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-s8200" xml:space="preserve">
              <pb o="218" file="0256" n="256" rhead="Apollonij Pergæi"/>
            les ſunt, & </s>
            <s xml:id="echoid-s8201" xml:space="preserve">axium abſciſſæ D G, H K æquales cum ſint latera oppoſita paralle-
              <lb/>
              <note position="left" xlink:label="note-0256-01" xlink:href="note-0256-01a" xml:space="preserve">ex prop@@.
                <lb/>
              huius.</note>
            logrammi D K; </s>
            <s xml:id="echoid-s8202" xml:space="preserve">ergo ordinatim ad axes applicatæ E G, & </s>
            <s xml:id="echoid-s8203" xml:space="preserve">I K æquales ſunt, & </s>
            <s xml:id="echoid-s8204" xml:space="preserve">
              <lb/>
            ablata communi I G, erit E I æqualis G K, ſeu D H; </s>
            <s xml:id="echoid-s8205" xml:space="preserve">erat autem intercepta
              <lb/>
            B I æqualis eidem D H; </s>
            <s xml:id="echoid-s8206" xml:space="preserve">igitur B I erit æqualis E I; </s>
            <s xml:id="echoid-s8207" xml:space="preserve">& </s>
            <s xml:id="echoid-s8208" xml:space="preserve">propterea punctum E
              <lb/>
            parabolæ E D F cadet ſuper punctum B parabolæ B A C; </s>
            <s xml:id="echoid-s8209" xml:space="preserve">ergo duæ parabolæ B
              <lb/>
              <note position="left" xlink:label="note-0256-02" xlink:href="note-0256-02a" xml:space="preserve">Maurol.
                <lb/>
              27. lib
                <lb/>
              Conic.</note>
            A C, & </s>
            <s xml:id="echoid-s8210" xml:space="preserve">E D F conueniunt in vno puncto, & </s>
            <s xml:id="echoid-s8211" xml:space="preserve">in eo ſe mutuo tangere non poſ-
              <lb/>
            ſunt; </s>
            <s xml:id="echoid-s8212" xml:space="preserve">igitur ſe mutuo ſecant. </s>
            <s xml:id="echoid-s8213" xml:space="preserve">Quare patet propoſitum.</s>
            <s xml:id="echoid-s8214" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s8215" xml:space="preserve">His demonſiratis manifeſtè percipitur, quod ex ſucceſſiua diminutione rectarũ
              <lb/>
            æquidiſtantium, inter coniſectiones interceptarum, deduci non poteſt, coniſe-
              <lb/>
            ctiones magis ad ſe ipſas propius accedere; </s>
            <s xml:id="echoid-s8216" xml:space="preserve">propterea quod in ij ſdem ſectionibus
              <lb/>
            aſymptoticis duci poßunt interceptæ rectæ lineæ inter ſe æquidiſtantes, quæ ſint
              <lb/>
            omnes æquales inter ſe, nimirum illæ, quæ parallelæ ſunt alicui communi dia-
              <lb/>
            metro, vel rectæ lineæ vertices earum coniungenti, vt in propoſitione 5. </s>
            <s xml:id="echoid-s8217" xml:space="preserve">additarũ
              <lb/>
            oſtenſum eſt. </s>
            <s xml:id="echoid-s8218" xml:space="preserve">Similiter aliæ interceptæ rectæ lineæ, inter ſe æquidiſtantes ſucceſſiuè
              <lb/>
            augentur aliæ verò ſucceſſiuè diminuuntur verſus caſdem partes, vt in propoſitione
              <lb/>
            3. </s>
            <s xml:id="echoid-s8219" xml:space="preserve">& </s>
            <s xml:id="echoid-s8220" xml:space="preserve">4. </s>
            <s xml:id="echoid-s8221" xml:space="preserve">addit. </s>
            <s xml:id="echoid-s8222" xml:space="preserve">oſtenſum eſt. </s>
            <s xml:id="echoid-s8223" xml:space="preserve">Et hoc nedũ verificatur in ſectionibus non congruen-
              <lb/>
            tibus, & </s>
            <s xml:id="echoid-s8224" xml:space="preserve">asymptoticis, ſed etiã in duabus æqualibus, & </s>
            <s xml:id="echoid-s8225" xml:space="preserve">inter ſe ſimilibus ſectioni-
              <lb/>
            bus ſe mutuo ſecantibus, dummodo earum axes paralleli ſint, in ijs enim inter-
              <lb/>
            ceptæ rectæ lineæ inter ſe æquidiſtantes, tendentes ad eaſdem partes, etiam illæ,
              <lb/>
            quæ proprius ad punctum occurſus ſcctionum conicarum accedunt, poßunt dimi-
              <lb/>
            nui, pariterque inter ſe æquales eße, & </s>
            <s xml:id="echoid-s8226" xml:space="preserve">quod mirum eſt poßunt ſemper magis
