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

34

NON-EUCLIDEAN GEOMETRY

[CH. II.

" We have seen that the sum of the angles of a rectilinear
triangle cannot be greater than r-. There still remains the
assumption that it may be equal to 7r or less than ir. Each of
these two can be adopted without any contradiction appearing
in the deductions made from it; and thus arise two geometries:
the one, the customary, it is that until now owing to its simplicity, agrees fully with all practical measurements; the
other, the imaginary, more general and therefore more difficult in its calculations, involves the possibility of a relation
between lines and angles.
"If we assume that the sum of the angles in a single rectilinear
triangle is equal to ir, then it will have the same value in all.
On the other hand, if we admit that it is less than 7r in a single
triangle, it is easy to show that as the sides increase, the sum of
the angles diminishes.
" In all cases, therefore, two lines can never intersect, when
they make with a third, angles whose sum is equal to 7r. It is
also possible that they do not intersect in the case when this
sum is less than T-, if, in addition, we assume that the sum
of the angles of a triangle is smaller than 7r.
" In relation to a line, all the lines of a plane can therefore
be divided into intersecting and not-intersecting lines. The
latter will be called parallel, if in the pencil of lines proceeding
from a point they form the limit between the two classes;
or, in other words, the boundary between the one and the
other.
" We imagine the perpendicular a dropped from a point to
a given line, and a parallel drawn from the same point to the
same line. We denote the angle between a and the parallel
by F (a). It is easy to show that the angle F (a) is equal to -
2
for every line, when the sum of the angles of a triangle is equal
to 7r; but, on the other hypothesis, the angle F (a) alters with
a, so that as a increases, it diminishes to zero, and it remains
always less than 7
2'
"To extend the meaning of F (a) to all lines a, on the latter
hypothesis, we shall take
F(0) =, F(-a)=sr —F(a).
In this way we can associate with every acute angle A a