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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8850Apollonij Pergæi K G eandem propor-
tionem habebit ad R

65[Figure 65] G, atque ad M K,
vnde R G æqualis e-
rit M K, vel F D,
quare eadem E I ad
K M, vel C D ad
D F, ſiue I C ad C
S eandem proportio-
nem habebit, quam
eadem E I ad R G,
vel I T ad B G (pro-
pter ſimilitudinem
triangulorum I E T,
& G R B) ergo com-
parando homologo-
rum ſummas in elli-
pſi, vel differentias
11Lem. 4. in hyperbola C T ad
B O, vel C H ad H
O (propter ſimilitu-
dinem triangulorum
C H T, & O H B)
eandem proportionẽ
habebit, quàm I C
ad C S, vel C D ad
D F, & diuidendo
in hyperbola, & cõ-
ponendo in ellipſi C O ad O H eandem proportionem habebit, quàm C F ad F D,
ſiue quàm habet latus tranſuerſum ad rectum; & propterea B H eſt breuiſsima
229. 10.
huius. linearum ex B ad axim cadentium.

Deinde educatur quilibet ramus E V ſupra, velinfr a breuiſecantem E B, qui
productus ſecet rectam I C in X, & C A in Z, atque S M in γ, & educatur ex
V recta V e perpendicularis ad axim, ſecans D F in c, & S M in e, atque
contingentem ſectionem in puncto B, ſcilicet ipſam B a ſecet in d. Et quia (vt
modo oſtenſum eſt) rectangulum F S æquale eſt rectangulo B G M, ſuntque pa-
riter oſtenſæ O C, A C, C a proportionales; ergo C a eſt quinta proportionalis poſt
quatuor præcedentes F C, N C, O C, A C continuè proportionales; & ideo F C ad
C O eſt, vt C O ad C a; ergo comparando homologorum differentias tam in hyper-
33Lem. 3. bola, quàm in ellipſi erit, F O ad O a, vt F C ad C O: eſt autem G B ad B O,
vt F C ad C O, vt antea oſtenſum eſt; ergo G B ad B O erit, vt F O ad O a; ſed
propter ſimilitudinem triangulorum B G b, B O a eſt G B ad B O, vt G b ad O a;
ergo F O, ſeu M G ad O a eandem proportionem habet, quàm G b ad eandem O a;
& propterea M G æqualis eſt G b; cumque M b ſecetur æqualiter in G, & inæqua-
liter in e (ex lemmate 6. huius) G b ad e b, ſeu B G, ad d e, propter ſimilitu-
dinem triangulorum B G b, & B O a, & multo magis B G ad V e portionem
ipſius d e habebit maiorem proportionem, quàm, e M ad G M; ergo

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