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-div296" type="section" level="1" n="98">
          <p>
            <s xml:id="echoid-s3124" xml:space="preserve">
              <pb o="75" file="0113" n="113" rhead="Conicor. Lib. V."/>
            A G, A H ſuper E A, & </s>
            <s xml:id="echoid-s3125" xml:space="preserve">D A. </s>
            <s xml:id="echoid-s3126" xml:space="preserve">Et quia A G tangit ſectionem, cadet
              <lb/>
            A H intra ſectionem, & </s>
            <s xml:id="echoid-s3127" xml:space="preserve">ducamus rectam B I tangentem ſectionem in
              <lb/>
              <note position="left" xlink:label="note-0113-01" xlink:href="note-0113-01a" xml:space="preserve">b</note>
              <note position="right" xlink:label="note-0113-02" xlink:href="note-0113-02a" xml:space="preserve">33. 34.
                <lb/>
              lib. 1.</note>
            B. </s>
            <s xml:id="echoid-s3128" xml:space="preserve">Quoniam ex D non educitur ad ſectionem A C vllus breuiſecans,
              <lb/>
            erit E A non maior dimidio erecti (49. </s>
            <s xml:id="echoid-s3129" xml:space="preserve">50. </s>
            <s xml:id="echoid-s3130" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s3131" xml:space="preserve">aut erit D E maior
              <lb/>
            quàm F (52. </s>
            <s xml:id="echoid-s3132" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s3133" xml:space="preserve">Iis poſitis vtique linea breuiſſima ex B educta abſcin-
              <lb/>
            dit cum A ex axi lineam maiorem, quàm A K (49. </s>
            <s xml:id="echoid-s3134" xml:space="preserve">50. </s>
            <s xml:id="echoid-s3135" xml:space="preserve">51. </s>
            <s xml:id="echoid-s3136" xml:space="preserve">52. </s>
            <s xml:id="echoid-s3137" xml:space="preserve">ex 5.)
              <lb/>
            </s>
            <s xml:id="echoid-s3138" xml:space="preserve">verùm linea breuiſſima continet cum tangente B I angulum rectum (29. </s>
            <s xml:id="echoid-s3139" xml:space="preserve">
              <lb/>
              <note position="left" xlink:label="note-0113-03" xlink:href="note-0113-03a" xml:space="preserve">c</note>
            30. </s>
            <s xml:id="echoid-s3140" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s3141" xml:space="preserve">igitur D B I eſt acutus, quare ſi centro D, interuallo D B cir-
              <lb/>
            culus deſcribatur, tunc B I cadit intra circulum, & </s>
            <s xml:id="echoid-s3142" xml:space="preserve">A H cadit extra id
              <lb/>
              <note position="left" xlink:label="note-0113-04" xlink:href="note-0113-04a" xml:space="preserve">d</note>
            ipſum, quia eſt perpendicularis ad D A; </s>
            <s xml:id="echoid-s3143" xml:space="preserve">igitur circulus ſecat coniſectio-
              <lb/>
            nem; </s>
            <s xml:id="echoid-s3144" xml:space="preserve">ſecet eam in L, & </s>
            <s xml:id="echoid-s3145" xml:space="preserve">iungamus L D, ducamuſque L G ſectionem,
              <lb/>
              <note position="right" xlink:label="note-0113-05" xlink:href="note-0113-05a" xml:space="preserve">33. 34.
