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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216178Apollonij Pergæi

240[Figure 240] diximus in 11. ex 6.) quod ſi ad abſciſſas A M, C O egrediantur quælibet
potentes, ad ſua abſciſſa eandẽ proportionẽ habebunt ſi abſciſſæ ad abſciſ-
ſas ſint in cadem proportione, & quod anguli à potentialibus, & ab-
11Defin. 7.
huius. ſciſſis contenti, erunt æquales in duabus ſectionibus: quare erit ſegmen-
tum H A G ſimile ſegmento I C K atque ſimiliter poſitum.

Deinde ijſdem ſignis in eiſdem figuris manẽtibus, vt prius de-
ſignatis ſupponatur, ſegmentum H A G ſimile ipſi K C I. Dico,
quod angulus E æqualis erit F, & M A ad A E erit, vt O C ad
C F.

Quoniam duo ſegmenta ſunt ſimilia erit angulus O æqualis M, & duo
22Defin. 7. anguli E A L, F C N illis æquales, ſunt quoque inter ſe æquales; ergo
duo anguli F, E, qui illis æquales ſunt, erunt inter ſe æquales, eoquod
A E, C F parallelæ ſunt G H, I K, & anguli N, L ſunt recti; ergo duo
triangula proportionis ſunt ſimilia, ideoque R A ad A L, nempe P A ad
3349. lib. 1.
11. lib. 1. duplam A E eſt, vt C S ad C N, nempe Q C ad duplam C F: & quia
G M poteſt P A in A M (12. ex 1.) & ſimiliter I O poteſt Q C in C O;
44b ergo P A ad G M eſt, vt Q C ad O I, & G M ad M A eſt, vt I O ad
O C; quia duo ſegmenta ſunt ſimilia, & E A ad A M eſt, vt C F ad C
O: & iam oſtenſum eſt, quod duo anguli E, F ſunt æquales. Et hoc erat
oſtendendum.

PROPOSITIO XVII.

DEinde ſectiones ſint hyperbolicæ, aut ellipticæ, & reliqua
55a ſupponantur, vt prius.

Educamus C c perpendicularẽ ſuper axim D F, & A a perpendicula-
rem ſuper axim B E; atque V, Y ſint duo centra. Ergo (propter ſimi-
litudinem duarum ſectionum) erit V a in a E ad quadratum A a

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