Full text: Pergaeus, Apollonius: Apollonii Pergaei Conicorvm Lib. V. VI. VII. paraphraste Abalphato Asphahanensi

Archimedis pleantur parallelogramma rectangula A L, A K, L B, B K, atque axe
F G, latere recto F N deſcribatur parabole F M ſecans H G in M; erit
igitur in parabola quadratum M G æquale rectangulo G F N ſub abſciſ-
ſa, & latere recto contento, ideoque idem quadratum F G ad rectangu-
lum N F G, atque ad quadratum M G eandem proportionem habebit: eſt vero quadratum F G ad rectangulum N F G, vt F G ad F N, cum
F G ſit illorum altitudo communis, nec non vt C F G ad C F N ſum-
pta nimirum C F communi altitudine, ergo rectangulum C F G ad C
F N eandem proportionem habebit, quam quadratum F G ad quadra-
tum M G, & permutando rectangulum C F G ad quadratum F G erit
vt rectangulum C F N ad quadratum G M, ſed vt rectangulum C F G
ad quadratum F G, ita eſt C F ad F G, & E A ad A C, igitur E A ad
A C erit vt rectangulum C F N ad quadratum G M, ſeu vt quadratum
E B, vel K G ad quadratum G M: eſt vero A C minor, quàm A E,
quæ triens eſt totius A B, igitur M G minor eſt, quàm G K. Poſtea
per B circa aſymptotos A C F deſcribatur hyperbole B K, quæ tran-
ſibit per punctum K, cum parallelogramma A F, & C K æqualia
ſint propter diagonalem C E G, quare punctum M paraboles cadet
intra hyperbolem B K, ſed parabole F M occurrit aſymptoto C F in ver-
tice F, & occurrit etiam aſymptoto C A in aliquo alio puncto, cum C
A ſit parallela axi F G paraboles, & hyperbole ſemper intra aſymptotos
incedat, igitur parabola F M bis hyperbolæ occurrit ſupra, & inſra pun-
ctum M: ſint occurſus X, à quibus ductis parallelis ad aſymptotos com-
pleantur parallelogramma R P, & A F, quæ erunt æqualia inter aſym-
ptotos, & hyperbolen conſtituta, & propterea C O S parallelogrammo-
rum diameter erit, & vna linca recta: & quia O A ad A C eſt vt C F
ad F S, ſiue vt rectangulum C F N ad rectangulum S F N: erat autem
quadratum E B æquale rectangulo C F N ex conſtructione, & quadra-

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