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

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Secentur diametrorum abſciſſæ G B, & H E in ijſdem rationibus in L, M,
N, O, & ab ijſdem punctis educantur baſibus æquiſtantes, ſeu ad diametros or-
dinatim applicatæ P Q, R S, T V, X Y [?] . Quoniam ex hypotheſi G B ad B I
eſt, vt H E ad E K; eſtque A G media proportionalis inter G B, & B I; pari-
terque D H media proportionalis eſt inter H E, & E K; igitur A G ad G B
eſt, vt D H ad H E; Et quoniam inuertendo L B ad B G eſt, vt N E ad E H,
atque B G ad B I poſita fuit, vt H E ad E K; ergo ex æquali ordinata L B ad
B I erit, vt N E ad E K, quare vt L B ad P L, mediã proportionalẽ inter L B,
& I B, ita erit N E ad N T mediam proportionalem inter N E, & E K. Eo-
dem modo oſtendetur, quod R M ad M B eandem proportionem habet, quàm X
O ad O E: & hoc ſemper continget in quibuslibet alijs diuiſionibus proportiona-
libus abſciſſarum, ſuntque anguli G, & H æquales; igitur ſegmenta A B C, & D E F ſimilia ſunt inter ſe. Quod erat oſtendendum.

216.1.

II. lib. I.
Defin. 7.
huius.

217. LEMMA VII.

S I in duobus ſegmentis A B C, & D E F hyperbolicis, aut ellipti-
cis, baſes A C, & D F cum diametris G B, & H E, æquales
angulos G, & H obliquos continentes, efficiant abſciſſas G B, & H E
proportionales lateribus rectis B I, & E K, atque tranſuerſis B Z, & E a, erunt ſegmenta ſimilia inter ſe.

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