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-div302" type="section" level="1" n="100">
          <head xml:id="echoid-head140" xml:space="preserve">PROPOSITIO LXVII.</head>
          <p>
            <s xml:id="echoid-s3180" xml:space="preserve">P Oſtea repetamus figuras, paraboles, & </s>
            <s xml:id="echoid-s3181" xml:space="preserve">hyperboles, & </s>
            <s xml:id="echoid-s3182" xml:space="preserve">
              <lb/>
              <note position="right" xlink:label="note-0114-01" xlink:href="note-0114-01a" xml:space="preserve">a</note>
            quoquot ſunt illius ſigna, & </s>
            <s xml:id="echoid-s3183" xml:space="preserve">ſupponamus quod ipſius D B
              <lb/>
            portio B K, ſit tantummodo linea breuiſſima; </s>
            <s xml:id="echoid-s3184" xml:space="preserve">Dico, quod D A
              <lb/>
            quoque minima eſt linearum egredientium ex D ad ſectionem
              <lb/>
              <note position="right" xlink:label="note-0114-02" xlink:href="note-0114-02a" xml:space="preserve">b</note>
            A C, & </s>
            <s xml:id="echoid-s3185" xml:space="preserve">illi propinquiores ſunt minores remotioribus.</s>
            <s xml:id="echoid-s3186" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3187" xml:space="preserve">Quia educitur ex D vnus tantum
              <lb/>
              <note position="right" xlink:label="note-0114-03" xlink:href="note-0114-03a" xml:space="preserve">c</note>
              <figure xlink:label="fig-0114-01" xlink:href="fig-0114-01a" number="96">
                <image file="0114-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0114-01"/>
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            breuiſecans erit menſura E A maior
              <lb/>
            dimidio erecti, & </s>
            <s xml:id="echoid-s3188" xml:space="preserve">D E æqualis F
              <lb/>
            Trutinæ (51. </s>
            <s xml:id="echoid-s3189" xml:space="preserve">52. </s>
            <s xml:id="echoid-s3190" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s3191" xml:space="preserve">vnde ſequi-
              <lb/>
            tur, quod lineæ breuiſſimæ eductæ
              <lb/>
            ab extremitatibus reliquorum ramo-
              <lb/>
            rum abſcindunt cum A ab axi line-
              <lb/>
            as maiores, quàm ſecant illi rami.
              <lb/>
            </s>
            <s xml:id="echoid-s3192" xml:space="preserve">Ducamus prius ad ſectionem B A
              <lb/>
            ramum D G, inde conſtat D G ma-
              <lb/>
              <note position="right" xlink:label="note-0114-04" xlink:href="note-0114-04a" xml:space="preserve">d</note>
            iorem eſſe, quàm D A (64. </s>
            <s xml:id="echoid-s3193" xml:space="preserve">65. </s>
            <s xml:id="echoid-s3194" xml:space="preserve">ex
              <lb/>
            5.) </s>
            <s xml:id="echoid-s3195" xml:space="preserve">Dico iam, quod D B maior eſt
              <lb/>
            illa, alioquin eſſet æqualis, vel mi-
              <lb/>
            nor illa, & </s>
            <s xml:id="echoid-s3196" xml:space="preserve">producamus D H ad ſectionem B G; </s>
            <s xml:id="echoid-s3197" xml:space="preserve">ergo D H maior eſt,
              <lb/>
            quàm D G, quia remotior eſt ab D A (64. </s>
            <s xml:id="echoid-s3198" xml:space="preserve">65. </s>
            <s xml:id="echoid-s3199" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s3200" xml:space="preserve">quare maior eſt,
              <lb/>
            quàm D B, & </s>
            <s xml:id="echoid-s3201" xml:space="preserve">ex illo ſecetur D I maior, quàm D B, & </s>
            <s xml:id="echoid-s3202" xml:space="preserve">minor, quàm,
