A history of Greek mathematics - Wilbourhall.org

436 PAPPUS OF ALEXANDRIA
to this diameter. Then R is determined by means of the
relation
in this way.
RG.GD.BG.GA = RE.EB:FE.EE (1)
Join DB, RA, meeting EF in K, L respectively.
Then, by similar triangles,
RG.GD.BG.GA = (RE : EL) . (BE : EK)
= RE.EB:KE.EL
Therefore, by ( 1
), FE.EE = KE . EL,
whence EL is determined, and therefore L. The intersection
of AL, BE determines R.
Next, in order to find the extremities P, P /
through V, W, we draw ED, RF meeting FP /
in M, JS T respectively.
Then, as before,
of the diameter
FW. WE:P'W. WP = FE.EE-.RE.ED, by the ellipse,
Therefore
= FW.WE-.NW.WM, by similar triangles.
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 f W.WP = VW.WQ'\ (13)
Q,
Q' are thus known, while P, P' remain to be found.
By (a)
P'V: VW= QV: VP,
W whence P' : VW= PQ : P V.
Therefore, by means of (/?),
PQ:PV=Q / W:WP,