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

122 NON-EUCLIDEAN GEOMETRY

[CH. V.

~ 71. The Element of Area in Cartesian Coordinates.
This result can be obtained from the expression found in
~ 70, by using the methods of the Calculus.

We have

= k tanh ek
x-=
e kI =cosh-.
k

[Cf. ~57
and ~ 69 (3).]

These are the equations connecting (x, y) and (e, r).
To find the element of area in Cartesian Coordinates (x, y),
we need only replace

e k dcdr y.
k D((, y)

by

After reduction, we obtain
cosh dx dy.
k

R

N

FIG. 83.

The result, however, can be found directly as follows:
Let P, Q be the points (x, y), (x + 8x, y + d/).
Let the Equidistant-Curves through P and Q with Ox as
base-line meet the ordinates at R and 8 (Fig. 83).