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

160

NON-EUCLIDEAN GEOMETRY [cH. VIII.

Thus, the geometry of these nominal points, lines, and planes
is identical with the ordinary Euclidean Geometry. Its
elements satisfy the same laws; every proposition valid in
the one is also valid in the other; and from the theorems
of the Euclidean Geometry those of the Nominal Geometry
can be inferred, and vice versa.
The plane geometry of the nominal points and lines described
in the preceding sections is a special case of the more general
plane geometry based upon the definitions of this section.
~ 97. The System of Circles orthogonal to a Fixed
Circle.
We proceed to discuss the geometry of the system of circles
orthogonal to a fixed circle, centre O and radius k. We shall
call this circle the fundamental circle. Then the system of
circles has power k2 with respect to O.
FIG. 107.
Let A and B be any two points within the fundamental
circle and A', B' the inverse points with respect to that circle.
Then A, A', B, B' are concyclic, and the circle which passes
through them cuts the fundamental circle orthogonally.
There is one and only one circle orthogonal to the fundamental
circle which passes through two different points within that
circle.
In discussing the properties of the family of circles orthogonal to the fundamental circle, we shall call the points within
that circle nominal points. The points on the circumference
of the fundamental circle are excluded from the domain of
the nominal points.*
* In this discussion the nominal points, etc., are defined somewhat
differently from the ideal points, etc., in the paper referred to on p. 156.