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

16

NON-EUCLIDEAN GEOMETRY

[OCH I.

In these proofs the infinity of the line is assumed. One of
them is as follows:
Let the sum of the angles of the triangle ABC be Xr + A, and
let A be the smallest angle.
Bisect BC at D and produce AD to E, making DE=AD.
Join BE.
Then from the triangles ADC C E
and BDE, we have
LCAD=L BED,
LACD=L DBE.
Thus the sum of the angles of
the triangle AEB is also equal to
r +oC, and one of the angles A B
BAD or AEB is less than or FIG. 10.
equal to z CAB.
Apply the same process to the triangle ABE, and we obtain
a new triangle in which one of the angles is less than or equal
to - L CAB, while the sum is again 7r + oc.
Proceeding in this way after n operations we obtain a triangle, in which the sum of the angles is 7r + oc, and one of the
angles is less than or equal to L CAB.
But we can choose n so large that 2o. > LCAB, by the
Postulate of Archimedes.
It follows that the sum of two of the angles of this triangle
is greater than two right angles, which is impossible (when the
length of the straight line is infinite).
Thus, we have Legendre's First Theorem that
The sum of the angles of a triangle cannot be greater than two
right angles.
Legendre also showed that
If the sum of the angles of a single triangle is equal to two
right angles, then the sum of the angles of every triangle is equal
to two right angles.
From these theorems it follows that
If the sum of the angles of a single triangle is less than two
right angles, then the sum of the angles of every triangle is less
than two right angles.