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

97, 98, 99] NOT-INTERSECTING NOMINAL LINES 163
circumference of each. Invert the circles from the point of
intersection R' of C1 and C2, which lies outside the fundamental
circle. Then the nominal lines C1 and C2 become two straight
lines Ca' and C2', through the inverse of R. Also the fundamental circle C inverts into a circle C', cutting Cl' and Cg' at
right angles, so that its centre is at the point of intersection
of these two lines. Again the circle C3 inverts into a circle C3',
cutting C' orthogonally. Hence its centre lies outside C'.
We thus obtain a curvilinear triangle in which the sum of
the angles is less than two right angles; and since the angles
in this triangle are equal to those in the nominal triangle, our
result is proved.
Finally, it can be shown that there is always one and only one
circle of the system which will cut two not-intersecting circles
of the system orthogonally. In other words, two not-intersecting
nominal lines have a common perpendicular.
All these results we have established in the Hyperbolic
Geometry. They could be accepted in the geometry of the
circles for that reason.
~ 99. As to the measurement of length, we define the nominal
length of a nominal segment as follows:
The nominal length of any nominal segment AB is equal to
lo, /AV BV\
"AU/BU/'
where U and V are the points where the circle which coincides
with the nominal line AB cuts the fundamental circle. (Cf.
Fig. 107.)
With this definition the nominal length of AB is the same
as that of BA. Also the nominal length of the complete line
is infinite. If C is any point on the nominal segment AB
between A and B, the nominal length of AB is the same as the
sum of the nominal lengths of AC and CB.
Let us consider what effect inversion with regard to a circle
of the system has upon the nominal points and lines.
Let A be a nominal point and A' the inverse of this point
in the fundamental circle.
Let the circle of inversion meet the fundamental circle in C,
and let its centre be D (Fig. 110).
Suppose A and A' invert into B and B'.
N.-E.G. L2