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-div804" type="section" level="1" n="247">
          <p style="it">
            <s xml:id="echoid-s9665" xml:space="preserve">
              <pb o="257" file="0295" n="295" rhead="Conicor. Lib. VI."/>
            rectangulum O V R ad quadratum V R, vt H E ad E I: </s>
            <s xml:id="echoid-s9666" xml:space="preserve">eſt verò rectangulum
              <lb/>
            H V E æquale rectangulo O V R (propterea quod duæ rect æ line æ O R, H E ſe ſe ſe-
              <lb/>
            cant intra circulum in V) igitur rectangulum H V E ad quadratum V R eandẽ
              <lb/>
            proportionẽ habet quàm H E ad E I; </s>
            <s xml:id="echoid-s9667" xml:space="preserve">cumq; </s>
            <s xml:id="echoid-s9668" xml:space="preserve">proportio rectanguli H V E ad qua.
              <lb/>
            </s>
            <s xml:id="echoid-s9669" xml:space="preserve">dratum V R compoſita ſit ex duabus rationibus, ipſius E V ad V R, ſeu R S ad
              <lb/>
            S E, (propter parallelogrammum V E S R), & </s>
            <s xml:id="echoid-s9670" xml:space="preserve">ex proportione H V ad V R,
              <lb/>
            quæ eadem eſt proportioni ipſius R S ad S T (propterea quod triangula H V R,
              <lb/>
            & </s>
            <s xml:id="echoid-s9671" xml:space="preserve">R S T ſimilia conſtituuntur ab æquidiſtantibus H V, R S, & </s>
            <s xml:id="echoid-s9672" xml:space="preserve">V R, S T)
              <lb/>
            quapropter duæ proportiones R S ad S E, & </s>
            <s xml:id="echoid-s9673" xml:space="preserve">R S ad S T componentes proportio-
              <lb/>
            nem quadrati R S ad rectangulum E S T eædem ſunt rationibus, ex quibus
              <lb/>
            componitur proportio rectanguli H V E ad quadratum V R; </s>
            <s xml:id="echoid-s9674" xml:space="preserve">& </s>
            <s xml:id="echoid-s9675" xml:space="preserve">ideo quadratum
              <lb/>
            R S ad rectangulum E S T eandem proportionem habebit, quàm rectangulum
              <lb/>
            H V E ad quadratum V R, ſeu eandem quàm habet H E ad E I; </s>
            <s xml:id="echoid-s9676" xml:space="preserve">igitur ſi fiat
              <lb/>
            conus, cuius vertex R, & </s>
            <s xml:id="echoid-s9677" xml:space="preserve">baſis circulus diametro E T, cuius planum perpen-
              <lb/>
            diculare ſit ad planum trianguli E R T, erit triangulum E R T iſoſcelium per
              <lb/>
            axim prædicti coni extenſum, atq; </s>
            <s xml:id="echoid-s9678" xml:space="preserve">ad ipſum ſectionis D E F planum eſt quo-
              <lb/>
            que perpendiculare, & </s>
            <s xml:id="echoid-s9679" xml:space="preserve">eius axis G E ſubtendit angulum E R H, qui deinceps
              <lb/>
            eſt angulo verticis; </s>
            <s xml:id="echoid-s9680" xml:space="preserve">igitur planum D E F in cono E R T generat hyperbolen,
              <lb/>
            cuius axis inclinatus eſt E H, & </s>
            <s xml:id="echoid-s9681" xml:space="preserve">erectus E I: </s>
            <s xml:id="echoid-s9682" xml:space="preserve">& </s>
            <s xml:id="echoid-s9683" xml:space="preserve">propterea conus E R T com-
              <lb/>
            prehendit hyperbolen D E F. </s>
            <s xml:id="echoid-s9684" xml:space="preserve">Rurſus ſi recta R X producatur quouſque ſecet
              <lb/>
            peripheriam circuli L E ex altera parte in puncto Y; </s>
            <s xml:id="echoid-s9685" xml:space="preserve">atque denuò coniungantur
              <lb/>
            rectæ lineæ E Y, & </s>
            <s xml:id="echoid-s9686" xml:space="preserve">H Y, quæ extendatur quouſquè conueniat cum recta linea
              <lb/>
            ex puncto E parallela ipſi O Y in puncto aliquo, quod concipiatur eſſe d; </s>
            <s xml:id="echoid-s9687" xml:space="preserve">fieri
              <lb/>
            poterit alius conus (cuius vertex Y, baſis circulus diametro E d erectus ad
              <lb/>
            planum trianguli) ſimilis cono E R T, ſiue A B C: </s>
            <s xml:id="echoid-s9688" xml:space="preserve">Oſtendetur ſicuti modo di-
