A history of Greek mathematics - Wilbourhall.org

372 PAPPUS OF ALEXANDRIA
There is,
says Pappus, on record an ancient proposition to
the following effect. Let successive circles be inscribed in the
dpftrjXos touching the semicircles and one another as shown
in the figure on p. 376, their centres being A, P, ... . Then,
Pi* Vv Vz
if
••• be the perpendiculars from the centres A, P, ...
on BG and d lf
c£ 2
, d 3
... the diameters of the corresponding
circles,
p1
= d 1
, p
2
=2d 2
, p
3
= Bd B
....
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, C 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
AC produced in a point E such that AB :BG = AE : EG.
This is easily proved, because the circular segments DL, LK
are similar, and CD is parallel to AG. Therefore
AB:BC = AK:GD = AE: EC.
Also KE.EL = EB 2 .
For AE:EC=AB:BC = AB:CF= (AE- AB) :
= BE:EF.
(EC- CF)
But AE:EC= KE : ED
Therefore KE .
EL
:
;
EL
.
Fig 1.
therefore KE:ED = BE: EF.
ED
= BE* :
BE
.
EF.
And EL. ED = BE. EF; therefore KE. EL = EB 2 .