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-div270" type="section" level="1" n="90">
          <pb o="69" file="0107" n="107" rhead="Conicor. Lib. V."/>
          <p style="it">
            <s xml:id="echoid-s2952" xml:space="preserve">Sit A D dimidium axis recti ſectionis ellipticæ A B C, &</s>
            <s xml:id="echoid-s2953" xml:space="preserve">c. </s>
            <s xml:id="echoid-s2954" xml:space="preserve">Sit A D di-
              <lb/>
              <note position="left" xlink:label="note-0107-01" xlink:href="note-0107-01a" xml:space="preserve">b</note>
            midium axis minoris, & </s>
            <s xml:id="echoid-s2955" xml:space="preserve">recti ellipſis A B C, ſitque menſura A E maior, quàm
              <lb/>
            A D, & </s>
            <s xml:id="echoid-s2956" xml:space="preserve">E A ad A D habeat maiorem, aut eandem proportionem, quàm habet
              <lb/>
            latus tranſuerſum B C ad eius rectum latus.</s>
            <s xml:id="echoid-s2957" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2958" xml:space="preserve">Ponatur ramus E F, & </s>
            <s xml:id="echoid-s2959" xml:space="preserve">producamus ex F, &</s>
            <s xml:id="echoid-s2960" xml:space="preserve">c. </s>
            <s xml:id="echoid-s2961" xml:space="preserve">Ducatur quilibet ramus
              <lb/>
              <note position="left" xlink:label="note-0107-02" xlink:href="note-0107-02a" xml:space="preserve">c</note>
            ſecans E F, & </s>
            <s xml:id="echoid-s2962" xml:space="preserve">ex F ad vtrumque axim perpendiculares F H, F I, quæ ſecent
              <lb/>
            eos in H, & </s>
            <s xml:id="echoid-s2963" xml:space="preserve">I. </s>
            <s xml:id="echoid-s2964" xml:space="preserve">Et quia D H minor eſt, quàm D A, habebit eadem E D ad
              <lb/>
            D H maiorem proportionem, quàm ad D A, & </s>
            <s xml:id="echoid-s2965" xml:space="preserve">componendo E H ad H D, ma-
              <lb/>
            iorem proportionem habebit, quàm E A ad A D; </s>
            <s xml:id="echoid-s2966" xml:space="preserve">eſt vero E F ad F G, vt E
              <lb/>
            H ad H D (propter parallelas D G, H F) nec non D I ad I G eſt, vt E F ad
              <lb/>
            F G (propter parallelas E D, I F) ergo D I ad I G maiorem proportionem ha-
              <lb/>
            bet, quàm E A ad A D: </s>
            <s xml:id="echoid-s2967" xml:space="preserve">habebat autem E A ad A D maiorem, aut eandem
              <lb/>
            proportionem, quàm latus tranſuer ſum B C ad eius rectum latus; </s>
            <s xml:id="echoid-s2968" xml:space="preserve">igitur D I ad
              <lb/>
            I G maiorem proportionem habebit, quàm latus tranſuer ſum B C ad eius rectum
              <lb/>
            latus: </s>
            <s xml:id="echoid-s2969" xml:space="preserve">fiat iam D I ad I K, vt latus tranſuer ſum B C ad eius latus rectum,
              <lb/>
            iungaturque F K, erit I K maior, quàm I G, & </s>
            <s xml:id="echoid-s2970" xml:space="preserve">F K linea breuiſsima, quæ ſe-
              <lb/>
              <note position="right" xlink:label="note-0107-03" xlink:href="note-0107-03a" xml:space="preserve">10. huius.</note>
            cat ſegmentum axis K B maius, quàm B G, vnde E F non erit breuiſcans.</s>
            <s xml:id="echoid-s2971" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div275" type="section" level="1" n="91">
          <head xml:id="echoid-head125" xml:space="preserve">Notæ in Propoſ. XLV.</head>
          <p style="it">
            <s xml:id="echoid-s2972" xml:space="preserve">SI autem fuerit ratio E A ad A D minor, quàm proportio figuræ, &</s>
            <s xml:id="echoid-s2973" xml:space="preserve">c.
