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="229" file="0267" n="267" rhead="Conicor. Lib. VI."/>
            ad L S eſt vt quadratum D Z ad quadratum Z F ; </s>
            <s xml:id="echoid-s8558" xml:space="preserve">igitur P G ad G R ean-
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            dem proportionem habet, quàm Z L ad L S, & </s>
            <s xml:id="echoid-s8559" xml:space="preserve">propterea figuræ ſectionem
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              <note position="right" xlink:label="note-0267-01" xlink:href="note-0267-01a" xml:space="preserve">ex 12.
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              huius.</note>
            erunt ſimiles; </s>
            <s xml:id="echoid-s8560" xml:space="preserve">ijs autẽ figuris æqualia oſtenſa ſunt quadrata dupliciũ O Y, & </s>
            <s xml:id="echoid-s8561" xml:space="preserve">K
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            Z, quæ ſuppoſitæ fuerunt æquales; </s>
            <s xml:id="echoid-s8562" xml:space="preserve">igitur figuræ P G R, & </s>
            <s xml:id="echoid-s8563" xml:space="preserve">Z L S ſimiles, & </s>
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            æquales ſunt inter ſe, atque diametri æquæ inclinatæ ſunt ad ordinatim ad eas
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            applicatas H I, M N; </s>
            <s xml:id="echoid-s8565" xml:space="preserve">igitur ſectiones H G I, M L N æquales ſunt inter ſe,
              <lb/>
              <note position="right" xlink:label="note-0267-02" xlink:href="note-0267-02a" xml:space="preserve">Prop. 10.
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              huius.</note>
            ſimiles, & </s>
            <s xml:id="echoid-s8566" xml:space="preserve">congruentes, quarum figuræ æquales ſunt quadratis duplicium inter-
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            ceptarum O Y, & </s>
            <s xml:id="echoid-s8567" xml:space="preserve">K Z, quod erat propoſitum.</s>
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        <div xml:id="echoid-div749" type="section" level="1" n="235">
          <head xml:id="echoid-head296" xml:space="preserve">LEMMA IX.</head>
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            <s xml:id="echoid-s8569" xml:space="preserve">S I in duobus conis A B C, D E F, baſes ſint in eodem plano, & </s>
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            duo triangula per axes A B C, D E F fuerint ſimilia, & </s>
            <s xml:id="echoid-s8571" xml:space="preserve">ſimi-
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            liter poſita, & </s>
            <s xml:id="echoid-s8572" xml:space="preserve">in eodem plano exiſtentia, erunt coni ſimiles inter ſe.</s>
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            <s xml:id="echoid-s8574" xml:space="preserve">Ducantur à verticibus A, & </s>
            <s xml:id="echoid-s8575" xml:space="preserve">D duæ rectæ A G, & </s>
            <s xml:id="echoid-s8576" xml:space="preserve">D H perpendiculares ad
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            baſes conorũ, & </s>
            <s xml:id="echoid-s8577" xml:space="preserve">à terminis axium A Y, & </s>
            <s xml:id="echoid-s8578" xml:space="preserve">D Z coniungantur rectæ lineæ Y G,
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            & </s>
            <s xml:id="echoid-s8579" xml:space="preserve">Z H. </s>
            <s xml:id="echoid-s8580" xml:space="preserve">Quoniã planum, in quo exiſtunt duo triangula A B C, D E F ſecat
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            planum, in quo baſes conorum iacent in vna recta linea, quæ baſis eſt vtriuſque
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            trianguli per axes conorum ducti; </s>
            <s xml:id="echoid-s8581" xml:space="preserve">ideoque B C, & </s>
            <s xml:id="echoid-s8582" xml:space="preserve">E F in directum conſtitutæ
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            erunt, & </s>
            <s xml:id="echoid-s8583" xml:space="preserve">circa angulos æquales B, & </s>
            <s xml:id="echoid-s8584" xml:space="preserve">E latera A B ad B C, atque D E ad E
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            F ſunt proportionalia ( propter triangulorum A B C, & </s>
            <s xml:id="echoid-s8585" xml:space="preserve">D E F ſimilitudinem)
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            erunt quoque ad conſequẽtium ſemiſſes proportionales, ſcilicet A B ad B Y erit,
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            vt D E ad E Z circa angulos æquales, & </s>
            <s xml:id="echoid-s8586" xml:space="preserve">propterea triangula A B Y, & </s>
            <s xml:id="echoid-s8587" xml:space="preserve">D E
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            Z ſimilia erunt: </s>
            <s xml:id="echoid-s8588" xml:space="preserve">& </s>
            <s xml:id="echoid-s8589" xml:space="preserve">ideò duo anguli B Y A, & </s>
            <s xml:id="echoid-s8590" xml:space="preserve">E Z D, externus interno, æqua-
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            les erunt inter ſe; </s>
            <s xml:id="echoid-s8591" xml:space="preserve">igitur Y A, & </s>
            <s xml:id="echoid-s8592" xml:space="preserve">Z D in eodem plano exiſtentes, parallelæ
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            erunt inter ſe; </s>
            <s xml:id="echoid-s8593" xml:space="preserve">ſunt quoque A G, D H inter ſe parallelæ ( cum ſint perpendicu-
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            lares ad idem planum baſium ) ergo duo anguli Y A G, & </s>
            <s xml:id="echoid-s8594" xml:space="preserve">Z D H æquales ſunt
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            inter ſe; </s>
            <s xml:id="echoid-s8595" xml:space="preserve">atquè anguli G, & </s>
            <s xml:id="echoid-s8596" xml:space="preserve">H æquales ſunt, nempe recti; </s>
            <s xml:id="echoid-s8597" xml:space="preserve">igitur in triangu-
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            lis A Y G, & </s>
            <s xml:id="echoid-s8598" xml:space="preserve">D Z H, duo poſtremi anguli A Y G, & </s>
            <s xml:id="echoid-s8599" xml:space="preserve">D Z H æquales </s>
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