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

101, 102, 103] THE PARALLEL POSTULATE

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inconsistency could arise in the Hyperbolic Plane Geometry,
provided no logical inconsistency can arise in the Euclidean
Plane Geometry. It could still be argued that such a contradiction might be found in the Hyperbolic Solid Geometry.
An answer to such an objection is forthcoming at once. The
geometry of the system of circles, all orthogonal to a fixed
circle, can be readily extended into a three-dimensional system.
The nominal points are the points inside a fixed sphere, excluding the points on the surface of the sphere from their domain.
The nominal lines are the circles through two nominal points
cutting the fixed sphere orthogonally. The nominal planes are
the spheres through three nominal points cutting the fixed
sphere orthogonally. The ordinary plane enters as a particular
case of these nominal planes, and so the plane geometry just
discussed is a special case of a plane geometry of this system.
With suitable definitions of nominal lengths, nominal parallels,
etc., we have a solid geometry exactly analogous to the Hyperbolic Solid Geometry. It follows that no logical inconsistency
could arise in the Hyperbolic Solid Geometry, since, if such did
occur, it would also be found in the interpretation of the
result in this Nominal Geometry, and therefore it would enter
into the Euclidean Geometry.
By this result our argument is complete. However far the
Hyperbolic Geometry is developed, no contradictory results
could be obtained. This system is thus logically possible,
and the axioms upon which it is founded are not contradictory.
Hence it is impossible to prove Euclid's Parallel Postulate,
since its proof would involve the denial of the Parallel Postulate
of Bolyai and Lobatschewsky.
~ 103. The System of Circles cutting a Fixed Circle
diametrally.
We shall now discuss the geometry of the system of circles
cutting a fixed circle centre, O and radius k, diametrally.
The points in which any circle of the system cuts the fixed
circle are to be at the extremities of some diameter. We
shall call the fixed circle, as before, the fundamental circle.
The system of circles with which we are to deal has power - k2
with respect to O.
Let A and B be any two points within the fundamental
circle, and A', B' the points on OA and OB, such that
OA. OA'= -k2 and OB. OB'= -k2.