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-div153" type="section" level="1" n="65">
          <pb o="39" file="0077" n="77" rhead="Conicor. Lib. V."/>
        </div>
        <div xml:id="echoid-div158" type="section" level="1" n="66">
          <head xml:id="echoid-head98" xml:space="preserve">PROPOSITIO LIV. LV.</head>
          <p>
            <s xml:id="echoid-s1977" xml:space="preserve">ITaque oſtenſum eſt, vti memorauimus, quod ex concurſu
              <lb/>
            duarum breuiſſimarum ad coniſectionem non egrediatur alia
              <lb/>
              <note position="left" xlink:label="note-0077-01" xlink:href="note-0077-01a" xml:space="preserve">a</note>
            breuiſecans præter illas duas, & </s>
            <s xml:id="echoid-s1978" xml:space="preserve">quod reliqui rami ex eorum
              <lb/>
            concurſu educti ad ſectionem habent proprietates ſuperiùs ex-
              <lb/>
            poſitas.</s>
            <s xml:id="echoid-s1979" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div160" type="section" level="1" n="67">
          <head xml:id="echoid-head99" xml:space="preserve">PROPOSITIO LVI.</head>
          <p>
            <s xml:id="echoid-s1980" xml:space="preserve">In ellipſi ramorum, ſecantium vtrumque axim, à concurſu vl-
              <lb/>
            tra centrum poſito egredientium, vnius tantum portio, inter
              <lb/>
            axim maiorem, & </s>
            <s xml:id="echoid-s1981" xml:space="preserve">ſectionem intercepta, erit linea breuiſsima,
              <lb/>
              <note position="left" xlink:label="note-0077-02" xlink:href="note-0077-02a" xml:space="preserve">a</note>
            ſiue menſura ipſam comparatam, nec non perpendicularis ipſam
              <lb/>
            trutinam ſuperet, æquet, vel ab ea deficiat.</s>
            <s xml:id="echoid-s1982" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1983" xml:space="preserve">SIt ſectio ellipſis A C B, & </s>
            <s xml:id="echoid-s1984" xml:space="preserve">axis maior tranſuerſus A B perpendicularis
              <lb/>
            E F, centrum D, & </s>
            <s xml:id="echoid-s1985" xml:space="preserve">ponamus D G ad G F, vt proportio figuræ, & </s>
            <s xml:id="echoid-s1986" xml:space="preserve">ſi-
              <lb/>
              <note position="left" xlink:label="note-0077-03" xlink:href="note-0077-03a" xml:space="preserve">b</note>
            militer E H ad H F, & </s>
            <s xml:id="echoid-s1987" xml:space="preserve">producamus per H rectam I H K parallelam ipſi A B,
              <lb/>
            & </s>
            <s xml:id="echoid-s1988" xml:space="preserve">per G rectã I G L ipſi
              <lb/>
              <figure xlink:label="fig-0077-01" xlink:href="fig-0077-01a" number="53">
                <image file="0077-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0077-01"/>
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            E F, quæ ſibi occurrant
              <lb/>
            in I, & </s>
            <s xml:id="echoid-s1989" xml:space="preserve">ducamus per
              <lb/>
              <note position="left" xlink:label="note-0077-04" xlink:href="note-0077-04a" xml:space="preserve">c</note>
            punctum E ſectionem
              <lb/>
              <note position="right" xlink:label="note-0077-05" xlink:href="note-0077-05a" xml:space="preserve">4. lib. 2</note>
            hyperbolen E M C cir-
              <lb/>
            ca duas eius continen-
              <lb/>
            tes L I, I K, quæ oc-
              <lb/>
            curret ſectioni A C B
              <lb/>
            ellipticæ, quia I L, I K
              <lb/>
            ſunt duæ cõtinentes ſe-
              <lb/>
            ctionem E M C, & </s>
            <s xml:id="echoid-s1990" xml:space="preserve">pro-
              <lb/>
            portio E H ad H F po-
              <lb/>
            ſita eſt, vt D G ad G F;
              <lb/>
            </s>
            <s xml:id="echoid-s1991" xml:space="preserve">
              <note position="left" xlink:label="note-0077-06" xlink:href="note-0077-06a" xml:space="preserve">d</note>
            ergo E H prima proportionalium in H I, nempe G F quartam, æquale
              <lb/>
            eſt D G ſecundæ in I G, nempe F H tertiam; </s>
            <s xml:id="echoid-s1992" xml:space="preserve">ergo punctum M eſt in il-
              <lb/>
            lius diametro, & </s>
            <s xml:id="echoid-s1993" xml:space="preserve">propterea ſectio hyperbole E M C tranſit per centrum
              <lb/>
            ſectionis ellipſis A C B; </s>
            <s xml:id="echoid-s1994" xml:space="preserve">quare duæ ſectiones ſe inuicem ſecant, ſitque
              <lb/>
            concurſus in C, & </s>
            <s xml:id="echoid-s1995" xml:space="preserve">producamus per E, C lineam occurrentem duabus con-
              <lb/>
              <note position="left" xlink:label="note-0077-07" xlink:href="note-0077-07a" xml:space="preserve">e</note>
            tinentibus ſectionem in L, K, & </s>
            <s xml:id="echoid-s1996" xml:space="preserve">producamus duas perpendiculares C N,
              <lb/>
            K O ſuper A B. </s>
            <s xml:id="echoid-s1997" xml:space="preserve">Et quia K C, L E ſunt æquales (16. </s>
            <s xml:id="echoid-s1998" xml:space="preserve">exſecundo) erit G F
              <lb/>
              <note position="right" xlink:label="note-0077-08" xlink:href="note-0077-08a" xml:space="preserve">8. lib. 2.</note>
            æqualis O N; </s>
            <s xml:id="echoid-s1999" xml:space="preserve">quare F O æqualis eſt ipſi G N; </s>
            <s xml:id="echoid-s2000" xml:space="preserve">atque E H ad H F, nempe
              <lb/>
              <note position="left" xlink:label="note-0077-09" xlink:href="note-0077-09a" xml:space="preserve">f</note>
            E K ad K P, ſeu F O (quæ eſt æqualis ipſi G N) ad O P eandem propor-
              <lb/>
            tionem habet, quàm D G ad G F, quę eſt ęqualis ipſi O N, & </s>
            <s xml:id="echoid-s2001" xml:space="preserve">ideo G N
              <lb/>
            ad O P eſt, vt D G ad O N, & </s>
            <s xml:id="echoid-s2002" xml:space="preserve">comparando homologum differentias D N
              <lb/>
              <note position="right" xlink:label="note-0077-10" xlink:href="note-0077-10a" xml:space="preserve">Lem. 3.</note>
            </s>
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