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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            <s xml:id="echoid-s8617" xml:space="preserve">
              <pb o="231" file="0269" n="269" rhead="Conicor. Lib. VI."/>
            ductionem K E, & </s>
            <s xml:id="echoid-s8618" xml:space="preserve">ab E ducatur A E B parallela 1 H, quæ ſecet G H in B:
              <lb/>
            </s>
            <s xml:id="echoid-s8619" xml:space="preserve">poſtea producatur H K, vt cumq; </s>
            <s xml:id="echoid-s8620" xml:space="preserve">in I, & </s>
            <s xml:id="echoid-s8621" xml:space="preserve">per I ducatur A 1 D parallela E G,
              <lb/>
            quæ ſecet B G in D; </s>
            <s xml:id="echoid-s8622" xml:space="preserve">& </s>
            <s xml:id="echoid-s8623" xml:space="preserve">in plano B X D C, diametris B G, B D, fiant duo
              <lb/>
            circuli, qui ſint baſes duorum conorum, quorum vertices A, & </s>
            <s xml:id="echoid-s8624" xml:space="preserve">E, & </s>
            <s xml:id="echoid-s8625" xml:space="preserve">in eo-
              <lb/>
            rum ſuperficiebus planum per X I C ductum, efficiat ſectiones C I X, & </s>
            <s xml:id="echoid-s8626" xml:space="preserve">F K
              <lb/>
            T. </s>
            <s xml:id="echoid-s8627" xml:space="preserve">Dico eas eße parabolas quæſitas. </s>
            <s xml:id="echoid-s8628" xml:space="preserve">Quoniam recta E G facta eſt parallela. </s>
            <s xml:id="echoid-s8629" xml:space="preserve">
              <lb/>
            ipſi A D; </s>
            <s xml:id="echoid-s8630" xml:space="preserve">igitur duo triangula A B D, & </s>
            <s xml:id="echoid-s8631" xml:space="preserve">E B G per axes conorum ducta ſi-
              <lb/>
            milia, & </s>
            <s xml:id="echoid-s8632" xml:space="preserve">ſimiliter poſita in eodem ſunt plano; </s>
            <s xml:id="echoid-s8633" xml:space="preserve">& </s>
            <s xml:id="echoid-s8634" xml:space="preserve">duo circuli baſium in eodem
              <lb/>
            ſunt plano; </s>
            <s xml:id="echoid-s8635" xml:space="preserve">ergo coni A B D, & </s>
            <s xml:id="echoid-s8636" xml:space="preserve">E B G ſimiles erunt: </s>
            <s xml:id="echoid-s8637" xml:space="preserve">poſtea quia triangula. </s>
            <s xml:id="echoid-s8638" xml:space="preserve">
              <lb/>
              <note position="right" xlink:label="note-0269-01" xlink:href="note-0269-01a" xml:space="preserve">Lem. 9.
                <lb/>
              huius.</note>
            A B D, & </s>
            <s xml:id="echoid-s8639" xml:space="preserve">E B G ſimilia ſunt, & </s>
            <s xml:id="echoid-s8640" xml:space="preserve">I K H communis diameter ſectionum ad
              <lb/>
            coincidentes baſes C X, F T æque inclinata, & </s>
            <s xml:id="echoid-s8641" xml:space="preserve">recta linea A E B à verticibus
              <lb/>
            conorum ducta parallelæ ſunt inter ſe, atque intercipiunt in angulis æqualibus
              <lb/>
            A B H, & </s>
            <s xml:id="echoid-s8642" xml:space="preserve">E B H communem portionem B H baſium triangulorum ſimilium.
              <lb/>
            </s>
            <s xml:id="echoid-s8643" xml:space="preserve">per axes; </s>
            <s xml:id="echoid-s8644" xml:space="preserve">ergo parabolæ C I X, & </s>
            <s xml:id="echoid-s8645" xml:space="preserve">F K T æquales ſunt inter ſe. </s>
            <s xml:id="echoid-s8646" xml:space="preserve">Secundò, quia
              <lb/>
              <note position="right" xlink:label="note-0269-02" xlink:href="note-0269-02a" xml:space="preserve">Prop. 10.
                <lb/>
              addit.</note>
            propter parallelas E B, K H ſunt triangula E B G, H K G ſimilia; </s>
            <s xml:id="echoid-s8647" xml:space="preserve">ergo qua-
              <lb/>
            dratum B G ad rectangulum B E G ſcilicet latus rectum parabolæ F K T ad K
              <lb/>
              <note position="right" xlink:label="note-0269-03" xlink:href="note-0269-03a" xml:space="preserve">11. lib. 1.</note>
            E eſt, vt quadratum H G ad rectangulum H K G, ſed latus rectum parabolæ
              <lb/>
            Z ad K E fuit vt qtadratum H G ad rectangulum H K G; </s>
            <s xml:id="echoid-s8648" xml:space="preserve">igitur duo latera
              <lb/>
            recta, parabole Z, atq; </s>
            <s xml:id="echoid-s8649" xml:space="preserve">parabole F K T ad eandem K E habent eandem pro-
              <lb/>
            portionem, & </s>
            <s xml:id="echoid-s8650" xml:space="preserve">propterea æqualia ſunt, & </s>
            <s xml:id="echoid-s8651" xml:space="preserve">diametri, ad baſes æque inclinatæ
              <lb/>
            ſunt ex conſtructione; </s>
            <s xml:id="echoid-s8652" xml:space="preserve">igitur parabole F K T, & </s>
            <s xml:id="echoid-s8653" xml:space="preserve">ei æqualis C I X erit æqua-
              <lb/>
              <note position="right" xlink:label="note-0269-04" xlink:href="note-0269-04a" xml:space="preserve">Prop. 10.
