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

66, 67] POLAR COORDINATES 115
It is simpler and more instructive to obtain the result
directly.
Let P, Q be the points (r, 0), (r + dr, 0 + 80).
N Ip
0.
FIG. 78.
Draw PN perpendicular to OQ.
Let PQ == s, PN=q, NQ=p, and ON =z.
Then, from the triangle PNQ, we have as before
ds2 =p2 + q2.
Also, from the triangle ONP, we have
sinh = sinh sin 80. [~ 63.]
Therefore q=k sinh- 80, to the lowest order.
Also, we have from the same triangle
cosh - = cosh - cosh,
Therefore r and z are nearly equal.
Put r=z+.