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

126

NON-EUCLIDEAN GEOMETRY

[OH. V-. 73]

Therefore sin = cos 3.
Therefore S=_ -/3\
Thus the area of a quadrilateral with three right angles and an
acute angle /3 is equal to k2
on this scale.
But a triangle ABC (Fig. 86) is equal in area to Saccheri's
Quadrilateral BCC'B', in which the angles at B and C are each
equal to half the sum of the angles of the triangle.
A
B F / \EC'
B H C
FIG. 86.
The triangle is thus equal in area to twice the quadrilateral
with three right angles, and an acute angle equal to |-(A + B + C).
Using the result just found, the area of the triangle ABC
on this scale of measurement is
72(7r - 2/),
where 2/3=A+B+C.
In other words, the area of the triangle is the product of
k2 and its defect.
Comparing this with ~ 52, we see why the particular unit of
area was chosen in ~ 70.