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

120 NON-EUCLIDEAN GEOMETRY [CH. V.
Let the area of ABB1A1 be denoted by A0.
Then (cf. ~ 48) the area of A1B1B2^A will be Aoe;
2
that of A2B2B3A3 will be AOe k, etc.
Thus the area of ABBnA
1 2 n\
=Ao(1l+e k+e + +...+e k )
_ n
1 -e
=A i ---
Therefore, as n - o, this area approaches a limit, namely
A- Ao
1-e k
This is the area of the region bounded by two axes of a
Limiting-Curve and an arc such that the tangent at one end
is parallel to the axis through the other end.
The unit of area has not yet been chosen in this discussion.
We now fix it so that the area denoted above by A will be
k2 the unit of area.
With this measurement
Ao0k2(1- -e).
Also the area of ABA^Bn will be k2 1- e-).
Next, let P be a point on AB, or AB produced, such that the
are AP=s.
Then area APP1A1: area ABBA A=s: k,
and area APPnA =ks(1 -e k).
Taking x, first, a rational number, and then treating the
irrational number x as the limit of a sequence of rational
numbers, we find from the above that the area bounded by the