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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            igitur G A æqualis eſt L D: </s>
            <s xml:id="echoid-s8522" xml:space="preserve">& </s>
            <s xml:id="echoid-s8523" xml:space="preserve">quia in triangulis ſimilibus rectangulum B A
              <lb/>
            C ad quadratum B C, ſeu A G ad latus rectum G R eandem proportionem ha-
              <lb/>
              <note position="left" xlink:label="note-0266-01" xlink:href="note-0266-01a" xml:space="preserve">11. lib. 1.</note>
            bet; </s>
            <s xml:id="echoid-s8524" xml:space="preserve">quàm rectangulum E D F ad quadratum E F, ſeu quàm D L habet ad la-
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            tus rectum L S; </s>
            <s xml:id="echoid-s8525" xml:space="preserve">igitur A G ad G R erit vt D L ad L S; </s>
            <s xml:id="echoid-s8526" xml:space="preserve">ſuntq; </s>
            <s xml:id="echoid-s8527" xml:space="preserve">A G, D L
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            oſtenſæ æquales ergo G R, & </s>
            <s xml:id="echoid-s8528" xml:space="preserve">L S latera recta æqualia ſunt, & </s>
            <s xml:id="echoid-s8529" xml:space="preserve">diametri ſectio-
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            num eſſiciunt angulos G O H, L K M æquales; </s>
            <s xml:id="echoid-s8530" xml:space="preserve">ergo parabolæ H G I, & </s>
            <s xml:id="echoid-s8531" xml:space="preserve">M L N
              <lb/>
              <note position="left" xlink:label="note-0266-02" xlink:href="note-0266-02a" xml:space="preserve">Prop 10.
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              huius.</note>
            æquales ſunt inter ſe.</s>
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            <s xml:id="echoid-s8533" xml:space="preserve">In hyperbolis verò, quoniam P G parallela eſt axi A Y, & </s>
            <s xml:id="echoid-s8534" xml:space="preserve">A V parallela,
              <lb/>
            eſt baſi B C, & </s>
            <s xml:id="echoid-s8535" xml:space="preserve">latera P B, & </s>
            <s xml:id="echoid-s8536" xml:space="preserve">A C ſunt communia; </s>
            <s xml:id="echoid-s8537" xml:space="preserve">igitur P V ad V A eſt vt
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            A Y ad Y B, & </s>
            <s xml:id="echoid-s8538" xml:space="preserve">G V ad V A eſt vt Y A ad Y C: </s>
            <s xml:id="echoid-s8539" xml:space="preserve">habet verò eadem A Y ad
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            æquales Y B, Y C eandem rationem ergò P V, & </s>
            <s xml:id="echoid-s8540" xml:space="preserve">G V ad eandem V A habent
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            eandem proportionem, & </s>
            <s xml:id="echoid-s8541" xml:space="preserve">ideo P V æqualis eſt V G, atq; </s>
            <s xml:id="echoid-s8542" xml:space="preserve">punctum V erit cen-
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            trum ſectionis, & </s>
            <s xml:id="echoid-s8543" xml:space="preserve">quadratum A Y æquale erit quadrato V O (propter paral-
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            lelogrammum V Y), & </s>
            <s xml:id="echoid-s8544" xml:space="preserve">quadratum V O æquale eſt rectangulo P O G cum qua-
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            drato V G; </s>
            <s xml:id="echoid-s8545" xml:space="preserve">pariterque quadratum C Y æquale eſt rectangulo C O B cum qua
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            drato O Y, & </s>
            <s xml:id="echoid-s8546" xml:space="preserve">habet quadratum A Y ad quadratum C Y eandem proportionem,
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            quàm latus tranſuer ſum P G ad latus rectum G R, ſeu eandem, quàm habet
              <lb/>
              <note position="left" xlink:label="note-0266-03" xlink:href="note-0266-03a" xml:space="preserve">21. lib.1.</note>
            rectangulum P O G ad rectangulum C O B, ergo diuidendo quadratum V G ad
              <lb/>
            quadratũ O Y eandem proportionem habebit, quàm quadratum A Y ad quadratũ
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            Y C, ſeu vt P G ad G R, ſeu vt quadratum P G ad rectangulum P G R,
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            & </s>
            <s xml:id="echoid-s8547" xml:space="preserve">ideo quadratum duplæ V G, ſeu P G eandem proportionem habebit ad re-
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            ctangulum P G R, atq; </s>
            <s xml:id="echoid-s8548" xml:space="preserve">ad quadratum duplæ ipſius Y O; </s>
            <s xml:id="echoid-s8549" xml:space="preserve">quare quadratum duplæ
              <lb/>
            ipſius O Y æquale erit figuræ ſectionis ſeu rectangulo P G R. </s>
            <s xml:id="echoid-s8550" xml:space="preserve">Eodem modo
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            oſtendetur X centrum hyperbolæ M L N, & </s>
            <s xml:id="echoid-s8551" xml:space="preserve">quadratum L Z ad quadratum du-
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            ple K Z eſſe vt quadratum D Z ad quadratum Z F, ſeu vt Z L ad L S, & </s>
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            ideo quadratum duplæ ipſius K Z æquale erit figuræ ſectionis, ſeu rectangulo Z
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            L S. </s>
            <s xml:id="echoid-s8553" xml:space="preserve">Tandem, quia propter ſimilitudinem triangulorum per axes, ſunt anguli
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            C, F æquales, & </s>
            <s xml:id="echoid-s8554" xml:space="preserve">anguli Y, Z pariter æquales ( cum ex hypotheſi diametri G O,
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            L K parallelæ axibus AY, D Z efficiant angulos G O C, L K F æquales); </s>
            <s xml:id="echoid-s8555" xml:space="preserve">ergo
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            A Y ad Y C erit vt D Z ad Z F, & </s>
            <s xml:id="echoid-s8556" xml:space="preserve">earum quadrata etiam proportionalia
              <lb/>
            erunt; </s>
            <s xml:id="echoid-s8557" xml:space="preserve">ſed P G ad G R eſt vt quadratum A Y ad quadratum Y C, atque Z </s>
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