436 
PAPPUS OF ALEXANDRIA 
to this diameter. Then R is determined by means of the 
relation 
RG.GD-.BG.GA = RH.HD-.FH.HE {!)' 
in this way. 
Join DB, BA, meeting EF in K, L respectively. 
Then, by similar triangles, 
RG.GD-.BG. G A = (RH : EL). {Dll : HK) 
= RH.HD.KH.HL. 
Therefore, by (1), FH. HE = KH.HL, 
* 
whence HL is determined, and therefore L. The intersection 
of AL, DH determines R. 
Next, in order to find the extremities P, P' of the diameter 
through V, W, we draw ED, RF meeting PP' in M, A 7 " respec 
tively. 
Then, as before, 
FW. WE: P'W. WP = FH. HE-.RH. HD, by the ellipse, 
= FW. WE : NW. WM, by similar triangles. 
Therefore P'W. WP = NW. WM- 
and similarly we can find the value of P'V. VP. 
Now, says Pappus, since P'W.WP and P'V.VP are given 
areas and the points V, W are given, P, P' are given. His 
determination of P, P' amounts (Prop. 14 following) to an 
elimination of one of the points and the finding of the other 
by means of an equation of the second degree. 
Take two points Q, Q' on the diameter such that 
P'V.VP=WV.VQ, (a) 
P'W.WP = VW.WQ' ] OS) 
Q, Q' are thus known, while P, P' remain to be found. 
By (a) P'V: VW= QV: VP, 
whence P'W:VW= PQ : PV. 
Therefore, by means of (/3), 
PQ-.PV= Q'W:WP,