•7 
372 
PAPPUS OF ALEXANDRIA 
k There is, says Pappus, on record an ancient proposition to 
the following effect. Let successive circles be inscribed in the 
dp(3r]Xos touching the semicircle^ and one another as shown 
in the figure on p. 376, their centres being A, P, 0 .... Then, if 
Pi> Pv Pz ••• be the perpendiculars from the centres A, P, 0... 
on BG and d 1 , d 2 , d ?j ... the diameters of the corresponding 
circles, 
Pi — d-y, = ^d%> Pz — 3d 3 .... 
He begins by some lemmas, the course of» which I shall 
reproduce as shortly as I can. 
I. If (Fig. 1) two circles with centres A, G of which the 
former is the greater touch externally at B, and another circle 
with centre G touches the two circles at K, L respectively, 
then KL produced cuts the circle BL again in D and meets 
AG produced in a point E such that AB :BG = AE: EG. 
This is easily proved, because the circular segments DL, LK 
are similar, and GD is parallel to AG. Therefore 
AB : BC = AK : GD = AE-.EC. 
Also KE. EL = EB 2 . 
For AE: EG = AB : BG = AB: GF = (.AE-AB): {EG- GF) 
• = BE: EF. 
But AE: EG = KE: ED; therefore KE; ED = BE: EF. 
Therefore KE. EL : EL . ED = BE 2, : BE. EF. 
And EL . ED — BE. EF; therefore KE. EL = EB 2 .