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

81] METHOD OF G]RARD AND MANSION 139

Then we know that pP < qQ <rR.
From rR cut off rP'=pP, and join QP'.
Then we have PQ =QP' and L QPp = L QP'r.
Therefore LQRP' > QP'R and QR <QP'.
Thus, if pq=qr, PQ>QR.
Therefore, for equal increments of x, we have decreasing
increments of r.
It follows from this that if P and Q are any two points upon
OL, such that OP < OQ <, and OM, ON are commensurable,
OM ON
OP OQ
When OM and ON are incommensurable, we reach the same
conclusion by proceeding to the limit.
Thus, from 0 to S, the ratio - continually increases.
r
IV. From 0 to S, the ratio -Y decreases.
r
First we consider points upon OL at equal distances along
that line.
L
KR/
H
0 P q r A
FIG. 99.
Let P, Q, and R be three such points, so that
PQ =QR.
From P and R draw PH and RK perpendicular to Qq.