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 SEVTEL TO EIILIt).

the rectangqles GOR. OS', OR'. OS are constant and
ejiual. S
Dem.-Let a, b de-
note the radii of the Th
circles; then we have
(Cor. ~3, Prop. 2),
a: b:: OS: OR;
therefore a: b OS. 0.0': OR. OS';
but OS. OS' = square of the tangent from 0 to the circle
whose radius is a, and is therefore constant. Hence,
since the three first terms of the proportioni are constant, the fourth term is constant.
In like manner, it may be proved that OR'. OS is a
fourth proportional to a, b and 05.0S';;.-.OR'.OS
is constant.
Prop. 3.-The six centres of similitude of three circles
lie three by three on four lines, called axes of similitude
of t he c ir cles.
Dem.-Let the radii of the ODs be denoted by a, b, c,
their centr es by A, B, C; the external centres of similitude by A', B', C', and their internal centres by
A!' B"1, ". INow, by Definition,
AC' a
B'A -
Hence the product of the three ratios on the right is
negative unity; and therefore the points A', B', C' are
collinear (C'or. 1, Prop. 4).