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

Apollonij Pergæi Quoniam O L ad L I oſtenſa fuit, vt Q M ad M K; atque prius oſtenſa ſuit
B L ad L I, vt E M ad M K; ergo inuertendo I L ad L B erit, vt K M ad
M E; & propterea ex æqualitate O L ad L B erit vt Q M ad M E; ſed B L
ad L N eſt, vt E M ad M P; igitur ex æqualitate O L ad L N erit vt Q M
ad M P; ſuntque duo anguli L, & M recti; ergo triangula O L N, & Q M P
æquiangula erunt; & propterea anguli O, & Qæquales inter ſe erunt; ſed quia
contingentes verticales V e, & γ f parallelæ ſunt or dinatim applicatis N A, P
D ad diametros V R, & γ S; igitur angulus V e B æqualis erit angulo N O L; pariterque angulus γ f E æqualis erit angulo P Q M; & propterea anguli e, & f æquales erunt inter ſe.

229.1.

c
ex 12.
huius.

Ergo propter ſimilitudinem duarum ſectionum T Z in Z e ad quadra-
tum Z V eandem proportionem habebit quàm X a in a f ad quadratum
a Y, & angulus e æqualis eſt angulo f; igitur V e T ſimile eſt Y f X,
& pariter, & c. Quoniam in ſectionibus ſimilibus V B, & γ E axes tranſuerſi
lateribus rectis proportionales ſunt, & ductæ ſunt ad axes ordinatim applicatæ
V Z, γ a, & contingentes V e, γ f, eſtque rectangulum T Z e ad quadratum
Z V, vt latus tranſuerſum ad rectum, pariterque rectangulum X a f ad qua-
dratum a γ, vt axis tranſuerſus ad erectum; igitur rectangulũ T Z e adqua-
dratum Z V eandem proportionem habet, quàm rectangulum X a f ad quadra-
tum a γ, & à verticibus V, γ duorum triangulorum V e T, & γ f X ductæ
ſunt ad baſes rectæ linæ V Z, γ a efficientes angulos rectos, cum ordinatim
applicatæ ſint ad axes; atque angulus V e Z oſtenſus eſt æqualis angulo γ f a,
igitur tertius angulus Z V e æqualis erit tertio angulo a γ f; & ideo duo triã-
gula V T e, & γ X f ſimilia erunt inter ſe: & propterea circa angulos æquales
T, & X latus e T ad T V eandem proportionem habebit, quàm f X ad X γ: cumque duæ contingentes verticales V e, γ f parallelæ ſint ordinatim applicatis
N A, & P D ad diametros V R, γ S, erit O e ad R V, vt e T ad T V; pa-
riterque Q f ad S γ erit, vt f X ad X r: erat autem e T ad T V, vt f X ad
X γ; igitur pariter O e ad R V eandem proportionem habebit, quàm Q f ad
S γ; ſed B I ad I A eſt, vt E K ad K D.

229.1.

d
12. huius.
37. lib. 1.
0242-01
Propof. 6
præmiſſ.
12. huius.
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