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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          <p style="it">
            <s xml:id="echoid-s12153" xml:space="preserve">Tertio ſi duplum O H ad H G minorem proportionem habuerit, quàm G H
              <lb/>
            ad H E, eodem progreſſu oſtendetur, quod duplum rectanguli ex differentia
              <lb/>
            ipſarum E H, & </s>
            <s xml:id="echoid-s12154" xml:space="preserve">G O in H O minus eſt quadratis ex G O, & </s>
            <s xml:id="echoid-s12155" xml:space="preserve">ex H O, quod erat
              <lb/>
            propoſitum.</s>
            <s xml:id="echoid-s12156" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1043" type="section" level="1" n="331">
          <head xml:id="echoid-head410" xml:space="preserve">LEMMA XIV.</head>
          <p style="it">
            <s xml:id="echoid-s12157" xml:space="preserve">Ilſdem poſitis ſit G E minimum ſegmentorum, dico quod duo qua-
              <lb/>
            drata ex E H, & </s>
            <s xml:id="echoid-s12158" xml:space="preserve">ex G E, ſcilicet ex maximo, & </s>
            <s xml:id="echoid-s12159" xml:space="preserve">minimo ſeg-
              <lb/>
            mentorum æqualia ſunt duobus quadratis ex O H, & </s>
            <s xml:id="echoid-s12160" xml:space="preserve">ex G O inter-
              <lb/>
            medijs ſegmentis vna cum duplo rectanguli ſub differentijs minimæ G
              <lb/>
            E à duabus intermedijs G O, & </s>
            <s xml:id="echoid-s12161" xml:space="preserve">H O.</s>
            <s xml:id="echoid-s12162" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s12163" xml:space="preserve">Fiat H a æqualis G E,
              <lb/>
              <figure xlink:label="fig-0388-01" xlink:href="fig-0388-01a" number="459">
                <image file="0388-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0388-01"/>
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            ergo O a erit differentia ipſa-
              <lb/>
            rum E H, & </s>
            <s xml:id="echoid-s12164" xml:space="preserve">G E, ſicuti O
              <lb/>
            E eſt differentia ipſarum G O,
              <lb/>
            & </s>
            <s xml:id="echoid-s12165" xml:space="preserve">G E. </s>
            <s xml:id="echoid-s12166" xml:space="preserve">Et quia duo quadra-
              <lb/>
            ta ex maximo, & </s>
            <s xml:id="echoid-s12167" xml:space="preserve">ex mini-
              <lb/>
            mo ſegmentorum, ſcilicet ex
              <lb/>
            H E, & </s>
            <s xml:id="echoid-s12168" xml:space="preserve">ex E G æqualia ſunt
              <lb/>
            duplo quadrati ex G D ſe-
              <lb/>
            miße totius, cũ duplo quadrati
              <lb/>
            ex E D intermedia ſectione;
              <lb/>
            </s>
            <s xml:id="echoid-s12169" xml:space="preserve">eſtque duplum quadrati ex E D ſemiſſe ipſius E a æquale duplo rectanguli E O
              <lb/>
            a ex inæqualibus ſegmentis vna cum duplo quadrati ex intermedia ſectione O
              <lb/>
            D, ergo duo quadrata ex G E, & </s>
            <s xml:id="echoid-s12170" xml:space="preserve">ex E H æqualia ſunt his omnibus ſpatijs,
              <lb/>
            ſcilicet duplo quadrati ex G D, & </s>
            <s xml:id="echoid-s12171" xml:space="preserve">duplo quadrati ex D O cum duplo rectan-
              <lb/>
            guli E O a, ſed duo quadrata ex inæqualibus ſegmentis G O, & </s>
            <s xml:id="echoid-s12172" xml:space="preserve">ex O H æqua-
              <lb/>
            lia ſunt duplo quadrati ex ſemiſſe totius G D cum duplo quadrati ex interme-
              <lb/>
            dia ſectione O D, igitur exceßus ſummæ quadratorum ex G E, & </s>
            <s xml:id="echoid-s12173" xml:space="preserve">ex E H,
              <lb/>
            ſupra ſummam quadratorum ex G O, & </s>
            <s xml:id="echoid-s12174" xml:space="preserve">O H æqualis eſt duplo rectanguli ex E
              <lb/>
            O a, quod erat oſtendendum.</s>
            <s xml:id="echoid-s12175" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div1045" type="section" level="1" n="332">
          <head xml:id="echoid-head411" xml:space="preserve">LEMMA XV.</head>
          <p style="it">
            <s xml:id="echoid-s12176" xml:space="preserve">IN ellypſi, cuius axis A C, erectus A F, diameter I L, eiuſq; </s>
            <s xml:id="echoid-s12177" xml:space="preserve">erectus
              <lb/>
            I K, & </s>
            <s xml:id="echoid-s12178" xml:space="preserve">latus C E, & </s>
            <s xml:id="echoid-s12179" xml:space="preserve">ſimiliter altera diameter Q P, cuius ere-
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            ctus P R, & </s>
            <s xml:id="echoid-s12180" xml:space="preserve">latus C O: </s>
            <s xml:id="echoid-s12181" xml:space="preserve">dico quod duplum rectanguli ex differentia
              <lb/>
            ipſarum E H, G O, in H O à duobus quadratis ex G O, & </s>
            <s xml:id="echoid-s12182" xml:space="preserve">ex </s>
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