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

168 NON-EUCLIDEAN GEOMETRY

[CH. VIII.

Further, it is obvious that as the point A moves away along
the perpendicular MA to the line BC (Fig. 108), the angle of
parallelism diminishes from - to zero in the limit.
2
We shall now prove that the angle of parallelism, II(p), for
the segment p, is given by
e - an
Consider a nominal line and a parallel to it through a point A.

MFI
FIG. 114.

Let AM (Fig. 114) be the perpendicular to the given line MU
and AU the parallel.
Let the figure be inverted from the point M', the radius of
inversion being the tangent from M' to the fundamental
circle.
Then we obtain a new figure (Fig. 115) in which the corresponding nominal lengths are the same, since the circle of
inversion is a circle of the system. The lines AM and MU
become straight lines through the centre of the fundamental
circle, which is the inverse of the point M. Also, the circle AU