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.

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A SEQUEL TO EUCLID.

(3). The two double points, and any twopairs of conjugate points, form a system in involution.
(4). Any line cutting three coaxal circles is cut in
involution.
DEF.-If a system of points in involution be joined to
any point P not on the line of collinearity of the points,
the six joining lines will have the anharmonic ratio of the
pencil formed by any four rays equal to the anharmonic
ratio of the pencil formed by their four conjugate rays.
Such a pencil is called a pencil in involution. The rays
passing through the double points are called the double
rays of the involution.
Prop. 7.-If four points be collinear, they belong to
three systems in involution.
Dem.-Let the four points be A, B, C, D; upon AB
and CD, as diameters, describe circles; then any circle
coaxal with these will intersect the line of collinearity
of A, B, C, D in a pair of points, which form an involution with the pairs A, B, C, D. Again, describe circles
on the segments AD, BC, and circles coaxal with them
will give us a second involution. Lastly, the circles
described on CA, BD will give us a third system. The
central points of these systems will be the points where
the radical axes of the coaxal systems intersect the line
of collinearity of the points.
Prop. 8.-The following examples will illustrate the
theory of involution:- A
(1). Any right line cut- D
ting the sides and diagonals
of a quadrilateral is cut in \
involution.
Dem.-Let ABCD be c — B
the quadrilateral, LL' the
transversal intersecting the diagonals in the points
N, N'. Join AN', CN'; then the anharmonic ratio of
the pencil (A. LMNN') = (A. DBON') = (C. DBON')
= (C. M'L'ON) = (C. L'M'N'N).