A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

124

124 ~A SEQUEL TO EUCLID).

Observation.-This method, though arrived at by the theory
of coaxal circles, is virtually the'same as Newton's 16th Lemma.
It is, however, somewhat simpler, as it does not employ conic sections, as is done in the -Principia. When I discovered it several
years ago, I was not aware to what an extent I had been anticipated.
Prop. 11. —ITf X, Y be two circles, AIB, A'B3' two chords
of X which are tangents to Y;A
then if the perpendiculars fromA
A, A' on the radical axis be de- g
noted by p. w, and the perpendiculars from 13, 13' by p',.7'
Y
AA':BB'::V'p+ VirV
Dem.-Let 0, 0' be the centres of the circles; then, by (1),
Prop. 1,
AD =V2. 00'..p, AD'=jV2.00'.'
AD +A'D' =/V2.00/ +~Vhr}.
But AD + A'D' is easily seen to be =AC + A'C;
AC + A'C = V2.00'f/pV w}.
In like manner,
BC + BI'C =1V2.00' ( VP+ V;-'
Hence,
AC + A'C: BC + BC::Ip ~ v'7r V '+ V;P.
Now, since the triangles AA'C, BB'C are equiangular,
we have
AC + A'C:13C ~ B'C:: AA': BB';

This theorem is very important, besides leading to an
immediate proof of.Poncelet'8T'iiheore~n. If we suppose