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

124

NON-EUCLIDEAN GEOMETRY

[Ca. v.

Let the circles through P and Q cut the radii at S and R,
forming the element PRQS.

Then we have

ar PS = ksinh E80, by ~ 69 (1),
PR = dr.

The figure PRQS becomes a rectangle in the limit.
Therefore the element of area in Polar Coordinates is
k sinh dr dO.
k
The area of the circle of radius a is thus given by
'a 27r r d,
|I 1k sinh - dl d0,
Jo Jo k

which becomes

27rk2(cosh ka- ),
47k2 sinh2a.
2k,

or

~73. The Area of a Triangle and of
with three Right Angles.

a Quadrilateral

c p B

C
m'

3

I

9y

x

0 a M A x
FIG. 85.
Let OABC be a quadrilateral with the sides a, m', c, l, as in
Fig. 85, and the angles at 0, A, C right angles; A lying on
the axis of x and C on the axis of y.