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

21] PARALLELS. LOBATSCHEWSKY

41

possible that there are other lines, such as AG, which do not
cut DC however far they are produced.
"In passing from the lines AF, which cut DC, to the lines AG,
which do not cut DC, we must come upon a line AH, parallel
to DC, that is to say, a line on one side of which the lines AG
do not meet the line DC, while, on the other side, all the lines
AF meet DC.
1 K Dl HDi
E' E
A f G
a2 a,
EB D F C
FIG. 13.
"The angle HAD, between the parallel AH and the perpendicular AD, is called (the angle of parallelism,) and we shall
denote it by II(p), p standing for the distance AD."
Lobatschewsky then shows that if; the angle of parallelism
were a right angle for the point A and this straight line BC,
the sum of the angles in every triangle would have to be two
right angles. Euclidean Geometry would follow, and the angle
of parallelism would be a right angle for any point and any
straight line.
On the other hand, if the angle of parallelism for the point A
and this straight line BC were an acute angle, he shows that
the sum of the angles in every triangle would have to be less
than two right angles, and the angle of parallelism for any
point and any straight line would be less than a right angle.
The assumption I(p) = serves as the foundation for the
ordinary geometry, and the assumption II(p) < 2 leads to the
new geometry, to which he gave the name Imaginary Geometry. In it two parallels can be drawn from any point to any
straight line.
In this argument Lobatschewsky relies upon the idea of