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.

A SEQUEL TO EUCLID.

Cor. 1.-The circle of self-inversion of a given circle
cuts it orthogonally.
Cor. 2.-Any three circles can be inverted into themselves, their circle of self-inversion being the circle
which cuts the three circles orthogonally.
Cor. 3.-If two circles be inverted into themselves,
the line joining their centres, namely ABCD, will be
inverted into a circle cutting both orthogonally; for
the line ABCD cuts the two circles orthogonally.
Cor. 4.-Any circle cutting two circles orthogonally
may be regarded as the inverse of the line passing
through their centres.
Cor. 5.-If ABCD be the line passing through the
centres of two circles, and A'B'C'D' any circle cutting
them orthogonally; then the points A', B', C', D' being
respectively the inverses of the points A, B, C, D, the
four lines AA', BB', CC', DD' will be concurrent.
Cor. 6. —Any three circles can be inverted into three
circles whose centres are collinear.
Prop. 5.-Any two circles can be inverted into two
egual circles.
Dem.-Let X, Y be the original Os, r and r' their
radii; let V, W be the inverse Os, p and p' their
radii; and let O be the
centre of inversion, and T,
T' the tangents from 0 to \
X and Y, and R the radius
of the circle of inversion.
Then, from the Demonstration of Prop. 1, we have /v w
r p:: T2:R2;
r: p':: T'2: R2.
Hence, since p = p', we have
r: r:: T: T'2;.~. the ratio of T2: T'2 is given; and, consequently, the
ratio of T: T' is given. Hence if a point be found,