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

93, 94] NOMINAL POINTS, LINES, PARALLELS 157

points. We define the angle between two nominal lines as
the angle between the circles with which the nominal lines
coincide at their common point.
With these definitions, two different nominal points A, B
in this Nominal Geometry always determine a nominal line
AB, just as two different ordinary points always determine
a straight line AB.
The nominal points and lines also satisfy the " axioms of
order,"* which express the idea of between-ness, when the
point 0 is excluded from the domain of the nominal points.
If this point were not excluded, we could not say that of any
three nominal points on a nominal line, there is always one,
and only one, which lies between the other two.
Proceeding to the question of parallels, we define parallel
nominal lines as follows:
The nominal line through a nominal point parallel to a given
nominal line is the circle of the system which passes through
the given point and touches at 0 the circle coinciding with the
given nominal line.
Referring to Fig. 106 we see that in the pencil of nominal
lines through A there is one nominal line which does not cut
BC, namely, the circle of the system which touches OBC at O.
C
0
MI
FIG. 106.
This nominal line does not cut the nominal line BC, for the
point O is excluded from the domain of the nominal points.
It is at right angles to AM, the nominal line through A perpendicular to the nominal line BC. Every nominal line through

* Cf. Hilbert, loc. cit. ~ 3.