              <lb/>
            D H, & </s>
            <s xml:id="echoid-s3203" xml:space="preserve">centro D interuallo D I deſcriptus circulus ſecabit ſectionem,
              <lb/>
            B G, ſecet eam in M, & </s>
            <s xml:id="echoid-s3204" xml:space="preserve">iungamus D M; </s>
            <s xml:id="echoid-s3205" xml:space="preserve">ergo D M, nempe D I, quæ
              <lb/>
              <note position="right" xlink:label="note-0114-05" xlink:href="note-0114-05a" xml:space="preserve">e</note>
            conceſſa fuit maior, quàm D B eſt etiam maior, quàm D H, propterea
              <lb/>
            quod eſt remotior ab D A, quàm D H (64. </s>
            <s xml:id="echoid-s3206" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s3207" xml:space="preserve">igitur D I maior eſt
              <lb/>
            quàm D H, quod eſt abſurdum; </s>
            <s xml:id="echoid-s3208" xml:space="preserve">quare D B maior eſt, quàm D H.</s>
            <s xml:id="echoid-s3209" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3210" xml:space="preserve">Patet etiam, quod D B minor ſit, quàm D C, alioquin eſſet vel illi
              <lb/>
              <note position="right" xlink:label="note-0114-06" xlink:href="note-0114-06a" xml:space="preserve">f</note>
            æqualis, aut maior, & </s>
            <s xml:id="echoid-s3211" xml:space="preserve">ducamus D N ad ſectionem C B; </s>
            <s xml:id="echoid-s3212" xml:space="preserve">ergo D N mi-
              <lb/>
            nor eſt, quàm D C, eò quod proximior eſt D A (64. </s>
            <s xml:id="echoid-s3213" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s3214" xml:space="preserve">quare mi-
              <lb/>
            nor eſt, quàm D B, & </s>
            <s xml:id="echoid-s3215" xml:space="preserve">fecetur D O ex D B maior, quàm D N, & </s>
            <s xml:id="echoid-s3216" xml:space="preserve">mi-
              <lb/>
            nor quàm D B, & </s>
            <s xml:id="echoid-s3217" xml:space="preserve">centro D, interuallo D O circulus deſcriptus ſecabit
              <lb/>
              <note position="right" xlink:label="note-0114-07" xlink:href="note-0114-07a" xml:space="preserve">g</note>
            ſectionem exempli gratia, in Q, & </s>
            <s xml:id="echoid-s3218" xml:space="preserve">iungamus D Q, igitur D Q minor eſt
              <lb/>
            quàm D N, ſed eſt æqualis D O, quæ ſuppoſita fuit maior, quàm D N,
              <lb/>
            ergo D Q maior eſt, quàm D N; </s>
            <s xml:id="echoid-s3219" xml:space="preserve">verum eſt minor illo, quod eſt abſur-
              <lb/>
            dum; </s>
            <s xml:id="echoid-s3220" xml:space="preserve">igitur D C non eſt minor D B, neque æqualis; </s>
            <s xml:id="echoid-s3221" xml:space="preserve">quare maior illa.
              <lb/>
            </s>
            <s xml:id="echoid-s3222" xml:space="preserve">eſt. </s>
            <s xml:id="echoid-s3223" xml:space="preserve">Atque ſic patet, quod D B minor ſit omnibus lineis egredientibus
              <lb/>
            ex D ad ſectionem B C, & </s>
            <s xml:id="echoid-s3224" xml:space="preserve">illi proximiores ex illa parte, minores ſunt
              <lb/>
            remotioribus. </s>
            <s xml:id="echoid-s3225" xml:space="preserve">Quapropter manifeſtum eſt, quod D A ſit minimus omni-
              <lb/>
            um ramorum egredientium ex D ad ſectionem A B C, & </s>
            <s xml:id="echoid-s3226" xml:space="preserve">reliqui proxi-
              <lb/>
            miores illi, minores ſunt remotioribus, quod erat oſtendendum.</s>
            <s xml:id="echoid-s3227" xml:space="preserve"/>
          </p>
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