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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          <pb o="36" file="0074" n="74" rhead="Apollonij Pergæi"/>
          <p>
            <s xml:id="echoid-s1896" xml:space="preserve">ITaque per C producamus C I parallelam perpendiculari E D, & </s>
            <s xml:id="echoid-s1897" xml:space="preserve">pona-
              <lb/>
              <note position="right" xlink:label="note-0074-01" xlink:href="note-0074-01a" xml:space="preserve">b</note>
            mus quamlibet duarum proportionum C F ad F D, & </s>
            <s xml:id="echoid-s1898" xml:space="preserve">E K ad K D, vt
              <lb/>
            proportio figuræ, & </s>
            <s xml:id="echoid-s1899" xml:space="preserve">educamus ex E, K rectas E I, K S parallelas ipſi C
              <lb/>
            AD, & </s>
            <s xml:id="echoid-s1900" xml:space="preserve">interponamus inter F C, C A duas medias proportionales C N,
              <lb/>
              <note position="left" xlink:label="note-0074-02" xlink:href="note-0074-02a" xml:space="preserve">Lem. 7.</note>
              <note position="right" xlink:label="note-0074-03" xlink:href="note-0074-03a" xml:space="preserve">c</note>
            C O, & </s>
            <s xml:id="echoid-s1901" xml:space="preserve">erigamus per O perpendicularem B O, quæ occurrat ſectioni in
              <lb/>
            B; </s>
            <s xml:id="echoid-s1902" xml:space="preserve">& </s>
            <s xml:id="echoid-s1903" xml:space="preserve">ponamus proportionem alicuius lineæ, vt Q ad B O compoſitam
              <lb/>
              <note position="right" xlink:label="note-0074-04" xlink:href="note-0074-04a" xml:space="preserve">d</note>
            ex C D ad D F, & </s>
            <s xml:id="echoid-s1904" xml:space="preserve">F O ad O C, & </s>
            <s xml:id="echoid-s1905" xml:space="preserve">ſit E D maior, quàm Q Trutina: </s>
            <s xml:id="echoid-s1906" xml:space="preserve">Di-
              <lb/>
            co, quod nulla breuiſecans egreditur ex E ad ſectionem, & </s>
            <s xml:id="echoid-s1907" xml:space="preserve">linea breuiſ-
              <lb/>
            ſima, egrediens ab extremitate cuiuslibet rami aſſignati, abſcindit cum
              <lb/>
            A ab axi maiorem lineam, quàm ſecant illi rami. </s>
            <s xml:id="echoid-s1908" xml:space="preserve">Producatur priùs E B
              <lb/>
              <note position="right" xlink:label="note-0074-05" xlink:href="note-0074-05a" xml:space="preserve">e</note>
            ſecans axim in H, & </s>
            <s xml:id="echoid-s1909" xml:space="preserve">quia E D maior eſt, quàm Q, ergo proportio E D
              <lb/>
              <note position="right" xlink:label="note-0074-06" xlink:href="note-0074-06a" xml:space="preserve">f</note>
            ad B O (quæ componitur ex E D ad D K, nempe I C ad C S, & </s>
            <s xml:id="echoid-s1910" xml:space="preserve">ex D
              <lb/>
            K, nempe G O ad O B) maior eſt proportione, quàm habet Q ad B O,
              <lb/>
            quæ ex hypotheſi componebatur ex C D ad D F, & </s>
            <s xml:id="echoid-s1911" xml:space="preserve">ex F O ad O C; </s>
            <s xml:id="echoid-s1912" xml:space="preserve">ſed
              <lb/>
              <note position="right" xlink:label="note-0074-07" xlink:href="note-0074-07a" xml:space="preserve">g</note>
            E D ad D K eſt, vt C D ad D F (quia quælibet earum eſt, vt proportio
              <lb/>
            figuræ compoſitæ, vel diuiſæ) remanet proportio O G ad O B maior ea,
              <lb/>
            quàm habet F O ad O C; </s>
            <s xml:id="echoid-s1913" xml:space="preserve">igitur O G in O C, nempe rectangulum C G
              <lb/>
              <note position="left" xlink:label="note-0074-08" xlink:href="note-0074-08a" xml:space="preserve">Lem. 5.
