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

32

NON-EUCLIDEAN GEOMETRY

[OH. II.

And lower down in the same passage:
"But there still remains the question, whether in some way
or other the considerations of space would not avail for the
establishment of E."
Indeed, owing to a mistake in his analysis, he thought for
a time that he had actually obtained a proof of the Euclidean
Hypothesis on these lines. But he discovered his error later.
From the fact that at one time he was willing to admit that,
with the aid of Solid Geometry, evidence against the logical
consistency of the Non-Euclidean Geometry might be obtained,
we must not imagine that he had failed to grasp the significance
of his earlier work. On the contrary, his argument shows that
he had seen more deeply into the heart of the matter than
Lobatschewsky himself. The latter, as we shall see below,
relied simply upon the formulae for the plane. Even when it
has been established that the Non-Euclidean Plane Geometry
is a perfectly logical and consistent system, the question still
remains, whether, somewhere or other, contradictory results
might not appear in the theorems of Solid Geometry.
This question, raised for the first time by Bolyai, was
settled many years later by Klein,* following upon some
investigations of Cayley. We shall give, in the last chapter
of this book, an elementary and rigorous demonstration
of the logical possibility of the Non-Euclidean Geometry of
Bolyai-Lobatschewsky, and shall show how the same argument
can be applied to the Non-Euclidean Geometry associated with
the name of Riemann.
~ 16. The Work of Lobatschewsky (1793-1856).
Nicolaus Lobatschewsky-Professor of Mathematics in the
University of Kasan-was a pupil of Bartels, the friend and
fellow-countryman of Gauss. As early as 1815 he was working
at the Theory of Parallels, and in notes of his lectures (1815 -1817), carefully preserved by one of his students, and now in
the Biblioteca Lobatschewskiana of the Kasan PhysicalMathematical Society, no less than three " proofs " of the
Parallel Postulate are to be found. From a work on Elementary
Geometry, completed in 1823, but never published, the MSS. of
which was discovered in 1898 in the archives of the University
of Kasan, we know that by that date he had made some
*Cf. "Uber die sogenannte Nicht-Euklidische Geometrie," Math.
Ann. vol. iv. (1871).