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-div278" type="section" level="1" n="92">
          <pb o="71" file="0109" n="109" rhead="Conicor. Lib. V."/>
        </div>
        <div xml:id="echoid-div281" type="section" level="1" n="93">
          <head xml:id="echoid-head129" xml:space="preserve">PROPOSITIO LXX.</head>
          <p>
            <s xml:id="echoid-s3017" xml:space="preserve">P Oſtea in ellipſi iungamus E H, A H, & </s>
            <s xml:id="echoid-s3018" xml:space="preserve">C
              <lb/>
              <figure xlink:label="fig-0109-01" xlink:href="fig-0109-01a" number="90">
                <image file="0109-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0109-01"/>
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            ſit extremitas axis recti; </s>
            <s xml:id="echoid-s3019" xml:space="preserve">erit A H minor
              <lb/>
            quàm E H (11. </s>
            <s xml:id="echoid-s3020" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s3021" xml:space="preserve">& </s>
            <s xml:id="echoid-s3022" xml:space="preserve">angulus EGH, nempe
              <lb/>
              <note position="left" xlink:label="note-0109-01" xlink:href="note-0109-01a" xml:space="preserve">c</note>
            A G F maior erit, quàm A G H, ſeu E G F,
              <lb/>
            ergo E F minor eſt, quàm F A, & </s>
            <s xml:id="echoid-s3023" xml:space="preserve">hoc erat
              <lb/>
            propoſitum.</s>
            <s xml:id="echoid-s3024" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div283" type="section" level="1" n="94">
          <head xml:id="echoid-head130" xml:space="preserve">PROPOSITIO LXXI.</head>
          <p>
            <s xml:id="echoid-s3025" xml:space="preserve">P Atet ex hoc, quod ſi producantur ex duo-
              <lb/>
              <note position="left" xlink:label="note-0109-02" xlink:href="note-0109-02a" xml:space="preserve">d</note>
            bus punctis contactus in ellipſi perpendi-
              <lb/>
            culares E M, A L, & </s>
            <s xml:id="echoid-s3026" xml:space="preserve">fuerit E M minor,
              <lb/>
            exempli gratia, tunc tangens educta ab eius
              <lb/>
            extremitate minor quoque eſt, quemadmodum demonſtrauimus, & </s>
            <s xml:id="echoid-s3027" xml:space="preserve">hoc
              <lb/>
            erat oſtendendum.</s>
            <s xml:id="echoid-s3028" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div285" type="section" level="1" n="95">
          <head xml:id="echoid-head131" xml:space="preserve">Notæ in Propoſit. LXVIII. LXIX. LXX.
            <lb/>
          & LXXI.</head>
          <p style="it">
            <s xml:id="echoid-s3029" xml:space="preserve">S I occurrant duæ tangentes alicui fectioni A B C, aut circulo, vt ſunt,
              <lb/>
              <note position="left" xlink:label="note-0109-03" xlink:href="note-0109-03a" xml:space="preserve">a</note>
            &</s>
            <s xml:id="echoid-s3030" xml:space="preserve">c. </s>
            <s xml:id="echoid-s3031" xml:space="preserve">Ideſt ſi coniſectionem A B C contingant duæ rectæ A F, E F in pun-
              <lb/>
            ctis A, & </s>
            <s xml:id="echoid-s3032" xml:space="preserve">E concurrentes in F, erit portio tangentis inter occurſum, & </s>
            <s xml:id="echoid-s3033" xml:space="preserve">conta-
              <lb/>
            ctum vertici B proximiorem intercepta, minor ea, quæ inter occur ſum, & </s>
            <s xml:id="echoid-s3034" xml:space="preserve">re-
              <lb/>
            motiorem à vertice contactum continetur: </s>
            <s xml:id="echoid-s3035" xml:space="preserve">oportet autem in ellipſi B verticem,
              <lb/>
            eſſe axis maioris. </s>
            <s xml:id="echoid-s3036" xml:space="preserve">Expungo verba, aut circulo, tanquam erronea, & </s>
            <s xml:id="echoid-s3037" xml:space="preserve">incaute
