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

158

NON-EUCLIDEAN GEOMETRY [CH. VII.

A making with AM an angle less than a right angle will cut
BC on the side of OAM in which the acute angle lies.
Therefore in the geometry of these nominal points and lines
the Euclidean Parallel Postulate holds.
~ 95. Before we can deal with the metrical properties of this
geometry, we require a measure of length. We define the
nominal length of a nominal segment as the length of the rectilinear
segment to which it corresponds.
From this definition it is not difficult to show that the
nominal length of a nominal segment is unaltered by inversion
with regard to a circle of the system; and that inversion with
regard to such a circle is equivalent to reflection of the nominal
points and lines in the nominal line which coincides with the
circle of inversion.
Now, if we invert successively with regard to two circles of
the system (i.e. if we reflect in two nominal lines one after
the other), we obtain what corresponds to a displacement in
two dimensions. A nominal triangle ABC takes up the position
A'B'C' after the first reflection; and from A'B'C' it passes to
the position A"B"C" in the second. The sides and angles
of A"B"C" (in our nominal measurement) are the same as the
sides and angles of the nominal triangle ABC, and the point
C" lies on the same side of A"B" as the point C does of AB.
Further, we can always fix upon two inversions which will
change a given nominal segment AB into a new position such
that A comes to A', and AB lies along a given nominal line
through A'. We need only invert first with regard to the
circle which "bisects " the nominal line AA' at right angles.
This brings AB into a position A'B", say. Then, if we invert
with regard to the circle of the system which bisects the angle
between A'B" and the given nominal line through A', the
segment AB is brought into the required position.
The method of superposition is thus available in the geometry
of the nominal points and lines. Euclid's argument can be
"translated" directly into the new geometry. We have
only to use the words nominal points, nominal lines, nominal
parallels, etc., instead of the ordinary points, lines, parallels,
etc., and we obtain from the ordinary geometry the corresponding propositions in the geometry of this family of circles.
It should perhaps be pointed out that the nominal circle
with centre A is an ordinary circle. For the orthogonal