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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9052Apollonij Pergæi in e M æquale eſt

67[Figure 67] rectangulo V e M,
alterius figuræ, & c.

Et ponamus re-
11d ctangulum F G cõ-
mune, & c. Scili-
cet, addatur in hy-
perbola, & aufera-
ratur in ellipſi com-
muniter rectangulis
F G.

Tandem proſe-
22e quamur ſuperiorẽ
demonſtrationem,
vt oſtendatur veri-
tas reliquarũ pro-
poſitionum, & c.

Demonſtratio ab
Apollonio breuitatis
gratia neglecta ſic
perficietur.

Quoniam rectã-
gulum E M æquale
eſt rectangulo V e
M, igitur vt E K ad
V e, ſeu K γ ad γ e
(propter ſimilitudinem triangulorum E K γ, & V e γ) ita erit e M ad M K,
& componendo, eadem e K habebit ad e γ, atque ad M K eandem proportionem,
ideoque e γ æqualis eſt M K; quare E I ad K M, ſeu I C ad C S eandem pro-
portionem habebit, quàm E I ad e γ, ſeu quàm I X ad e V (propter ſimilitudi-
nem triangulorum I E X, & e γ V) quare comparando homologorum differentias
in hyperbola, & eorundem ſummas in ellipſi C X ad c V, vel C Z ad Z c (propter
33Lem. 3. ſimilitudinem triangulorum C Z X, c Z V) habebit eandem proportionem, quàm I
C ad C S, vel C D ad D F, & diuidendo in hyperbola, & componendo in ellipſi C c
ad c Z eandem proportionem habebit, quàm C F ad F D, ſeu quàm habet latus
449. 10.
huius. tranſuerſum ad rectum, & propterea recta linea V Z eſt breuiſsima omnium,
quæ ex V ad axim A D duci poſſunt.

Iiſdem prorſus verbis oſtenſum erit, quod recta linea l m ſit breuiſsima om-
nium cadentium ex puncto l ad axim, ſi nimirum apponãtur caracteres prioris
caſus, vt patet in ſecunda, & quarta figura.

Iiſdem poſitis oſtendendum eſt, ramum B E, interceptum inter duos breuiſe-
cantes E V, non eſſe breuiſecantem, atque lineam breuiſsimam ex B ad axim
A D extenſam cadere ſupra ramum B E verſus verticem A.

Quoniam rectangulum B G M maius eſt rectangulo O G M, atque oſtenſum ſuit
rectangulum E M æquale rectangulo O G M; ergo rectangulum B G M maius eſt
rectangulo E M, & propterea E K ad B G, ſeu K R ad R G (propter ſimilitudi-
55Lem. 5. nem triangulorum) minorem proportionem habet, quàm G M ad M K, &

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