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-div317" type="section" level="1" n="106">
          <pb o="81" file="0119" n="119" rhead="Conicor. Lib. V."/>
          <p style="it">
            <s xml:id="echoid-s3370" xml:space="preserve">Deinde patebit, quemadmodum demonſtrauimus, &</s>
            <s xml:id="echoid-s3371" xml:space="preserve">c. </s>
            <s xml:id="echoid-s3372" xml:space="preserve">Quia D M fa-
              <lb/>
              <note position="left" xlink:label="note-0119-01" xlink:href="note-0119-01a" xml:space="preserve">f</note>
            cta eſt maior, quàm D B, & </s>
            <s xml:id="echoid-s3373" xml:space="preserve">minor quàm D A, eſtque circuli radius D N
              <lb/>
            æqualis D M; </s>
            <s xml:id="echoid-s3374" xml:space="preserve">ergo punctum M cadit intra coniſectionem, N vero extra ip-
              <lb/>
            ſam; </s>
            <s xml:id="echoid-s3375" xml:space="preserve">& </s>
            <s xml:id="echoid-s3376" xml:space="preserve">propterea circulus M L N ſectionem conicam ſecabit alicubi, vt in L,
              <lb/>
            & </s>
            <s xml:id="echoid-s3377" xml:space="preserve">portio circuli M L intra coniſectionem A L incidet: </s>
            <s xml:id="echoid-s3378" xml:space="preserve">rurſus ducatur radius
              <lb/>
            D L, & </s>
            <s xml:id="echoid-s3379" xml:space="preserve">L G coniſectionem tangens in L erit, vt priùs angulus D L G acu-
              <lb/>
              <note position="right" xlink:label="note-0119-02" xlink:href="note-0119-02a" xml:space="preserve">33. 34.
                <lb/>
              lib. 1.</note>
            tus; </s>
            <s xml:id="echoid-s3380" xml:space="preserve">& </s>
            <s xml:id="echoid-s3381" xml:space="preserve">ideo L G cadit intra circulum L M, & </s>
            <s xml:id="echoid-s3382" xml:space="preserve">propterea intra coniſectionem
              <lb/>
            A L, ſed eadem L G cadit extra ipſam, quia eam contingit in L, quod eſt ab-
              <lb/>
            ſurdum; </s>
            <s xml:id="echoid-s3383" xml:space="preserve">quare ramus D A non eſt maior, quàm D B; </s>
            <s xml:id="echoid-s3384" xml:space="preserve">ſed priùs neque illi
              <lb/>
            æqualis erat; </s>
            <s xml:id="echoid-s3385" xml:space="preserve">igitur ramus terminatus D A minor eſt quolibet ramo ſecante
              <lb/>
            D B infra ipſum poſito, & </s>
            <s xml:id="echoid-s3386" xml:space="preserve">propterea minimus erit omnium ſecantium.</s>
            <s xml:id="echoid-s3387" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s3388" xml:space="preserve">Poſtea dico, quod D C maior eſt, quàm D B, &</s>
            <s xml:id="echoid-s3389" xml:space="preserve">c. </s>
            <s xml:id="echoid-s3390" xml:space="preserve">Demonſtratio ſe-
              <lb/>
              <note position="left" xlink:label="note-0119-03" xlink:href="note-0119-03a" xml:space="preserve">g</note>
            cundæ partis huius propoſitionis, quàm Apollonius innuit (quia conſtructione,
              <lb/>
            ac progreſſu ſimili ſuperiori perſici poteſt) hac ratione reſtituitur. </s>
            <s xml:id="echoid-s3391" xml:space="preserve">Demonſtran-
              <lb/>
            dum eſt quemlibet ramum D B vertici A proximiorem eße minorem quolibet
              <lb/>
            ramo D C remotiore. </s>
            <s xml:id="echoid-s3392" xml:space="preserve">Ducantur recta C P contingens ſectionem in C, & </s>
            <s xml:id="echoid-s3393" xml:space="preserve">O B
              <lb/>
            tangens ſectionem in B, & </s>
            <s xml:id="echoid-s3394" xml:space="preserve">recta B R perpendicularis ad ramum D B; </s>
            <s xml:id="echoid-s3395" xml:space="preserve">& </s>
            <s xml:id="echoid-s3396" xml:space="preserve">ſi
              <lb/>
            quidem ramus D C non concedatur maior, quàm D B, ſit primo ei æqualis, ſi
              <lb/>
            fieri poteſt, & </s>
            <s xml:id="echoid-s3397" xml:space="preserve">centro D interuallo D C deſcribatur circulus C P R, qui tran-
              <lb/>
            ſibit per punctum B, ob æqualitatem radiorum D C, D B; </s>
            <s xml:id="echoid-s3398" xml:space="preserve">& </s>
            <s xml:id="echoid-s3399" xml:space="preserve">quia (ex Lem-
              <lb/>
            mate nono) angulus D C P verticem reſpiciens, eſt acutus, recta C P cadet