              <lb/>
            </s>
            <s xml:id="echoid-s2974" xml:space="preserve">
              <note position="left" xlink:label="note-0107-04" xlink:href="note-0107-04a" xml:space="preserve">a</note>
            Habeat E A ad A D minorẽ proportionem, quàm latus tranſuer ſum B C ad
              <lb/>
            eius rectum latus, & </s>
            <s xml:id="echoid-s2975" xml:space="preserve">fiat E H ad H D, vt latus tranſuer ſum ad rectum; </s>
            <s xml:id="echoid-s2976" xml:space="preserve">ha-
              <lb/>
            bebit E H ad H D maiorem proportionem, quàm E A ad A D, & </s>
            <s xml:id="echoid-s2977" xml:space="preserve">diuidendo
              <lb/>
            eadem E D ad D H habebit maiorem proportionem, quàm ad D A; </s>
            <s xml:id="echoid-s2978" xml:space="preserve">& </s>
            <s xml:id="echoid-s2979" xml:space="preserve">pro-
              <lb/>
            pterea D H minor erit, quàm D A; </s>
            <s xml:id="echoid-s2980" xml:space="preserve">vnde ex puncto H ſi eleuetur H F perpen-
              <lb/>
            dicularis ad D A intra ſectionem cadet, & </s>
            <s xml:id="echoid-s2981" xml:space="preserve">ſecabit eam alicubi, vt in F: </s>
            <s xml:id="echoid-s2982" xml:space="preserve">duca-
              <lb/>
            tur poſtea ex F recta F E, quæ ſecet axim in G, & </s>
            <s xml:id="echoid-s2983" xml:space="preserve">F I perpendicularis ad axim
              <lb/>
            B C eum ſecans in I. </s>
            <s xml:id="echoid-s2984" xml:space="preserve">Et quoniam, propter parallelas G D, F H, eſt E F ad F
              <lb/>
            G, vt E H ad H D, pariterque, propter parallelas E D, I F, eſt D I ad I G, vt
              <lb/>
            E F ad F G, quare D I ad I G eandem proportionem habet, quàm E H ad H
              <lb/>
            D, ſeu quàm latus tranſuer ſum B C ad eius latus rectum; </s>
            <s xml:id="echoid-s2985" xml:space="preserve">& </s>
            <s xml:id="echoid-s2986" xml:space="preserve">propterea F G eſt
              <lb/>
              <note position="right" xlink:label="note-0107-05" xlink:href="note-0107-05a" xml:space="preserve">10. huius.</note>
            breuiſsima.</s>
            <s xml:id="echoid-s2987" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2988" xml:space="preserve">Et quoniam iam eductæ ſunt ex E duæ breuiſecantes, &</s>
            <s xml:id="echoid-s2989" xml:space="preserve">c. </s>
            <s xml:id="echoid-s2990" xml:space="preserve">Textus Ara-
              <lb/>
              <note position="left" xlink:label="note-0107-06" xlink:href="note-0107-06a" xml:space="preserve">b</note>
            bicus vſque ad finem propoſitionis eſt omnino corruptus, cum ſupponat propoſi-
              <lb/>
            tionem non demonſtratam, vt in propoſitione 56. </s>
            <s xml:id="echoid-s2991" xml:space="preserve">notaui; </s>
            <s xml:id="echoid-s2992" xml:space="preserve">Itaque, ſic eum reſti-
              <lb/>
            tui poſſe cenſeo. </s>
            <s xml:id="echoid-s2993" xml:space="preserve">Quoniam ex conſurſu E breuiſsimæ F G, & </s>
            <s xml:id="echoid-s2994" xml:space="preserve">ſemiaxis recti
              <lb/>
            minoris D A rami educti ad ſectionem F A ſecant axis ſegmenta vſque ad
              <lb/>
            verticem B maiora, quàm abſcindant breuiſsimæ ab eorum terminis ad axim
              <lb/>
            ductæ, ſcilicet breuiſsimæ cadunt ſupra ramos (ex Lemmate 8. </s>
            <s xml:id="echoid-s2995" xml:space="preserve">præmiſſo) ſimi-
              <lb/>
            liter rami ex concur ſu E ad ſectionem B F ducti cadunt ſupra breuiſsimas ab
              <lb/>
            eorum terminis ad axim extenſas (ex eodem Lemmate 8.) </s>
            <s xml:id="echoid-s2996" xml:space="preserve">& </s>
            <s xml:id="echoid-s2997" xml:space="preserve">hoc erat oſten-
              <lb/>
            dendum.</s>
            <s xml:id="echoid-s2998" xml:space="preserve"/>
          </p>
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