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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196158Apollonij Pergæi

MONITVM.

IN principio huius libri monuimus, definitionem ſimilium conicarum
ſectionum, quæ circunfertur, vitioſam eſſe; quod hic oſtendendum
ſuſcepimus: ſed prius hæc demonſtranda ſunt.

LEMMA II.

IN duabus coniſectionibus A B, E F eiuſdem nominis ſint axium
figuræ G B D, K F I ſimiles inter ſe, ideſt tranſuerſa latera G B,
K F proportionalia ſint lateribus rectis B D, F I : duci debent in ſingu-
lis ſectionibus ſeries applicatarum ad axes, ita vt axium abſciſſæ (quæ
proportionales ſunt inter ſe) ad conterminas potentiales non ſint in ĳſdem
rationibus.

Sumantur duæ abſciſſæ B C, F H, quarum C B ad B D habeat maiorem pro-
portionem, quàm habet H F ad F I, & C B, H F ſecentur proportionaliter in
R, V. , & per ea puncta ducantur ad axes ordinatim applicatæ A C, E H, Q
R, T V. Quoniam quadratum A C ad rectangulum G C B eandem proportio-

214[Figure 214] nem babet, quàm latus rectum D B ad tranſuerſum G B, pariterq; quadratum
1121. lib. 5. E H ad rectangulum K H F eſt vt I F ad F K; atq; D B ad B G ex hypotheſi,
eſt vt I F ad F K; ergo quadratum A C ad rectangulum G C B eandem pro-
portionem habet quàm quadratum E H ad rectangulum K H F : & quia G B
ad B D eſt vt K F ad F I, & D B ad B C minorem proportionẽ habet quàm
I F ad F H, ergo ex æquali G B ad B C, minorem proportionem habet quàm
K F ad F H, & componendo in hyperbola, & diuidendo in ellipſi G C ad C B
ſeu rectangulum G C B ad quadratum B C minorem proportionẽ habebit quàm
K H ad H F, ſeu quàm rectangulum K H F ad quadratum F H : erat autem
quadratum A C ad rectangulum G C B vt quadratum E H ad rectangulum K
H F ; igitur ex æquali, quadratum A C, ad quadratum C B minorem propor-
tionem habet quàm quaàratum E H ad quadratum H F, & ideo A C ad C

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