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-div594" type="section" level="1" n="197">
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        <div xml:id="echoid-div596" type="section" level="1" n="198">
          <head xml:id="echoid-head252" xml:space="preserve">LEMMA IV.</head>
          <p style="it">
            <s xml:id="echoid-s6213" xml:space="preserve">SI G B ad B D maiorem proportionem habuerit, quàm K F ad F
              <lb/>
            I: </s>
            <s xml:id="echoid-s6214" xml:space="preserve">Dico in ſingulis ſectionibus reperiri non poſſe binas axium ab-
              <lb/>
            ſciſſas inter ſe proportionales, quæ ad conterminas potentiales ſint in eiſ-
              <lb/>
            dem rationibus.</s>
            <s xml:id="echoid-s6215" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6216" xml:space="preserve">Si enim fieri poteſt, ſit A C ad
              <lb/>
              <figure xlink:label="fig-0198-01" xlink:href="fig-0198-01a" number="216">
                <image file="0198-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0198-01"/>
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            C B, vt E H ad H F, & </s>
            <s xml:id="echoid-s6217" xml:space="preserve">Q R ad
              <lb/>
            R B ſit, vt T V ad V F, atque C
              <lb/>
            B ad B R ſit vt H F ad F V; </s>
            <s xml:id="echoid-s6218" xml:space="preserve">con-
              <lb/>
            iungantur rectæ G D, K I quæ ſecẽt
              <lb/>
            ordinatas in S, P, X, L; </s>
            <s xml:id="echoid-s6219" xml:space="preserve">& </s>
            <s xml:id="echoid-s6220" xml:space="preserve">ſecen-
              <lb/>
            tur C a æqualis R S, & </s>
            <s xml:id="echoid-s6221" xml:space="preserve">H b æqualis
              <lb/>
            V X, ſuntq; </s>
            <s xml:id="echoid-s6222" xml:space="preserve">æquidiſtantes; </s>
            <s xml:id="echoid-s6223" xml:space="preserve">ergo co-
              <lb/>
            niungentes S a, R C æquales ſunt,
              <lb/>
            & </s>
            <s xml:id="echoid-s6224" xml:space="preserve">parallelæ, & </s>
            <s xml:id="echoid-s6225" xml:space="preserve">ſic etiam coniun-
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            gentes X b, & </s>
            <s xml:id="echoid-s6226" xml:space="preserve">V H, quare quadratum A C, ſeu rectangulum P C B ad qua-
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            dratum C B eandem proportionem habet, quàm quadratum E H, ſeu rectangu-
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              <note position="left" xlink:label="note-0198-01" xlink:href="note-0198-01a" xml:space="preserve">12. 13.
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              lib. 1.</note>
            lum L H F ad quadratum H F; </s>
            <s xml:id="echoid-s6227" xml:space="preserve">ideoque P C ad C B eandem proportionem ha-
              <lb/>
            bet, quàm L H ad H F; </s>
            <s xml:id="echoid-s6228" xml:space="preserve">eſt verò C B ad B R, vt H F ad F V, & </s>
            <s xml:id="echoid-s6229" xml:space="preserve">per conuerſio-
              <lb/>
            nem rationis C B ad C R eſt vt H F ad H V, ergo ex æquali C P ad C R eſt
              <lb/>
            vt L H ad H V: </s>
            <s xml:id="echoid-s6230" xml:space="preserve">Eodem modo oſtendetur, quod S R, ſeu a C ad R C eſt, vt
              <lb/>
            X V, ſeu b H ad V H; </s>
            <s xml:id="echoid-s6231" xml:space="preserve">erat autem P C ad C R vt L H ad H V; </s>
            <s xml:id="echoid-s6232" xml:space="preserve">ergo a P dif-
              <lb/>
            ferentia ipſarum S R, P C ad G R, ſeu ad S a eſt vt b L differentia ipſarum
              <lb/>
            X V, L H ad H V, ſeu ad X b; </s>
            <s xml:id="echoid-s6233" xml:space="preserve">eſtque D B ad B G vt P a ad S a (propter pa-
              <lb/>
            rallelas a S, C G, & </s>
            <s xml:id="echoid-s6234" xml:space="preserve">parallelas a P, & </s>
            <s xml:id="echoid-s6235" xml:space="preserve">B D) pariterque I F ad F K eſt vt L
              <lb/>
            b ad b X, ergo D B ad B G eandem proportionem habet, quàm I F ad F K;
              <lb/>
            </s>
            <s xml:id="echoid-s6236" xml:space="preserve">quod eſt contra hypotheſim, non ergo binæ axium abſciſſæ inter ſe proportionales
              <lb/>
            reperiri poſſunt in ſectionibus A B, & </s>
            <s xml:id="echoid-s6237" xml:space="preserve">E F, quæ ad conterminas potentiales ſint
              <lb/>
            in eiſdem rationibus; </s>
            <s xml:id="echoid-s6238" xml:space="preserve">quod erat oſtendendum.</s>
            <s xml:id="echoid-s6239" xml:space="preserve"/>
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        <div xml:id="echoid-div598" type="section" level="1" n="199">
          <head xml:id="echoid-head253" xml:space="preserve">COROLLARIVM.</head>
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            <s xml:id="echoid-s6240" xml:space="preserve">HInc conſtat in duabus ſectionibus eiuſdem nominis ſi axium figuræ G B D,
              <lb/>
            & </s>
            <s xml:id="echoid-s6241" xml:space="preserve">K F I non ſuerint ſimiles, neque ſectiones A B, & </s>
            <s xml:id="echoid-s6242" xml:space="preserve">E F, ſimiles eſſe.
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            </s>
            <s xml:id="echoid-s6243" xml:space="preserve">Nam eſt impoſſibile, vt omnes, ideſt infinitæ axium abſciſſæ inter ſe proportio-
              <lb/>
            nales ad conterminas potentiales ſint in eiſdem rationibus, cum neque bine in
              <lb/>
            ſingulis reperiri poſſint ex hac propoſitione.</s>
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