              <lb/>
            augeri. </s>
            <s xml:id="echoid-s8227" xml:space="preserve">Si igitur æquidiſtantes interceptæ ſunt menſuræ diſtantiarũ duarum ſe-
              <lb/>
            ctionum, eædem coniſectiones cenſeri debent modo parallelæ, & </s>
            <s xml:id="echoid-s8228" xml:space="preserve">æqualibus inter-
              <lb/>
            uallis inter ſe diſtantes, modo ad eaſdem partes ſtringi, & </s>
            <s xml:id="echoid-s8229" xml:space="preserve">coanguſtari, & </s>
            <s xml:id="echoid-s8230" xml:space="preserve">ſi-
              <lb/>
            mul dilatari magis, ac magis, quod omnino videtur abſurdum. </s>
            <s xml:id="echoid-s8231" xml:space="preserve">Non igitur ex
              <lb/>
            eo qnod omnes interceptæ rectæ lineæ inter ſe æquidiſtantes ſunt æquales inter ſe;
              <lb/>
            </s>
            <s xml:id="echoid-s8232" xml:space="preserve">propterea ſectiones ipſæ crunt parallelæ, & </s>
            <s xml:id="echoid-s8233" xml:space="preserve">asymptoticæ, & </s>
            <s xml:id="echoid-s8234" xml:space="preserve">ſemper æquali in-
              <lb/>
            teruallo ad inuicem ſeparatæ; </s>
            <s xml:id="echoid-s8235" xml:space="preserve">neque ex eo quod prædictæ parallelæ magis augẽ-
              <lb/>
            tur, vel diminuuntur interualla augeri, vel ſtringi cenſendum eſt.</s>
            <s xml:id="echoid-s8236" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s8237" xml:space="preserve">Et præcipuè præſtantiſſimus Gregorius à Sancto Vincentio neſcio an iure de-
              <lb/>
            monſtrationem propoſitionis 14. </s>
            <s xml:id="echoid-s8238" xml:space="preserve">libri 2. </s>
            <s xml:id="echoid-s8239" xml:space="preserve">ipſiuſmet Apollonij inſufficientem repu-
              <lb/>
            tauerit, propterea quod Apollonius deduxit rectas lineas hyperbolen compræbendẽ-
              <lb/>
            tes, quæ aſymptoti vocantur ſemper magis, ac magis ſectioni viciniores fieri ex eo
              <lb/>
            quod rectæ lineæ inter ſe æquidiſtãtes, interceptæ inter rectas asymptotos vocatas,
              <lb/>
            & </s>
            <s xml:id="echoid-s8240" xml:space="preserve">hyperbolen contentam ſucceſſiuè ſemper magis, ac magis diminuantur; </s>
            <s xml:id="echoid-s8241" xml:space="preserve">& </s>
            <s xml:id="echoid-s8242" xml:space="preserve">è
              <lb/>
            contra aßeruit cum Cardano, & </s>
            <s xml:id="echoid-s8243" xml:space="preserve">quodam Rabino Moſe diſtantiam hyperbolæ à re-
              <lb/>
            ctis asymptotis ſumi debere, non à quibu ſcunque rectis lineis interceptis inter
              <lb/>
            ſe parallelis, ſed tantummodo à rectis lineis perpendicularibus ad aſymptotos,
              <lb/>
            quæ ſolummodo, inquiunt ipſi, diſtantias determinant; </s>
            <s xml:id="echoid-s8244" xml:space="preserve">at reuera hæc animad-
              <lb/>
            nerſio non videtur neceßaria: </s>
            <s xml:id="echoid-s8245" xml:space="preserve">perinde enim eſt conſiderare rectas lineas ab hy-
              <lb/>
            perbole ad vnam rectam lineam continentium ductas, quæ efficiat cum illa an-
              <lb/>
            gulos æquales, ac ſi perpendiculares eßent ad eandem: </s>
            <s xml:id="echoid-s8246" xml:space="preserve">at quando rectæ lineæ in-
              <lb/>
            terceptæ ſunt inter ſe æquidiſtantes, tunc omnes efficiunt ſuper rectam lineam
              <lb/>
            continentem hyperbolen angulos æquales ad eaſdem partes; </s>
            <s xml:id="echoid-s8247" xml:space="preserve">& </s>
            <s xml:id="echoid-s8248" xml:space="preserve">propterca (ex inæ-
              <lb/>
            qualitate prædictarum æquidiſt antium) optimè concluditur cum Apollonio inæ-
              <lb/>
            qualitas perpendicularium, ſeu diſtantiarum. </s>
            <s xml:id="echoid-s8249" xml:space="preserve">Quando verò conſiderantur duæ
              <lb/>
            lineæ curuæ veluti ſunt duæ parabolæ, vel duæ hyperbolæ, vel ellipſes, tunc </s>
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