              <lb/>
            intra circulum C P R; </s>
            <s xml:id="echoid-s3400" xml:space="preserve">ſed cadit extra coniſectionem, cum ſit contingens; </s>
            <s xml:id="echoid-s3401" xml:space="preserve">igi-
              <lb/>
            tur portio circularis peripheriæ C P ducitur extra coniſectionem C Q B: </s>
            <s xml:id="echoid-s3402" xml:space="preserve">rur-
              <lb/>
            ſus, quia angulus D B O eſt obtuſus (ex nono Lemmate, cum verticem A non reſpi-
              <lb/>
            ciat) ergo R B perpendicularis ad D B cadit intra coniſectionẽ, cum B O poſita ſit eã
              <lb/>
            contingens: </s>
            <s xml:id="echoid-s3403" xml:space="preserve">cadit verò eadem B R extra circulum B R Q, cum ſit perpendicu-
              <lb/>
            laris ad circuli radium D B; </s>
            <s xml:id="echoid-s3404" xml:space="preserve">igitur circuli portio B R intra coniſectionem ca-
              <lb/>
            det: </s>
            <s xml:id="echoid-s3405" xml:space="preserve">ſed priùs eiuſdem circuli portio C P extra eandem ſectionem ducebatur;
              <lb/>
            </s>
            <s xml:id="echoid-s3406" xml:space="preserve">igitur idem circulus ſecat coniſectionem alicubi, vt in Q, ducaturque denuo
              <lb/>
            ramus D Q, & </s>
            <s xml:id="echoid-s3407" xml:space="preserve">Q O contingens ſectionem in Q; </s>
            <s xml:id="echoid-s3408" xml:space="preserve">Vnde (ex nono Lemmate)
              <lb/>
              <note position="right" xlink:label="note-0119-04" xlink:href="note-0119-04a" xml:space="preserve">33. 34.
                <lb/>
              lib. 1.</note>
            angulus D Q O erit acutus; </s>
            <s xml:id="echoid-s3409" xml:space="preserve">& </s>
            <s xml:id="echoid-s3410" xml:space="preserve">propterea recta Q O intra circuli portionem;
              <lb/>
            </s>
            <s xml:id="echoid-s3411" xml:space="preserve">Q R conſtituta intra coniſectionem cadet, quod eſt abſurdum; </s>
            <s xml:id="echoid-s3412" xml:space="preserve">recta enim Q
              <lb/>
            O extra coniſectionem Q A cadit, quàm contingit in Q; </s>
            <s xml:id="echoid-s3413" xml:space="preserve">non ergo ramus D
              <lb/>
            C æqualis eſt ipſi D B. </s>
            <s xml:id="echoid-s3414" xml:space="preserve">Sit ſecundò D C minor, quàm D B (ſi fieri poteſt) ſe-
              <lb/>
            ceturque D T minor quàm D B, ſed maior quàm D C; </s>
            <s xml:id="echoid-s3415" xml:space="preserve">& </s>
            <s xml:id="echoid-s3416" xml:space="preserve">centro D interuallo
              <lb/>
            D T deſcribatur circulus T Q S; </s>
            <s xml:id="echoid-s3417" xml:space="preserve">is quidem ad partes B cadet intra, ad par-
              <lb/>
            tes vero C extra coniſectionem; </s>
            <s xml:id="echoid-s3418" xml:space="preserve">& </s>
            <s xml:id="echoid-s3419" xml:space="preserve">propterea eam alicubi ſecabit, vt in Q; </s>
            <s xml:id="echoid-s3420" xml:space="preserve">
              <lb/>
            & </s>
            <s xml:id="echoid-s3421" xml:space="preserve">ducto ramo D Q, & </s>
            <s xml:id="echoid-s3422" xml:space="preserve">Q O contingente ſectionem in Q, erit angulus D Q
              <lb/>
              <note position="right" xlink:label="note-0119-05" xlink:href="note-0119-05a" xml:space="preserve">Lem. 9.</note>
            O acutus, & </s>
            <s xml:id="echoid-s3423" xml:space="preserve">ideo recta Q O cadet intra circulum T Q, & </s>
            <s xml:id="echoid-s3424" xml:space="preserve">propterea intra
              <lb/>
            coniſectionem, quod eſt abſurdum; </s>
            <s xml:id="echoid-s3425" xml:space="preserve">Q O enim cadit extra ſectionem Q A,
              <lb/>
            quàm contingit in Q; </s>
            <s xml:id="echoid-s3426" xml:space="preserve">non ergo ramus D C minor eſt, quàm D B, ſed neque
              <lb/>
            æqualis priùs oſtenſus fuit; </s>
            <s xml:id="echoid-s3427" xml:space="preserve">igitur quilibet ramus D B vertici A propinquior
              <lb/>
            minor eſt quolibet ramo remotiore D C, quod erat oſtendendum.</s>
            <s xml:id="echoid-s3428" xml:space="preserve"/>
          </p>
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