              <lb/>
            ctum eſt, quod idem planum H D F eſſiciet in cono γ d E eandem hyperbolen
              <lb/>
            D E F.</s>
            <s xml:id="echoid-s9689" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9690" xml:space="preserve">Inde demonſtrabitur quod E H ad E I neceſſe eſt, vt habeat eandem
              <lb/>
              <note position="left" xlink:label="note-0295-01" xlink:href="note-0295-01a" xml:space="preserve">h</note>
            proportionem, quàm O e ad e Z: </s>
            <s xml:id="echoid-s9691" xml:space="preserve">& </s>
            <s xml:id="echoid-s9692" xml:space="preserve">hoc eſt abſurdum, &</s>
            <s xml:id="echoid-s9693" xml:space="preserve">c. </s>
            <s xml:id="echoid-s9694" xml:space="preserve">quia conus
              <lb/>
            Z E f continet hyperbolen D E F neceſſariò eius axis tranſuerſus E H ſubten-
              <lb/>
            det angulum H Z E, qui deinceps eſt anguli verticis trianguli per axim; </s>
            <s xml:id="echoid-s9695" xml:space="preserve">& </s>
            <s xml:id="echoid-s9696" xml:space="preserve">
              <lb/>
            propter ſimilitudinẽ conorũ rectorum, ſunt triangula per axes A B C, E R T, & </s>
            <s xml:id="echoid-s9697" xml:space="preserve">
              <lb/>
            E Z f ſimilia inter ſe, & </s>
            <s xml:id="echoid-s9698" xml:space="preserve">anguli verticales B, Z, & </s>
            <s xml:id="echoid-s9699" xml:space="preserve">R æquales erunt inter ſe;
              <lb/>
            </s>
            <s xml:id="echoid-s9700" xml:space="preserve">ideo conſequentes anguli M B C, & </s>
            <s xml:id="echoid-s9701" xml:space="preserve">H R E, nec non H Z E æquales erunt in-
              <lb/>
            ter ſe, & </s>
            <s xml:id="echoid-s9702" xml:space="preserve">ſubtenduntur ab eadem recta linea H E; </s>
            <s xml:id="echoid-s9703" xml:space="preserve">ergo in eodem circuli ſeg-
              <lb/>
            mento conſiſtunt: </s>
            <s xml:id="echoid-s9704" xml:space="preserve">& </s>
            <s xml:id="echoid-s9705" xml:space="preserve">propterea punctum Z in circuli peripheria H Z E cadit. </s>
            <s xml:id="echoid-s9706" xml:space="preserve">
              <lb/>
            Poſtea (vt in propoſitione 53. </s>
            <s xml:id="echoid-s9707" xml:space="preserve">primi libri, & </s>
            <s xml:id="echoid-s9708" xml:space="preserve">in hac eadem propoſitione demon-
              <lb/>
            ſtrauit Apollonius) conſtat quod H E ad E I habet eandem proportionem, quàm
              <lb/>
            O e ad e Z; </s>
            <s xml:id="echoid-s9709" xml:space="preserve">& </s>
            <s xml:id="echoid-s9710" xml:space="preserve">prius O V ad V R erat vt H E ad E I; </s>
            <s xml:id="echoid-s9711" xml:space="preserve">ergo O V ad V R eã-
              <lb/>
            dem proportionem habet quàm O e ad e Z; </s>
            <s xml:id="echoid-s9712" xml:space="preserve">ſed quia punctum Z non cadit in
              <lb/>
            R, neque in γ alias conus E Z f non eſſet alius à præcedentibus E R T, & </s>
            <s xml:id="echoid-s9713" xml:space="preserve">E
              <lb/>
            γ d; </s>
            <s xml:id="echoid-s9714" xml:space="preserve">ergo O e ad e Z non habet eandem proportionem, quàm O V ad V R, quod
              <lb/>
            eſt abſurdum.</s>
            <s xml:id="echoid-s9715" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9716" xml:space="preserve">Et demonſtrabitur quod O V ad V R ſit vt H E ad E I, &</s>
            <s xml:id="echoid-s9717" xml:space="preserve">c. </s>
            <s xml:id="echoid-s9718" xml:space="preserve">Repeta-
              <lb/>
              <note position="left" xlink:label="note-0295-02" xlink:href="note-0295-02a" xml:space="preserve">i</note>
            tur denuo conſtructio primi caſus huius propoſitionis, vt fiat conus rectus L E
              <lb/>
            K ſim lis cono A B C, tunc quidem quadratum L P ad quadratum E P habe-
              <lb/>
            bit eandem proportionem, quàm O N ad N L, ſeu quàm quadratum B Q </s>
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