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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204166Apollonij Pergæi

Notæ in Propoſit. XII.

SVpponamus itaque ſectiones A B, E F, earum inclinati, vel tran-
11a ſuerſi B a, F b, & erecti eorum B D, F I ordinationes, & propoſitio-
nes, vti diximus, & c. Ideſt. Sint axes inclinati, ſiue tranſuerſi B a, F b, &
maneant ſigna, ordinationes, & proportiones eædem, quæ in præcedenti propoſi-
tione; ſcilicet fiat C B ad B D, vt H F ad F I, & quia D B ad B a eſt vt I
F ad F b ( propter ſimilitudinem figurarum D B a, I F b ) ergo ex æquali C
B ad B a erit vt H F ad F b; & comparando antecedentes ad ſummas termino-
rum in hyperbola, & ad differentias in ellipſi erit B C ad C a vt F H ad H b:
poſtea diuidantur tam B C, quàm F H in ĳſdem rationibus in punctis K, L,
M, N, & educantur ordinatim applicatæ, ſeu æquidiſtantes baſibus O P, Q R,
A S, T V, X r, E Z.

Quoniam figura ſectionis A B ſimilis eſt figuræ ſectionis E F erit qua-
22b dratum H E ad H b in H F, vt quadratum A C ad C a in C B, & b H
in H F ad quadratum H F, vt C a in C B ad quadratnm C B ( nam po-
ſuimus H F ad F b, vt C B ad B a, & c.) Quouiam in figuris, ſeu rectan-
gulis ſimilibus D B a, & I F b habet D B ad B a eandem proportionem, quàm
3321. lib. I. I F ad F b, & vt D B ad B a, ita eſt quadratum A C ad rectangulum B C a,
pariterque vt I F ad F b ita eſt quadratum E H ad rectangulũ F H b ſed ( ſi-
cut in præcedenti nota dictum eſt) C a ad C B, ſeu rectangulum B C a ad qua-
dratum C B eandem proportionem habet, quàm H b ad H F, ſeu quàm rectan-
gulum F H b ad quadratum F H; igitur ex æqualitate quadratum A C ad qua-
dratum C B eandem proportionem habet, quàm quadratum E H ad quadratum
H F.

Atque quadratum H F ad H F in H b eſt vt quadratum C B ad B C in
44C C a (eo quod H F ad F b poſita fuit C B ad B a), ergo ex æqualitate, & c.
Ideſt ſumã tur axium abſcißæ C B, H F, quæ ſint proportionales lateribus rectis
B D, & F I, ſeu proportionales ſint lateribus tranſuerſis B a, & F b, & ſecẽtur
abſciſſæ B C, & F H proportionaliter in punctis K, L, M, N, & per puncta
diuiſionum ducantur ordinatim applicatæ A C, Q L, E H, X N, & c. Quia ſe-
ctiones A B, E F ſupponuntur ſimiles; ergo ex definitione 2. huius A C ad C B
eandem proportionem habebit, quàm E H ad H F, nec non Q L ad L B erit vt
X N ad N F; & ideo quadratum A C ad quadratum C B eandem proportionẽ
habet, quàm quadratum E H ad quadratum H F; & quia ex conſtructione,
iuxta leges definitionis 2. vt C B ad B a ita erat H F ad F b, & comparando
antecedentes ad terminorũ ſummas in hyperbolis, & ad differentias in ellipſibus,
habebit B C ad C a, ſeu quadratum B C ad rectangulum B C a eandẽ propor-
tionem quàm F H habet ad H b, ſeu quàm quadratum F H habet ad rectangu-
lum F H b; ergo ex æqualitate quadratum A C ad rectangulum B C a eãdem
proportionem habet, quàm quadratum E H ad rectangulum F H b; eſt verò la-
tus rectum D B ad latus tranſuerſum B a, vt quadratum A C ad

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