s xml:id="echoid-s3898" xml:space="preserve">ſic reliqui omnes rami inter puncta G, & </s>
            <s xml:id="echoid-s3899" xml:space="preserve">A cadentes; </s>
            <s xml:id="echoid-s3900" xml:space="preserve">quare (ex Corollario
              <lb/>
            propoſitionum 64. </s>
            <s xml:id="echoid-s3901" xml:space="preserve">& </s>
            <s xml:id="echoid-s3902" xml:space="preserve">65.) </s>
            <s xml:id="echoid-s3903" xml:space="preserve">ramus E A minor erit quolibet ramo E O inter verti-
              <lb/>
            cem A, & </s>
            <s xml:id="echoid-s3904" xml:space="preserve">G cadente: </s>
            <s xml:id="echoid-s3905" xml:space="preserve">rurſus, quoniam breuiſecans E G conſtituit cum tangente
              <lb/>
              <note position="right" xlink:label="note-0133-07" xlink:href="note-0133-07a" xml:space="preserve">29. 30.
                <lb/>
              huius.</note>
            angulũ E G Q rectum; </s>
            <s xml:id="echoid-s3906" xml:space="preserve">quare ex concurſu E ad ſectionis peripheriam G A omnes
              <lb/>
              <note position="right" xlink:label="note-0133-08" xlink:href="note-0133-08a" xml:space="preserve">Lem. 12.</note>
            rami cadentes efficiunt cum tangentibus angulos, verticem A reſpicientes, acutos,
              <lb/>
            & </s>
            <s xml:id="echoid-s3907" xml:space="preserve">vnus tantummodo E G Q eſt rectus; </s>
            <s xml:id="echoid-s3908" xml:space="preserve">igitur (ex Coroll. </s>
            <s xml:id="echoid-s3909" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s3910" xml:space="preserve">67. </s>
            <s xml:id="echoid-s3911" xml:space="preserve">huius) ramus
              <lb/>
            E O vertici A propinquior minor eſt remotiore E G; </s>
            <s xml:id="echoid-s3912" xml:space="preserve">Quapropter ramus breuiſecãs
              <lb/>
            E G maximus eſt omnium ramorum ſecantium ad peripheriam A B C cadentium.</s>
            <s xml:id="echoid-s3913" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s3914" xml:space="preserve">At adhuc non conſtat, ramum E C minimum eſſe prædictorum ramorum om-
              <lb/>
            nium, niſi priùs oſtendatur, E C minorem eſſe quolibet ramo ad peripheriam
              <lb/>
            A G educto: </s>
            <s xml:id="echoid-s3915" xml:space="preserve">& </s>
            <s xml:id="echoid-s3916" xml:space="preserve">hoc etiam ob ſui facilitatem neglectum fuit ab Apollonio. </s>
            <s xml:id="echoid-s3917" xml:space="preserve">Abſol-
              <lb/>
            uetur tamen hac ratione.</s>
            <s xml:id="echoid-s3918" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s3919" xml:space="preserve">Quoniam perpendicularis E F cadit inter C, & </s>
            <s xml:id="echoid-s3920" xml:space="preserve">D, igitur A F maior eſt, quàm
              <lb/>
            C F, & </s>
            <s xml:id="echoid-s3921" xml:space="preserve">F E eſt communis circa angulos rectos in triangulis C F E, A F E, igi-
              <lb/>
            tur C E minor eſt, quàm E A: </s>
            <s xml:id="echoid-s3922" xml:space="preserve">eſtque E A minor quolibet alio E O inter A, & </s>
            <s xml:id="echoid-s3923" xml:space="preserve">G
              <lb/>
            cadente, igitur E C minor eſt omnium ramorum cadentium ad peripheriam A G,
              <lb/>
            ſed priùs minor oſtenſus fuit reliquis omnibus cadentibus ad peripheriam C B G;
              <lb/>
            </s>
            <s xml:id="echoid-s3924" xml:space="preserve">igitur ramus E C minimus eſt omnium ſecantium, quod erat oſtendendum.</s>
            <s xml:id="echoid-s3925" xml:space="preserve"/>
          </p>
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