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

CHAPTER VIII.

THE CONSISTENCY OF THE NON-EUCLIDEAN GEOMETRIES
AND THE IMPOSSIBILITY OF PROVING THE PARALLEL
POSTULATE.
~ 91. As we have already seen, the discovery of the NonEuclidean Geometries arose from the attempts to prove
Euclid's Parallel Postulate. Bolyai and Lobatschewsky did
a double service to Geometry. They showed why these
attempts had failed, and why they must always fail; for they
succeeded in building up a geometry as logical and consistent
as the Euclidean Geometry, upon the same foundations, except
that for the Parallel Postulate of Euclid, another incompatible
with it was substituted. They differed from almost all their
predecessors in their belief that, proceeding on these lines, they
would not meet any contradiction; and they held that the
system of geometry built upon their Parallel Postulate was a
fit subject of study for its own sake.
The question naturally arises: How can one be certain
that these Non-Euclidean Geometries are logical and consistent systems? How can we be sure that continued study
would not after all reveal some contradiction, some inconsistency? Saccheri thought he had found such in the Hyperbolic Geometry; but he was mistaken. Even Bolyai, many
years after the publication of the Appendix, was for a time of
the opinion that he had come upon a contradiction, and that
the sought-for proof of the Euclidean Hypothesis was in his
hands. He, too, was mistaken.
Of course, it is not sufficient simply to point to the fact that
these geometries-developed into a large body of doctrine as
they have been-do not offer in any of their propositions the
contradiction which the earlier workers in those fields were
convinced they must contain. We must be sure that, proceeding further on these lines, such contradiction could never be