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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197159Conicor. Lib. VI. minorem proportionem habebit, quàm E H ad H F. Poſtea quia C B ad B R
erat vt H F ad F V, & prius G B ad B C minorẽ proportionem habebat, quàm
K F ad F H, ergo ex æquali G B ad B R minorem proportionem habet, quàm
K F ad F V, & componendo in hyperbola, & diuidenào in ellipſi G R ad R B,
ſeu rectangulum G R B ad quadratum B R minorem proportionem habet, quàm
K V ad V F, ſeu rectangulum K V F ad quadratum V F ; ſed propter ſimili-
tudinem figurarum, vt prius quadratum Q R ad rectangulum G R B eſt vt qua-
dratũ T V ad rectangulum K V F; ergo ex æquali quadratum Q R ad quadra-
tum R B minorem proportionem habet, quàm quadratum T V ad quadratum
V F, & Q R ad R B minorem proportionem habebit, quàm T V ad V F. Et
ſic reliquæ omnes abſciſſæ : quapropter patet propoſitum.

COROLLARIVM.

HInc conſtat in duabus ſimilibus coniſectionibus duci poſſe duas ſeries appli-
catarum ad axes, itaut abſciſſæ axium, quæ inter ſe proportionales ſunt,
ad ſuas potentiales nonſint in ĳſdem rationibus. Quandoquidẽ ex prima parte
propoſitionis 12. quotieſcunque axium figuræ ſimiles ſunt etiam ſectiones ipſæ
ſunt ſimiles.

LEMMA III.

IN ĳſdem figuris habeat G B ad B D maiorem proportionem, quàm
K F ad F I duci debent duæ ordinatim ad axes applicatæ, quæ ad
conterminas abſciſſas eandem proportionem habeant.

Ducatur quælibet ordinata E H, producanturq; vt ſecet coniunctam K I in
L, & vt D B ad B G ita fiat L H ad H N, atq; fiat G C ad B C, vt N H ad
H F, ducaturque ordinata A C; quæ producta ſecet coniunctam G D in P. Di-
co A C, & E H eße quæſitas. Quoniam quadratum A C ad rectangulum G C
1121. lib. 1. B eandem proportionem habet, quàm D B ad B G, ſeu L H ad H N, & rectã.
gulum G C B ad quadratum B C eſt

215[Figure 215] vt G C ad C B, ſeu vt N H ad H
F, ergo ex æqualitate quadratum
A C ad quadratum C B eſt vt L H
ad H F, ſeu vt rectangnlum L H
F ad quadratum H F; vel potius
vt quadratum E H ad quadratum
2212. 13.
lib. 1. H F; ideoque A C ad C B erit vt
E H ad H F. Quod erat propoſi-
tum.

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