                <lb/>
              lib, 1.</note>
            tangentem. </s>
            <s xml:id="echoid-s3146" xml:space="preserve">Pater (vt dictũ eſt) quod D L G ſit acutus; </s>
            <s xml:id="echoid-s3147" xml:space="preserve">ergo L G cadit
              <lb/>
              <note position="left" xlink:label="note-0113-06" xlink:href="note-0113-06a" xml:space="preserve">e</note>
            intra circulum B L A, ſed cadit extra, quod eſt abſurdum; </s>
            <s xml:id="echoid-s3148" xml:space="preserve">ergo B D
              <lb/>
            non eſt æqualis ipſi A D. </s>
            <s xml:id="echoid-s3149" xml:space="preserve">Neque minor illo eſſe poteſt; </s>
            <s xml:id="echoid-s3150" xml:space="preserve">quia ſi ſecetur
              <lb/>
            D M maior, quàm D B, & </s>
            <s xml:id="echoid-s3151" xml:space="preserve">minor, quàm D A, & </s>
            <s xml:id="echoid-s3152" xml:space="preserve">centro D, interuallo
              <lb/>
            D M, circulus M L N deſcribatur, tunc D N, nempe D M maior eſt,
              <lb/>
            quàm D B, & </s>
            <s xml:id="echoid-s3153" xml:space="preserve">propterea circulus N L M ſecat coniſectionem. </s>
            <s xml:id="echoid-s3154" xml:space="preserve">Subinde,
              <lb/>
              <note position="left" xlink:label="note-0113-07" xlink:href="note-0113-07a" xml:space="preserve">f</note>
            patebit (quemadmodũ demoſtrauimus) quod D B non ſit minor, quàm
              <lb/>
            D A; </s>
            <s xml:id="echoid-s3155" xml:space="preserve">igitur D B maior eſt, quàm D A.</s>
            <s xml:id="echoid-s3156" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3157" xml:space="preserve">Poſtea dico, quod D C maior eſt, quàm D B; </s>
            <s xml:id="echoid-s3158" xml:space="preserve">quia demonſtrauimus,
              <lb/>
              <note position="left" xlink:label="note-0113-08" xlink:href="note-0113-08a" xml:space="preserve">g</note>
            angulũ D B O eſſe obtuſum, & </s>
            <s xml:id="echoid-s3159" xml:space="preserve">patet, quod D C P eſt acutus, & </s>
            <s xml:id="echoid-s3160" xml:space="preserve">proce-
              <lb/>
            dendo trito iam itinere demonſtrabimus, quod Q O neceſſe eſt, vt cadat
              <lb/>
            intra circulum C Q B. </s>
            <s xml:id="echoid-s3161" xml:space="preserve">Et quod ſi fuerit D C minor, quàm D B, aut æ-
              <lb/>
            qualis, neceſſe eſt, vt Q O cadat intra circulum C Q B; </s>
            <s xml:id="echoid-s3162" xml:space="preserve">ſed cecidit ex-
              <lb/>
            tra, quod eſt abſurdum; </s>
            <s xml:id="echoid-s3163" xml:space="preserve">igitur D C maior eſt, quàm D B, & </s>
            <s xml:id="echoid-s3164" xml:space="preserve">D B ma-
              <lb/>
            ior, quàm D A, quod erat oſtendendum.</s>
            <s xml:id="echoid-s3165" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div299" type="section" level="1" n="99">
          <head xml:id="echoid-head139" xml:space="preserve">PROPOSITIO LXVI.</head>
          <p>
            <s xml:id="echoid-s3166" xml:space="preserve">IN ſectione elliptica A B C,
              <lb/>
              <figure xlink:label="fig-0113-01" xlink:href="fig-0113-01a" number="95">
                <image file="0113-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0113-01"/>
              </figure>
            cuius axis maior A C eius
              <lb/>
            centrum D, & </s>
            <s xml:id="echoid-s3167" xml:space="preserve">D B dimidium
              <lb/>
            recti, duci nequeat ex E ad
              <lb/>
            quadrantem A B breuiſecans,
              <lb/>
            & </s>
            <s xml:id="echoid-s3168" xml:space="preserve">producatur perpendicularis
              <lb/>
            E F; </s>
            <s xml:id="echoid-s3169" xml:space="preserve">Dico punctum F cadere
              <lb/>
            inter D A.</s>
            <s xml:id="echoid-s3170" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3171" xml:space="preserve">Quia ſi caderet inter C, D du-
              <lb/>
              <note position="left" xlink:label="note-0113-09" xlink:href="note-0113-09a" xml:space="preserve">a</note>
            ci poſſet ex E ad ſectionem A B
              <lb/>
              <note position="left" xlink:label="note-0113-10" xlink:href="note-0113-10a" xml:space="preserve">b</note>
            aliqua breuiſecans (56. </s>
            <s xml:id="echoid-s3172" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s3173" xml:space="preserve">quod eſt contra ſuppoſitionem. </s>
            <s xml:id="echoid-s3174" xml:space="preserve">Deinde
              <lb/>
            patet, quemadmodum demonſtrauimus in parabola, & </s>
            <s xml:id="echoid-s3175" xml:space="preserve">hyperbola, quod
              <lb/>
              <note position="right" xlink:label="note-0113-11" xlink:href="note-0113-11a" xml:space="preserve">pr. 64. 65.
                <lb/>
              huius.</note>
            E A minima ſit linearum, & </s>
            <s xml:id="echoid-s3176" xml:space="preserve">ramorum ad ſectionem B A cadentium, & </s>
            <s xml:id="echoid-s3177" xml:space="preserve">
              <lb/>
            propinquior illi, minor ſit remotiore, & </s>
            <s xml:id="echoid-s3178" xml:space="preserve">hoc erat propoſitum.</s>
            <s xml:id="echoid-s3179" xml:space="preserve"/>
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