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-div598" type="section" level="1" n="199">
          <pb o="161" file="0199" n="199" rhead="Conicor. Lib. VI."/>
        </div>
        <div xml:id="echoid-div599" type="section" level="1" n="200">
          <head xml:id="echoid-head254" xml:space="preserve">LEMMAV.</head>
          <p style="it">
            <s xml:id="echoid-s6245" xml:space="preserve">IN eiſdem figuris rurſus G B ad B D maiorem proportionem habeat,
              <lb/>
            qnàm K F ad F 1 : </s>
            <s xml:id="echoid-s6246" xml:space="preserve">Dico quod minimè reperiri poſſunt axium ab-
              <lb/>
            ſcißæ erectis proportionales, quæ habeant eandem rationem ad contermi-
              <lb/>
            nas potentiales.</s>
            <s xml:id="echoid-s6247" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s6248" xml:space="preserve">Secentur quælibet abſciſſæ, B C, F H ita vt C B ad B D ſit vt H F ad F I,
              <lb/>
            & </s>
            <s xml:id="echoid-s6249" xml:space="preserve">ducantur ordinatim ad axes applicatæ A C, E H, quæ productæ ſecent, con-
              <lb/>
            iunctas G D, K I in P, L, atque fiat γ B ad B D vt K F ad F I, iungatur-
              <lb/>
            que γ D ſecans A P in M. </s>
            <s xml:id="echoid-s6250" xml:space="preserve">Manifeſtum eſt rectam C M inæqualem eſſe C P,
              <lb/>
            (propterea quod γ B minor eſt, quàm G B, cum ad eandem B D minorem pro-
              <lb/>
            portionem habeat, quàm G B, ideoque punctum Y
              <unsure/>
            , & </s>
            <s xml:id="echoid-s6251" xml:space="preserve">recta γ D cadent intra,
              <lb/>
            triangulum G B D, & </s>
            <s xml:id="echoid-s6252" xml:space="preserve">punctum M intra ipſum cadet, aut extra G D pro-
              <lb/>
            ductam). </s>
            <s xml:id="echoid-s6253" xml:space="preserve">Quoniam D B ad B γ eſt vt I F ad F K, & </s>
            <s xml:id="echoid-s6254" xml:space="preserve">erat C B ad B D vt
              <lb/>
            H F ad F I ; </s>
            <s xml:id="echoid-s6255" xml:space="preserve">ergo ex æquali C B ad B γ erit vt H F ad F K, & </s>
            <s xml:id="echoid-s6256" xml:space="preserve">comparando
              <lb/>
            terminorum ſummas in hyperbola, & </s>
            <s xml:id="echoid-s6257" xml:space="preserve">differentias in ellipſi ad antecedentes, γ C
              <lb/>
            ad C B erit vt K H ad H F; </s>
            <s xml:id="echoid-s6258" xml:space="preserve">eſt verò M C ad C R
              <unsure/>
            vt L H ad H K (eoquod
              <lb/>
            triãgula M C R
              <unsure/>
            , & </s>
            <s xml:id="echoid-s6259" xml:space="preserve">L H K ſimilia ſunt triangulis ſimilibus B D Y
              <unsure/>
            , I F K,) ergo
              <lb/>
            ex æquali M C ad C B erit vt L H ad H F, & </s>
            <s xml:id="echoid-s6260" xml:space="preserve">rectangulum M C B ad quadra-
              <lb/>
            tum C B eandem proportionem habebit, quàrn rectangulum L H F ad quadra-
              <lb/>
            tũ H F; </s>
            <s xml:id="echoid-s6261" xml:space="preserve">ſed rectangulũ M C B æquale nõ eſt rectangulo P C B (cum M C oſtenſa
              <lb/>
            ſit inæqualis P C); </s>
            <s xml:id="echoid-s6262" xml:space="preserve">ergo rectangulum P C B, ſeu quadratum A C ad quadratum
              <lb/>
              <note position="right" xlink:label="note-0199-01" xlink:href="note-0199-01a" xml:space="preserve">12. 13.
                <lb/>
              lib. 1.</note>
            C B non eandem proportionem habet, quàm rectangulum L H F, ſeu quadratum
              <lb/>
            E H ad quadratum H F; </s>
            <s xml:id="echoid-s6263" xml:space="preserve">& </s>
            <s xml:id="echoid-s6264" xml:space="preserve">propterea A C ad C B non eandem proportionem
              <lb/>
            habebit quàm E H ad H F. </s>
            <s xml:id="echoid-s6265" xml:space="preserve">Idem oſtendetur in reliquis omnibus abſciſſis ſimi-
              <lb/>
            liter poſitis. </s>
            <s xml:id="echoid-s6266" xml:space="preserve">Quare patet propoſitum.</s>
            <s xml:id="echoid-s6267" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div601" type="section" level="1" n="201">
          <head xml:id="echoid-head255" xml:space="preserve">COROLLARIVM I.</head>
          <p style="it">
            <s xml:id="echoid-s6268" xml:space="preserve">MAnifeſtum eſt in coniſectionibus non ſimilibus duci poſſe duas ſeries appli-
              <lb/>
            catarum ad axes, itaut abſciſſæ ſimiles, ſeu proportionales inter ſe adcõ-
              <lb/>
            terminas potentiales non ſint in ijſdem rationibus.</s>
            <s xml:id="echoid-s6269" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div602" type="section" level="1" n="202">
          <head xml:id="echoid-head256" xml:space="preserve">COROLLARIVM II.</head>
          <p style="it">
            <s xml:id="echoid-s6270" xml:space="preserve">Colligitur pariter conuertendo, quod in duabus ſectionibus eiuſdem nominis
              <lb/>
            ſi duæ ſeries abſciſſarum ſimilium in axibus poſitæ fuerint, & </s>
            <s xml:id="echoid-s6271" xml:space="preserve">in vna ſe-
              <lb/>
            rie abſciſſæ ad conterminas potentiales maiorem proportionem habeant, quàm in
              <lb/>
            altera ſerie, fieri poteſt vt ſiguræ axium non ſint inter ſe ſimiles: </s>
            <s xml:id="echoid-s6272" xml:space="preserve">Quod verifi-
              <lb/>
            catur ſaltem in caſu præcedentis propoſitionis.</s>
            <s xml:id="echoid-s6273" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s6274" xml:space="preserve">His præmiſſis, quoniam paſſo in definitione poſita eſſentialiter conuenit defini-
              <lb/>
            to eſt impoſſibile, vt eidem ſubiecto definito competant duæ paſſiones diuerſæ, & </s>
            <s xml:id="echoid-s6275" xml:space="preserve">
              <lb/>
            inter ſe oppoſitæ, exempli gratia, fieri non poteſt, vt in triangulis ſimilibus </s>
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