The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

166

NON-EUCLIDEAN GEOMETRY [cH. vmI.

The preceding results may be summed up as follows:
Inversion with regard to any circle of the system has the same
effect upon the nominal points and lines as reflection in the
nominal line with which the circle of inversion coincides.
U
0 u
C
v
FIo. 112.
The argument of ~ 95 can now be applied to the geometry
of this family of circles. Successive inversion with regard to
two circles of the system corresponds to a displacement in
two dimensions. We can always fix upon two circles of the
system which will change a nominal segment AB into a new
position, such that A coincides with P and AB lies along a given
nominal line through P. The method of superposition is thus
available in this geometry, and any theorems in the Hyperbolic
Geometry involving congruence of linear segments can be at
once " translated " into it.
~ 100. We notice that the definition of the nominal length
of a segment fixes the nominal unit of length. We may take
this unit segment on one of the diameters of the fundamental
circle, since these lines are also nominal lines of the system.
Let it be the segment OP (Fig. 113).
Then we must have
log o-0/ P-U -;
i, PU\ ta i PU
that is, log; that is, e.
PV~~~~~~~ p-V =e