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.

139

three false positions, and finding double points, as follows:Let A, B, C be the centres of the three given circles; Z
the required tangential circle; a, /3, y the points of contact: then the triangles ABC, a/3y are in perspective,
the centre of Z being their centre of perspective, and the
axis of similitude of the three given circles being their
axis of perspective. Let A', B', C' be the three centres

C'

of similitude. Then take any three points P, P', P" in
the circle A; join them to the point B', cutting the circle
C in the points Q, Q', Q": again, join these points to A',
and let the joining lines cut the circle B in the points
RI, R', IR": lastly, join R, R', t" to C', cutting the circle
A in the points 7r, rr', r"; then a will be such that the
anharmonic ratio (aPP'P") will be equal to the anharmonic ratio (a 7r'7r"). Hence the problem is solved.