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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382343Conicor. Lib. VII.
PROPOSITIO XXXXI.
IN ellipſi A B C ſit A C axis maior, & y O minor, & ſint P
Q, &
S T duæ aliæ diametri, ſitque A F erectus ipſius A
C, &
P R erectus ipſius P Q, & O f ipſius y O. Dico quod
C F minor eſt, quàm Q R, &
Q R, quàm T Z, & T Z,
quàm y f.
Ducantur A N, A X ordinatim applicatæ ad diametros P Q, S T,
&
duæ ad axim perpendiculares N M, X V, & interceptæ A G, C H.
Quia quadratum A C ad quadratum y O, nempe A C ad A F eandem,
11b proportionem habet, quàm C G ad G A, ſeu ad C H habebit quadra-
22Defin. 1.
huius.
tum C A ad quadratum C F ſummæ ipſius C A, eiuſque erecti eandem
proportionem, quàm quadratum C G, nempe C G in A H ad quadra-
tum G H:
& quadratum A C ad quadratum y O eandem proportionem,
habet, quàm G C in C H ad quadratum C H:
eſtquè quadratum y O ad
quadratum ſummæ y f, vt quadra-
452[Figure 452] tum C H ad quadratum H G;
er-
go quadratum A C ad quadratum
y f eſt, vt C G in C H minorem
ad quadratum H G;
ſed quadra-
tum A C ad quadratum C F ean-
dem proportionem habet, quàm.
G C in maiorem A H ad quadra-
tum G H;
igitur A C ad C F ma-
iorem proportionem habet, quàm
ad y f:
& propterea C F ſumma,
A C, &
erecti illius minor eſt,
quàm y f, quæ eſt ſumma y O, &

erecti illius.
Et quoniam C G in,
M H, quod minus eſt, quàm C G
in A H ad quadratum H G eandem
proportionem habet, quàm qua-
dratum A C ad quadratum Q R
ſummæ diametri, &
erecti ipſius
P Q (16.
ex 7.) quare quadratum
A C ad quadratnm C F maiorem
proportionem babebit, quàm ad
quadratum Q R, &
propterea C F
minor erit, quam Q R.
Et quoniam
C G in V H ad quadratum H G eſt vt quadratum A C ad quadratum
T Z ad quàm ordinatim applicatur A X (16.
ex 7.) erit C F minor quàm
T Z:
cumque C G in H M ad quadratum H G maiorem proportionem,
habeat, quàm G C in V H ad quadratum idipſum H G habebit

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