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

140

NON-EUCLIDEAN GEOMETRY

[CH. VTI.

Then we know that Pp < Hq and Rr < Kq (~ 79).
But QH = QK. Therefore Qq - Hq = Kq - Qq.
It follows from the above that Qq - Pp > Rr - Qq.
Therefore, for equal increments of r we have diminishing
increments of y.
It follows from this that if P and Q are any two points upon
OL, such that OP < OQ < &, and OP, OQ are commensurable,
PP > Qq
OP OQ
When OP and OQ are incommensurable, we obtain the same
result by proceeding to the limit.
Thus, as P moves along OL from 0 towards S, the ratio
Y continually decreases.
r
V. When r tends to zero, the ratio x: r tends towards a finite
limit from above, and the ratio y: r tends towards a finite limit
from below.
From (III.) we know that x: r continually decreases as r
tends to zero, so that this ratio has a limit, finite or zero.
From (IV.) we know that y: r continually increases as r
tends to zero, so that this ratio either has a finite limit, not
zero, or becomes infinite.
But from the quadrilateral whose sides are (x,, x', y') we
have x > x'. (Fig. 95.) Thus x: r > x': r.
But, by (IV.), x': r either has a finite limit, not zero, or
becomes infinite, as r tends to zero.
Therefore the limit of x: r cannot be zero, and must be some
finite number. Also x: r approaches this limit from above.
But it follows from the preceding argument that y': r has
a finite limit, not zero.
Also we know that y < y', and thus y: r < y': r.
It follows that y: r has a finite limit, not zero, and it
approaches this from below.
These two limits Lt (), Lt () are chosen as the sine
r-0\/ ->0 r
and cosine of the acute angle which OL makes with OA,* and
the other ratios follow in the usual way.
* These limits are functions of the angle. It can be shown that they
are continuous, and that with a proper unit of angle they are given
by the usual exponential expressions. Cf. Coolidge, loc. cit. p. 53.