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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232194Apollonij Pergæi

262[Figure 262] ad E K, nempe P M ad D K potentia, nempe M Q ad Q K, & per con-
uerſionem rationis O L ad L I erit, vt Q M ad M K: eſtque I L ad L B,
vt K M ad M E; ergo O L ad L B eſt, vt Q M ad M E, & L B ad L N
eſt, vt E M ad M P (propter ſimilitudinem duarum ſectionum) ergo ex
11c22Defin. 2. æqualitate O L ad L N erit, vt Q M ad M P; ſuntque M, & L duo an-
guli recti; ergo N L O ſimile eſt P M Q; & per R, S ſemipartitiones ip-
ſarum N A, D P ducamus ipſas T V, X Y parallelas duobus axibus, &
ex duobus punctis V, Y, educamus perpendiculares V Z, Y a ſuper duos
axes. Et quia N O ad O A eſt, vt P Q ad Q D comparando antecedẽ-
tes ad ſemiſſes differentiarum terminorum vel ad ſemiſummas eorũ fiet N
33d O ad R O, nempe N L ad L T, quæ eſt æqualis ipſi V Z, nempe L B
ad B Z longitudine (19. ex 1.) vt P Q ad Q S, nempe P M ad X M æ-
qualem ipſi Y a, nempe longitudine, vt M E ad E a (19. ex 1) igitur
4420. lib. 1. comparando differentias terminorum ad antecedentes, erit Z L ad L B,
vt a M ad M E, & L B ad L O eſt, vt M E ad M Q; ergo ex æqualitate
L Z ad L O, nempe N b ad N O eſt, vt M a ad M Q, nempe P c ad P Q
55Ibidem. crat autem prius N R ad N O, vt S P ad P Q, & comparando ſemisũ-
66e mas, vel ſemidifferentias terminorum ad eorundem differentias O R ad
R b erit, vt Q S ad S c, & R b ad R V eſt, vt S c ad S Y; quia
duo triangula V R b, Y S c ſunt ſimilia; ergo R O ad R V eandem pro-
portionem habet, quàm Q S ad S Y; ſed tangens in V perueniens ad L O
77f æqualis eſt O R, cui parallela eſt; quia cadit inter duas lineas parallelas;
& ſimiliter tangens in Y parallela eſt S Q, & ei æqualis; ergo V R ab-
ſciſſa ad tangentem eſt, vt abſciſſa S Y ad eius tangentem, & angulus Q
æqualis eſt angulo O; igitur duo ſegmenta N V A, P Y D ſunt ſimilia
(16. ex 6.) & pariter duo ſegmenta A B C, D E F, atque duo ſegmen-
88g ta N B, P E ſunt ſimilia inter ſe, & ſimiliter poſita.

Deinde ponamus aliud ſegmentum P d. Dico non eſſe ſimile alicui
99h prædictorum ſegmentorum, quia non abſcinduntur à duabus ordinationi-
bus vnius axis (18. ex 6.) . Et hoc erat oſtendendum.

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