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

[CH. III

CHAPTER III.
THE HYPERBOLIC PLANE GEOMETRY.
~ 21. In this chapter we proceed to the development of the
Plane Geometry of Bolyai and Lobatschewsky-the Hyperbolic
Geometry. We have already seen that we are led to it by the
consideration of the possible values for the sum of the angles
of a triangle, at any rate when the Postulate of Archimedes
is adopted. This sum cannot be greater than two right angles,
assuming the infinity of the straight line. If it is equal to
two right angles, the Euclidean Geometry follows. If it is
less than two right angles, then two parallels can be drawn
through any point to a straight line.
It is instructive to see how Lobatschewsky treats this
question in the Geometrische Untersuchungen,,* one of his later
works, written when his ideas on the best presentation of this
fundamental point were finally determined.
"All straight lines in a plane which pass through the same
point," he says in ~ 16, " with reference to a given straight
line, can be divided into two classes, those wuhich cut the line,
and those which do not cut it. That line which forms the
boundary between these two classes is said to be parallel to the
given line.
" From the point A (Fig. 13) draw the perpendicular AD
to the line BC, and at A erect the perpendicular AE to the line
AD. In the right angle EAD either all the straight lines going
out from A will meet the line DC, as, for example, AF; or some
of them, as the perpendicular AE, will not meet it.
" In the uncertainty whether the perpendicular AE is the
only line which does not meet DC, let us assume that it is
* Geometrische Unter suchungen zur Theorie der Parallellinien (Berlin,
1840). English translation by Halsted (Austin, Texas, 1891).