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

138 NON-EUCLIDEAN GEOMETRY [cH. vii.
Therefore L pP'Q and L P'Qq are equal acute angles, which is
impossible.
L
0 P q A
FIG. 97
Thus, as the point P moves along OL from O towards S, y
continually increases.
III. From 0 to S, the ratio x continually increases.
r
First, consider points upon OL corresponding to equal segments on OA.
L
R
0 P q r A
FIG. 98.
Let P, Q, R be three such points, so that
pq =qr.