                <lb/>
              huius.</note>
            lis eidem parabolæ Z. </s>
            <s xml:id="echoid-s8654" xml:space="preserve">Tertiò quia ſectionum plano, & </s>
            <s xml:id="echoid-s8655" xml:space="preserve">communi diametro I
              <lb/>
            K H æquidiſtat cummune lateris A E B, in quo duo coni ſe ſe contingunt; </s>
            <s xml:id="echoid-s8656" xml:space="preserve">ergo
              <lb/>
            latus A E B nunquàm occurret plano C I X: </s>
            <s xml:id="echoid-s8657" xml:space="preserve">ſed duæ ſuperficies conicæ tantum-
              <lb/>
            modò ſe ſe tangunt in latere A E B, & </s>
            <s xml:id="echoid-s8658" xml:space="preserve">reliquis omnibus in locis ſeparatæ ſunt;
              <lb/>
            </s>
            <s xml:id="echoid-s8659" xml:space="preserve">igitur duæ parabolæ C I X, F K T in illo plano poſitæ per contactum A E B
              <lb/>
            non tranſeunte, & </s>
            <s xml:id="echoid-s8660" xml:space="preserve">extenſæ in duabus conicis ſuperficiebus nunquàm conuenien-
              <lb/>
            tibus, erunt asymptoticæ. </s>
            <s xml:id="echoid-s8661" xml:space="preserve">Quartò quia duæ parabole C I X, F K T æquales
              <lb/>
            ſunt, & </s>
            <s xml:id="echoid-s8662" xml:space="preserve">ſimiliter poſitæ circa communem diametrum I K H; </s>
            <s xml:id="echoid-s8663" xml:space="preserve">ergo earum di-
              <lb/>
              <note position="right" xlink:label="note-0269-05" xlink:href="note-0269-05a" xml:space="preserve">Propof. 7.
                <lb/>
              addit.</note>
            ſtantiæ ſemper magis, ac magis diminuuntur quouſque ſint minores qualibet
              <lb/>
            recta linea data. </s>
            <s xml:id="echoid-s8664" xml:space="preserve">Quod erat faciendum.</s>
            <s xml:id="echoid-s8665" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s8666" xml:space="preserve">Data hyperbola Z duos conos ſimiles exhibere, vt idem planum in,
              <lb/>
              <note position="right" xlink:label="note-0269-06" xlink:href="note-0269-06a" xml:space="preserve">PRO 1.
                <lb/>
              12.
                <lb/>
              Addit</note>
            eis efſiciat duas hyperbolas æquales, & </s>
            <s xml:id="echoid-s8667" xml:space="preserve">ſimiles datæ, quæ aſymptoticæ
              <lb/>
            ſint, & </s>
            <s xml:id="echoid-s8668" xml:space="preserve">ſibi ipſis ſemper viciniores fiant, non tamen interuallo minore
              <lb/>
            recta linea data.</s>
            <s xml:id="echoid-s8669" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s8670" xml:space="preserve">In quolibet plano fiat angulus H I M æqualis angulo inclinationis diametri,
              <lb/>
            & </s>
            <s xml:id="echoid-s8671" xml:space="preserve">baſis datæ hyperboles Z, & </s>
            <s xml:id="echoid-s8672" xml:space="preserve">per M I extenſo quolibet alio plano ducatur in
              <lb/>
            eo B I C perpendicularis ad M I K; </s>
            <s xml:id="echoid-s8673" xml:space="preserve">& </s>
            <s xml:id="echoid-s8674" xml:space="preserve">ſumpto quolibet puncto O in recta linea
              <lb/>
            I H producta, ducatur à puncto O in plano per O I B extenſo recta linea O A
              <lb/>
            parallela ipſi B I, & </s>
            <s xml:id="echoid-s8675" xml:space="preserve">ſecetur O A æqualis ſemiſſi potentis figuram ſectionis Z,
              <lb/>
            cuius rectum latus ad tranſuerſum eandem proportionem habeat quàm quadra-
              <lb/>
            tum A O ad quadratum O H; </s>
            <s xml:id="echoid-s8676" xml:space="preserve">atque à puncto A àucatur recta linea A D G
              <lb/>
            parallela ipſi H I, & </s>
            <s xml:id="echoid-s8677" xml:space="preserve">coniungatur A H, quæ ſecent rectam lineam G I in pun-
              <lb/>
            ctis G, & </s>
            <s xml:id="echoid-s8678" xml:space="preserve">C, & </s>
            <s xml:id="echoid-s8679" xml:space="preserve">ſectur recta linea G B æqualis G C iungaturq; </s>
            <s xml:id="echoid-s8680" xml:space="preserve">A B, & </s>
            <s xml:id="echoid-s8681" xml:space="preserve">à
              <lb/>
            quolibet puncto D in recta A G ſumpto ducãtur in eodem plano A B C duæ re-
              <lb/>
            ctæ lineæ D E, & </s>
            <s xml:id="echoid-s8682" xml:space="preserve">D F @ parallelæ lateribus A B, & </s>
            <s xml:id="echoid-s8683" xml:space="preserve">A C; </s>
            <s xml:id="echoid-s8684" xml:space="preserve">eruntque </s>
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