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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              <pb o="235" file="0273" n="273" rhead="Conicor. Lib. VI."/>
            soincidentibus angulos æquales I D H, & </s>
            <s xml:id="echoid-s8785" xml:space="preserve">V a T & </s>
            <s xml:id="echoid-s8786" xml:space="preserve">cum ipſis D d, & </s>
            <s xml:id="echoid-s8787" xml:space="preserve">a b etiã
              <lb/>
            parallelis inter ſe continebunt angulos æquales I D d, & </s>
            <s xml:id="echoid-s8788" xml:space="preserve">V a b, eruntque in-
              <lb/>
            terceptæ D d, a b æquales ( cum ſint latera oppoſita parallelogrammi D b);
              <lb/>
            </s>
            <s xml:id="echoid-s8789" xml:space="preserve">
              <note position="right" xlink:label="note-0273-01" xlink:href="note-0273-01a" xml:space="preserve">Prop. 10.
                <lb/>
              addit.
                <lb/>
              huius.</note>
            igitur hyperbole H I K, & </s>
            <s xml:id="echoid-s8790" xml:space="preserve">T V e æquales ſunt inter ſe, & </s>
            <s xml:id="echoid-s8791" xml:space="preserve">ſimiles atq; </s>
            <s xml:id="echoid-s8792" xml:space="preserve">earum
              <lb/>
            figuris æqualia ſunt quadrata ex duplis interceptarum D d, & </s>
            <s xml:id="echoid-s8793" xml:space="preserve">a b. </s>
            <s xml:id="echoid-s8794" xml:space="preserve">Et quia
              <lb/>
            triangula A G O, N G P ſunt ſimilia in eodem plano, ſuntque pariter duo cir-
              <lb/>
            culi baſium in vno plano extenſi; </s>
            <s xml:id="echoid-s8795" xml:space="preserve">igitur coni A B C, & </s>
            <s xml:id="echoid-s8796" xml:space="preserve">N L Q ſimiles ſunt
              <lb/>
              <note position="right" xlink:label="note-0273-02" xlink:href="note-0273-02a" xml:space="preserve">Lem. 9.
                <lb/>
              huius.</note>
            inter ſe. </s>
            <s xml:id="echoid-s8797" xml:space="preserve">Secundo quia vt quadratum A d ad rectangulum G d O, ſeu ad re-
              <lb/>
            ctangulum B d C ita eſt latus tranſuerſum ad rectum ſectionis H I K, & </s>
            <s xml:id="echoid-s8798" xml:space="preserve">(ex
              <lb/>
            conſtructione) in eadem proportione erat latus tranſuer ſum ad rectum hyperbo-
              <lb/>
            les X, atque anguli I D K, & </s>
            <s xml:id="echoid-s8799" xml:space="preserve">A d O æquales ſunt inter ſe (propterea quod
              <lb/>
            D I, d A parallelæ ſunt, pariterque D K, d O parallelæ ſunt inter ſe, cum
              <lb/>
            communes ſectiones ſint plani baſis, & </s>
            <s xml:id="echoid-s8800" xml:space="preserve">duorum planorum æquidiſtantium K I
              <lb/>
            H, & </s>
            <s xml:id="echoid-s8801" xml:space="preserve">O A G): </s>
            <s xml:id="echoid-s8802" xml:space="preserve">& </s>
            <s xml:id="echoid-s8803" xml:space="preserve">erat angulus inclinationis diametri, & </s>
            <s xml:id="echoid-s8804" xml:space="preserve">baſis hyperbolæ X æ-
              <lb/>
            qualis angulo A d O; </s>
            <s xml:id="echoid-s8805" xml:space="preserve">igitur diametri ſectionum X, & </s>
            <s xml:id="echoid-s8806" xml:space="preserve">H I K ad ſuas baſes
              <lb/>
            æque inclinantur, & </s>
            <s xml:id="echoid-s8807" xml:space="preserve">habebant latera earundem figurarum proportionalia; </s>
            <s xml:id="echoid-s8808" xml:space="preserve">ſuntq;
              <lb/>
            </s>
            <s xml:id="echoid-s8809" xml:space="preserve">prædictæ figuræ æquales, cum ſint æquales quadrato ex dupla interceptæ D d vt
              <lb/>
            dictum eſt: </s>
            <s xml:id="echoid-s8810" xml:space="preserve">igitur ſectiones H I K, & </s>
            <s xml:id="echoid-s8811" xml:space="preserve">X ſimiles ſunt inter ſe, & </s>
            <s xml:id="echoid-s8812" xml:space="preserve">æquales; </s>
            <s xml:id="echoid-s8813" xml:space="preserve">
              <lb/>
              <note position="right" xlink:label="note-0273-03" xlink:href="note-0273-03a" xml:space="preserve">10. 12.
