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

142

NON-EUCLIDEAN GEOMETRY

[CH. VII.

which the sides x', y include the obtuse angle, if y' is kept fixed
and x tends to zero, the ratio x': x tends to a finite limit p (y')
from above, and this ratio is less than 5 (y).
At

0 x M A
FIG. 101.
As in ~81 we find that x': x continually decreases as x
tends to zero. It must have a limit, which may be zero or
some number less than unity.
Produce MP, and draw M'Q perpendicular to MP.
From ~ 82 we know that as x decreases, the ratio M'Q
increases.

0 x M
FIG. 102.

It must therefore have a finite limit, not zero, or become
infinite.
But M'P
But Q <-, since M'Q < M'P.
x x
Thus Lt (- ) cannot be zero.
L~oo /