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

110 NON-EUCLIDEAN GEOMETRY

[CH. IV.

If a different unit is adopted, so that the ratio of the arc
AB to the arc A1B1 is a, any number greater than unity, we
have the equation
sx =sa-x instead of s= se-x.
1
Putting a=ek,
this gives s = se k.
This parameter k will enter into all the equations of the
a c a
preceding sections, so that sinh cosh etc, will replace
sinha, cosha, etc.
And the equation for the Angle of Parallelism will be
p
tan2I (P)=e k.
The Euclidean Geometry now appears as a special case of
the Hyperbolic Geometry, for if we let k->oo, the formulae
of this Non-Euclidean Geometry reduce to those of the
Euclidean.
In the first place, since
p
tan II(p)-e k,
the angle of parallelism becomes - when k->oo.
2
Further, the equations connecting the sides and angles of a
right-angled triangle, viz.
a. c
sh inh sinh sin A,
b 1 a c
sinh = tanh cot X,
cosh - = cot X cot /u,
c a b
cosh -= Cosh - cosh,
k-a k o k
cos X = cosh T- sin ft,
a c
tanh = tanh - cos
I Iccos,