                <lb/>
              huius.</note>
            ideoque reliqua ſectio T V d, quæ æqualis, & </s>
            <s xml:id="echoid-s8814" xml:space="preserve">congruens oſtenſa eſt ipſi H I K,
              <lb/>
            erit quoque ſimilis, & </s>
            <s xml:id="echoid-s8815" xml:space="preserve">æqualis eidem hyperbolæ X. </s>
            <s xml:id="echoid-s8816" xml:space="preserve">Tertiò quoniam plana H I
              <lb/>
            K, & </s>
            <s xml:id="echoid-s8817" xml:space="preserve">G A O æquidiſtantia ſunt, nunquam conuenient; </s>
            <s xml:id="echoid-s8818" xml:space="preserve">& </s>
            <s xml:id="echoid-s8819" xml:space="preserve">ideo plannum H I K
              <lb/>
            nunquam lateri A N G alterius plani occurret; </s>
            <s xml:id="echoid-s8820" xml:space="preserve">ſed ſuperficies conicæ ſe ſe tan-
              <lb/>
            tummodo tangunt in communi latere A N G, & </s>
            <s xml:id="echoid-s8821" xml:space="preserve">alibi perpetuo ſeparatæ incedunt;
              <lb/>
            </s>
            <s xml:id="echoid-s8822" xml:space="preserve">igitur duæ ſectiones H I K, & </s>
            <s xml:id="echoid-s8823" xml:space="preserve">T V e in plano E I K exiſtentes, quæ infinitè
              <lb/>
            producuntur in ſuperficiebus conicis, nunquam ſe ſe mutuo ſecant; </s>
            <s xml:id="echoid-s8824" xml:space="preserve">igitur ſectio-
              <lb/>
            nes ipſæ aſymptoticæ ſunt. </s>
            <s xml:id="echoid-s8825" xml:space="preserve">Quartò ducantur rectæ lineæ G E, O F, P R tan-
              <lb/>
            gentes circulos in extremitatibus communis diametri G P O, quæ parallelæ erunt
              <lb/>
            inter ſe (cum perpendiculares ſint ad communem diametrum G P O): </s>
            <s xml:id="echoid-s8826" xml:space="preserve">poſtea
              <lb/>
            producantur plana E G A, F O A, R P N tangentia conos in lateribus G A,
              <lb/>
            O A, & </s>
            <s xml:id="echoid-s8827" xml:space="preserve">P N, & </s>
            <s xml:id="echoid-s8828" xml:space="preserve">extendantur quouſque ſecent planum conicæ ſectionis H I Kin
              <lb/>
            rectis lineis E S M, F M, R S. </s>
            <s xml:id="echoid-s8829" xml:space="preserve">Et quoniam duo plana æquidiſtantia G A O,
              <lb/>
            et E M F efficiunt in eodem plano E G A, vtrumque conum contingente, duas
              <lb/>
            rectas lineas G A, E M æquidiſtantes inter ſe: </s>
            <s xml:id="echoid-s8830" xml:space="preserve">pari ratione in plano tangente
              <lb/>
            F O A erunt rectæ lineæ F M, et O A parallelæ inter ſe: </s>
            <s xml:id="echoid-s8831" xml:space="preserve">ſimili modo in plano
              <lb/>
            R P N erunt P N, et R S inter ſe æquidiſtantes, cumque A O, et N P paral-
              <lb/>
            lelæ ſint, erunt quoque F M, et R S inter ſe æquidiſtantes; </s>
            <s xml:id="echoid-s8832" xml:space="preserve">ſuntque E M, et
              <lb/>
            M F aſymptoti continentes hyperbolen E I K pariterq; </s>
            <s xml:id="echoid-s8833" xml:space="preserve">rectæ lineæ E S, S R ſunt
              <lb/>
              <note position="right" xlink:label="note-0273-04" xlink:href="note-0273-04a" xml:space="preserve">Maurol.
                <lb/>
              lib. 3. de
                <lb/>
              lin. horar.
                <lb/>
              ca. 6. 7.</note>
            aſymptoti hyperboles T V e: </s>
            <s xml:id="echoid-s8834" xml:space="preserve">quare duæ hyperbolæ H I K, et T V e, ſimiles ei-
              <lb/>
            dem X, et æquales, & </s>
            <s xml:id="echoid-s8835" xml:space="preserve">ſimiliter poſitæ, quarum duæ asymptoti F M, R S æqui-
              <lb/>
            diſtantes ſunt; </s>
            <s xml:id="echoid-s8836" xml:space="preserve">reliquæ verò E M, & </s>
            <s xml:id="echoid-s8837" xml:space="preserve">E S coincidunt (cum exiſtant in eodem
              <lb/>
            plano tangente E A), & </s>
            <s xml:id="echoid-s8838" xml:space="preserve">angulus ab eis contenctus E M F, vel E S R eſt acu-
              <lb/>
            tus (cum æqualis ſit acuto angulo ab asymptotis ſectionis X contento, propter ſi-
              <lb/>
              <note position="right" xlink:label="note-0273-05" xlink:href="note-0273-05a" xml:space="preserve">Propoſ. 6.
                <lb/>
              addit.
                <lb/>
              huius.</note>
            militudinẽ ſectionũ, vt ab alijs oſtenſum eſt): </s>
            <s xml:id="echoid-s8839" xml:space="preserve">poterit ergo duciramus breuiſſimus
              <lb/>
            in ſectione T V e adpartes V e qui æquidiſtãs ſit rectæ lineæ V I vertices ſectionũ
              <lb/>
            coniungenti: </s>
            <s xml:id="echoid-s8840" xml:space="preserve">eritque illius breuiſſimæ portio inter ſectiones compræhenſa diſtantia
              <lb/>
              <note position="right" xlink:label="note-0273-06" xlink:href="note-0273-06a" xml:space="preserve">Propoſ. 8.
                <lb/>
              addit.
                <lb/>
              huius.</note>
            omniũ maxima; </s>
            <s xml:id="echoid-s8841" xml:space="preserve">& </s>
            <s xml:id="echoid-s8842" xml:space="preserve">propterea interualla ſectionũ ad vtraſq; </s>
            <s xml:id="echoid-s8843" xml:space="preserve">partes maximæ diſtã-
              <lb/>
            tiæ ſucceſſiuè diminuuntur, & </s>
            <s xml:id="echoid-s8844" xml:space="preserve">ad partes æquidiſtantiũ asymptotorũ F M, R S </s>
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