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.

BOOK VI.

99

such that the tangents drawn from it to the two Os
X, Y will be in the ratio of the square roots of their
radii, and if X, Y, be inverted from that point, their
inverses will be equal. It will be seen, in the next
Section, that the locus of 0 is a circle coaxal with X
and Y.
Cor. 1.-Any three circles can be inverted into three
equal circles.
Cor. 2.-Hence can be inferred a method of describing a circle to touch any three circles.
Cor. 3.-If any two circles be the inverses of two
others, then any circle touching three out of the four
circles will also touch the fourth.
Cor. 4.-If any two points be the inverses of two
other points, the four points are concyclic.
Prop. 6.-If A and B be any two points, 0 a centre
of inversion; and if the inverses of A, B be the points
A', B', andp, p', the perpendiculars from 0 on the lines
AB, A'B'; then AB: A'B':: p: p.
Dem.-Since 0 is the centre of inversion, we have
OA. OA'= OB. OB';
therefore OA OB:: OB': OA'.
And the angle 0 is common to the two As AOB, A'OB';.'. the As are equiangular. Hence the Proposition is
proved.
Prop. 7. —f A, B, 0... L be any number of collinear points, we have
AB + B + CD. + LA = O.
(Since LA is measured backwards, it is regarded as
negative.) Now, let p be the.L from any point 0 on
the line AL; and, dividing by p, we have
AB 3BC CD LA
+ p - + -. - = 0.
P P P i
Let the whole be inverted from 0; and, denoting the
2