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

154 NON-EUCLIDEAN GEOMETRY

[CH. VIII.

discovered. If we can prove this to be the case, then we know
that Euclid's Parallel Postulate cannot be demonstrated.
~ 92. There are several ways by which it is possible to
establish the fact that the Hyperbolic and Elliptic Geometries
are as logical and consistent as the Euclidean Geometry.*
Lobatschewsky, and to some extent Bolyai, relied upon
the formulae of the Hyperbolic Plane Trigonometry. These
are identical with the formulae of Spherical Trigonometry,
if the radius of the sphere is imaginary. If the ordinary
Spherical Trigonometry offers no contradiction, their geometry could not do so. However, this proof is not complete
in itself, for it leaves aside the domain of Solid Geometry, and
does not establish the impossibility of the difficulty appearing
in that field. (Cf. Chapter II. ~~ 15, 17.)
The most important of all the proofs of the consistency of
the Non-Euclidean Geometries is that due to Cayley and
Klein. In it one passes beyond the elementary regions within
the confines of which this book is meant to remain. Other
proofs are analytical. The assumptions of geometry are
translated into the domain of number. Any inconsistency
would then appear in the arithmetical form of the assumptions
or in the deductions from them. This form of proof also seems
to lie outside the province of this book.
Finally, there are a number of geometrical proofs, depending
upon concrete interpretations of the Non-Euclidean Geometries in the Euclidean. The earliest of these-due to
Beltrami, and dealing with the Hyperbolic Geometryrequires a knowledge of the Geometry of Surfaces. But an
elementary representation of the Hyperbolic Plane and Space
in the Euclidean was given by Poincare.
" Let us consider," he says, " a certain plane, which I shall
call the fundamental plane, and let us construct a kind of
dictionary by making a double series of terms written in two
columns, and corresponding each to each, just as in ordinary
dictionaries the words in two languages which have the same
signification correspond to one another:
Space. - -The portion of space situated above
the fundamental plane.
* For a discussion on more advanced lines, cf. Sommerville's NonEuclidean Geometry, ch. v. and vi. (London, 1914).