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

94, 95, 96] EXTENSION TO SOLID GEOMETRY 159
trajectories of the circles of the system through A (i.e. of the
nominal lines through A) is the family of coaxal circles with
O and A as Limiting Points. The nominal lengths of the
nominal segments from A to the points where one of these
circles cuts the pencil of lines will be the same.
~ 96. The argument sketched in the preceding sections can
be extended to Solid Geometry. Instead of the system of
circles lying in one plane and all passing through the point 0,
we have now to deal with the system of spheres all passing
through the point 0.
The nominal point is the same as the ordinary point, but the
point O is excluded from the domain of the nominal points.
The nominal line through two nominal points is the circle
passing through 0 and these two points.
The nominal plane through three nominal points is the sphere
passing through 0 and these three points.
The nominal line through a point A parallel to a nominal
line BC is the circle through A which lies on the sphere through
0, A, B and C, and touches the circle OBC at the point 0.
It is clear that a nominal line is determined by two different
nominal points, just as a straight line is determined by two
different ordinary points. The nominal plane is determined
by three different nominal points, not on a nominal line, just
as an ordinary plane is determined by three different ordinary
points not on a straight line. If two points of a nominal line
lie on a nominal plane, then all the points of that line lie on
that plane. The intersection of two nominal planes is a nominal
line, etc.
The measurement of angles in the new geometry is the same
as that in the ordinary geometry; the angle between two
nominal lines is defined as the angle between the circles with
which these lines coincide at their intersection. The measurement of length is as before. Inversion in a sphere through 0
is equivalent to reflection in the nominal plane coinciding with
that sphere. Displacements, being point-transformations
according to which every point of the domain is transformed
into a point of the domain, in such a way that nominal lines
remain nominal lines, and nominal lengths and angles are
unaltered, will be given by an even number of inversions in the
spheres of the system.