                <lb/>
              præmiſſ.</note>
            maius eſt, quàm B O in O F: </s>
            <s xml:id="echoid-s1914" xml:space="preserve">& </s>
            <s xml:id="echoid-s1915" xml:space="preserve">ponamus rectangulum F G commune,
              <lb/>
              <note position="right" xlink:label="note-0074-09" xlink:href="note-0074-09a" xml:space="preserve">h</note>
            erit rectangulum F S maius, quàm B G in G M; </s>
            <s xml:id="echoid-s1916" xml:space="preserve">eſt verò rectangulum
              <lb/>
            F S æquale rectangulo E M (eo quod E K ad K D, nempe ad F M eſt, vt
              <lb/>
            S M ad M K, quia quælibet earum eſt, vt proportio figuræ; </s>
            <s xml:id="echoid-s1917" xml:space="preserve">itaque re-
              <lb/>
              <note position="right" xlink:label="note-0074-10" xlink:href="note-0074-10a" xml:space="preserve">i</note>
            ctangulum E M maius eſt, quàm M G in G B, & </s>
            <s xml:id="echoid-s1918" xml:space="preserve">propterea E K ad B G,
              <lb/>
              <note position="left" xlink:label="note-0074-11" xlink:href="note-0074-11a" xml:space="preserve">ibidem.</note>
            nempe K R ad R G maiorem rationem habet, quàm G M ad M K, ergo
              <lb/>
            componendo, patet, quod K M, nempe D F maior eſt, quàm G R, & </s>
            <s xml:id="echoid-s1919" xml:space="preserve">
              <lb/>
            ideo E I ad K M, nempe C D ad D F, ſeu I C ad C S minorem propor-
              <lb/>
            tionem habet, quàm E I ad G R, quæ eſt, vt I T ad B G, propter ſimi-
              <lb/>
            litudinem duorum triangulorum E I T, B G R, ergo I T ad B G maiorem
              <lb/>
              <note position="right" xlink:label="note-0074-12" xlink:href="note-0074-12a" xml:space="preserve">K</note>
            rationem habet, quàm I C ad C S, ſeu ad O G; </s>
            <s xml:id="echoid-s1920" xml:space="preserve">& </s>
            <s xml:id="echoid-s1921" xml:space="preserve">comparando homo-
              <lb/>
              <note position="left" xlink:label="note-0074-13" xlink:href="note-0074-13a" xml:space="preserve">Lem. 4.
                <lb/>
              præm</note>
            logorum differentias in hyperbola, & </s>
            <s xml:id="echoid-s1922" xml:space="preserve">eorum ſummas in ellipſi, habebit
              <lb/>
            C T ad B O, nempe C H ad H O maiorem rationem, quàm I C ad C S,
              <lb/>
            nempe C D ad D F, & </s>
            <s xml:id="echoid-s1923" xml:space="preserve">diuidendo in hyperbola, & </s>
            <s xml:id="echoid-s1924" xml:space="preserve">componendo in elli-
              <lb/>
            pſi C O ad O H, habebit maiorem proportionem quàm C F ad F D, quæ
              <lb/>
            eſt, vt proportio figuræ; </s>
            <s xml:id="echoid-s1925" xml:space="preserve">igitur breuiſſima egrediens ex B (9. </s>
            <s xml:id="echoid-s1926" xml:space="preserve">10. </s>
            <s xml:id="echoid-s1927" xml:space="preserve">ex quinto)
              <lb/>
            abſcindit cum A maiorem lineam, quàm A H.</s>
            <s xml:id="echoid-s1928" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1929" xml:space="preserve">Poſteà educamus ex E lineam occurrentem ſectioni in V, & </s>
            <s xml:id="echoid-s1930" xml:space="preserve">produca-
              <lb/>
            mus eam, quouſque occurrat C I ad X, & </s>
            <s xml:id="echoid-s1931" xml:space="preserve">ducamus per B lineam tan-
              <lb/>
              <note position="right" xlink:label="note-0074-14" xlink:href="note-0074-14a" xml:space="preserve">l</note>
            gentem ſectionem, quæ occurrat inclinato, ſiue tranſuerſæ in a, & </s>
            <s xml:id="echoid-s1932" xml:space="preserve">per V
              <lb/>
            ducamus perpendicularem ſuper axim, cui occurrat ad c, & </s>
            <s xml:id="echoid-s1933" xml:space="preserve">occurrat tan-
              <lb/>
            genti B a in d; </s>
            <s xml:id="echoid-s1934" xml:space="preserve">& </s>
            <s xml:id="echoid-s1935" xml:space="preserve">quoniam O G ad O B, quemadmodum demonſtraui-
              <lb/>
            mus, maiorem proportionem habet, quàm F O ad O C, ponamus fO ad
              <lb/>
            O B, vt F O ad O C, & </s>
            <s xml:id="echoid-s1936" xml:space="preserve">per f producamus f g h parallelam axi A D: </s>
            <s xml:id="echoid-s1937" xml:space="preserve">Et
              <lb/>
              <note position="right" xlink:label="note-0074-15" xlink:href="note-0074-15a" xml:space="preserve">m</note>
            quia f O ad O B eſt, vt F O ad O C, erit rectangulum f O C æquale B O
              <lb/>
            in O F, & </s>
            <s xml:id="echoid-s1938" xml:space="preserve">ponamus rectangulum f F communiter fiet B f in f g æquale g
              <lb/>
              <note position="right" xlink:label="note-0074-16" xlink:href="note-0074-16a" xml:space="preserve">n</note>
            F in F C, & </s>
            <s xml:id="echoid-s1939" xml:space="preserve">quia C O inuerſa in trutinatam C a æquale eſt quadrato C
              <lb/>
            A dimidij inclinati, ſiue tranſuerſæ (39. </s>
            <s xml:id="echoid-s1940" xml:space="preserve">ex primo) erit O C ad C A, vt
              <lb/>
            C A ad C a; </s>
            <s xml:id="echoid-s1941" xml:space="preserve">igitur C a eſt linea quinta proportionalis aliarum quatuor
              <lb/>
              <note position="left" xlink:label="note-0074-17" xlink:href="note-0074-17a" xml:space="preserve">37. primi.</note>
              <note position="right" xlink:label="note-0074-18" xlink:href="note-0074-18a" xml:space="preserve">o</note>
            linearum proportionalium aſſignatarum; </s>
            <s xml:id="echoid-s1942" xml:space="preserve">ergo F C ad C O eſt, vt C O </s>
          </p>
        </div>
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