# Full text: Pergaeus, Apollonius: Apollonii Pergaei Conicorvm Lib. V. VI. VII. paraphraste Abalphato Asphahanensi

Quoniam facta conuenienti ſuperpoſitione axis A M ſuper axim D
O, cadet quoque ſectio A B ſuper ſectionem D E: ſi enim non cadit ſu-
per illam, ſumatur (ſi fieri poteſt) eius punctum B, extra ſectionem. D E cadens; & producatur ad axim perpendicularis B L vſque ad P: & perficiatur planum A P applicatum comparatum; & ſecetur D N æqua-
lis A L, & erigatur per N ad axim perpendicularis N E, & producatur
vſque ad R, perficiendo planum D R applicatum comparatum; Et quia
A I æqualis eſt D K, & A L æqualis D N: erit planum I L, æquale pla-
no K N; cumque G I, H K ſint duæ figuræ ſimiles, & æquales, pariter-
que I P, K R; ergo duo plana A P, D R ſunt æqualia: & propterea E
N, B L, quæ illa ſpatia poſſunt (13. 14. ex 1.) ſunt æquales. Si autem
ſuperponatur axis axi cadet B L ſuper E N, eoquod duo anguli N, & L
ſunt æquales; igitur B cadit ſuper E, quod prius cadere non concedeba-
tur: & hoc eſt abſurdum. Quapropter ſectio ſectioni æqualis eſt.

### 170.1.

b
12. 13.
lib. I.

Deinde ponamus duas ſe-
ctiones æquales, vtique con-
gruet ſectio A B ſectioni D E,
& axis A L axi D N, quia ſi
in hyperbola duo axes, & in
ellipſi tres axes, quod eſt ab-
ſurdum (52. 53. ex 2.) Et fi-
at A L æqualis D N, & reli-
qua perficiantur, vt prius ca-
dent duo puncta L, B ſuper
N, E; ideoque B L æqualis
erit E N; & poterunt æqua-
lia rectangula A P, D R applicata ad æquales A L, D N (13. 14. ex 1.) ergo L P æqualis eſt N R. Similiter ponatur A M æqualis D O, & edu-
cantur C M Q, F O S duæ ordinationes, oſtendetur, quod M Q æqua-
lis eſt O S, & L M æqualis N O; & propterea duo plana P Q, R S ſunt
æqualia, & ſimilia; igitur duo plana G P, H R ſunt æqualia, & ſimilia,
& L P oſtenſa eſt æqualis N R: ergo G L æqualis eſt H N, & A L æ-
qualis D N; & propterea G A æqualis eſt D H, & A I æqualis D K.

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