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

130 NON-EUCLIDEAN GEOMETRY [OH. VI.
We shall denote the constant distance OA by A, so that
with this notation the length of the line is 4|.
Thus two other assumptions of the ordinary geometry are
contradicted in this geometry:
Two straight lines enclose a space;
Two points do not always determine a straight line.
0

A

B

0,
FIG. 89.
Through the two poles of a line an infinite number of lines
can be drawn, just as through the two ends of a diameter of a
sphere an infinite number of great circles can be drawn.
It is now clear that the argument which Euclid employs in
I. 16 is not valid in this geometry. The exterior angle of a
triangle is greater than either of the interior and opposite
angles only when the corresponding median is inferior to 3.
If this median is equal to E, the exterior angle is equal to
the interior angle considered; if it is greater than C, the
exterior angle is less than the interior angle considered.
Also, as I. 16 was essential to the proof of I. 27, it is now
evident why in this geometry that theorem does not hold. Of
course, if I. 16 did hold, there would have to be at least one
parallel to a line through any point outside it. In a limited