              <lb/>
            ab aliquo textui ſuperaddita. </s>
            <s xml:id="echoid-s3038" xml:space="preserve">Circulum enim tangentes ab eodem puncto ductæ
              <lb/>
            inæquales eſſe nequeunt.</s>
            <s xml:id="echoid-s3039" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s3040" xml:space="preserve">Et ducamus A D in parabola, & </s>
            <s xml:id="echoid-s3041" xml:space="preserve">hyperbola, &</s>
            <s xml:id="echoid-s3042" xml:space="preserve">c. </s>
            <s xml:id="echoid-s3043" xml:space="preserve">Et ducamus A D in
              <lb/>
              <note position="left" xlink:label="note-0109-04" xlink:href="note-0109-04a" xml:space="preserve">b</note>
            parabola, & </s>
            <s xml:id="echoid-s3044" xml:space="preserve">hyperbola perpendicularem ſuper axim B D, ſecantem eum in D,
              <lb/>
            atque G F H in I; </s>
            <s xml:id="echoid-s3045" xml:space="preserve">cumque in parabola diameter F G I ſit parallela axi B D,
              <lb/>
            erit angulus A I G rectus æqualis interno, & </s>
            <s xml:id="echoid-s3046" xml:space="preserve">oppoſito ad eaſdem partes, angu-
              <lb/>
            lo D; </s>
            <s xml:id="echoid-s3047" xml:space="preserve">in hyperbola vero cum triangulum H D I ſit rectangulum in D, erit ex-
              <lb/>
            ternus A I G obtuſus, eſtque in triangulo G I A angulus externus A G F maior
              <lb/>
            interno, & </s>
            <s xml:id="echoid-s3048" xml:space="preserve">oppoſito A I G, recto in parabola, & </s>
            <s xml:id="echoid-s3049" xml:space="preserve">obtuſo in hyperbola; </s>
            <s xml:id="echoid-s3050" xml:space="preserve">erit quo-
              <lb/>
            que angulus F G A obtuſus in parabola, & </s>
            <s xml:id="echoid-s3051" xml:space="preserve">hyperbola.</s>
            <s xml:id="echoid-s3052" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s3053" xml:space="preserve">Et angulus E G H, &</s>
            <s xml:id="echoid-s3054" xml:space="preserve">c. </s>
            <s xml:id="echoid-s3055" xml:space="preserve">Zuia F H eſt diameter ſecans bifariam E A in
              <lb/>
              <note position="left" xlink:label="note-0109-05" xlink:href="note-0109-05a" xml:space="preserve">c</note>
              <note position="right" xlink:label="note-0109-06" xlink:href="note-0109-06a" xml:space="preserve">30. ex 2.
                <lb/>
              Com.</note>
            G; </s>
            <s xml:id="echoid-s3056" xml:space="preserve">ergo triangula E G H, & </s>
            <s xml:id="echoid-s3057" xml:space="preserve">A G H habent àuo latera ægualia E G, A G, & </s>
            <s xml:id="echoid-s3058" xml:space="preserve">
              <lb/>
              <note position="right" xlink:label="note-0109-07" xlink:href="note-0109-07a" xml:space="preserve">11. huius.</note>
            G H, commune; </s>
            <s xml:id="echoid-s3059" xml:space="preserve">eſtque H E, vertici B axis maioris ellipſis propinquior, maior
              <lb/>
            remotiore H A; </s>
            <s xml:id="echoid-s3060" xml:space="preserve">ergo angulus E G H maior erit angulo A G H; </s>
            <s xml:id="echoid-s3061" xml:space="preserve">eſtque angulus
              <lb/>
            A G F æqualis E G H maiori, & </s>
            <s xml:id="echoid-s3062" xml:space="preserve">E G F æqualis minori A G H; </s>
            <s xml:id="echoid-s3063" xml:space="preserve">igitur angulus
              <lb/>
            A G F maior eſt angulo E G F, & </s>
            <s xml:id="echoid-s3064" xml:space="preserve">latera circa inæquales angulos ſunt æqualia
              <lb/>
            ſingula ſingulis, ergo tangens A F maior eſt, quàm E F.</s>
            <s xml:id="echoid-s3065" xml:space="preserve"/>
          </p>
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