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.

Hence the line A1A,,_l passes through a, the pole of
AoAo'. Similarly, A2A,,_2 passes through a, &c. Hence
the proposition is proved.
CoR.-The vertices AA,,l_, A2A,_2, ic., form a system
of points in involution, and points Ao, Ao' are the double
points. Hencee the vertices of a harmonic polygon form
as many systems in involution as it has symmedian lines.
Prop. 5.-If a transversal through the symmedian
point K cuts the sides of the polygon in the points R1, R2.... R. and if a point P be taken on it so that 1/KR1 +
1/KR2... = n/KP, the locus of P is the polar of K with
respect to the circumcircle.
Dem.-Let a be the pole of the symmedian chord
AoAo'. Join Ka. It is easy to see that a is one position
of P. For if n be even, the sides may be distributed
in pairs, so that the points K, a are harmonic conjugates
to the points in which each pair of sides may be cut by
the line Ka. Hence,
1/KRi + 1/KR,, = 2/Ka,
1/KR2 + l/KRni = 2/Ka, &c.