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

12

NON-EUCLIDEAN GEOMETRY

[CH. I.

This discovery, which was made about 1823-1830, does not
detract from the value of Euclid's work. The Euclidean
Geometry is not to be replaced by the Non-Euclidean Geometries. The latter have thrown light upon the true nature of
Geometry as a science. They have also shown that Euclid's
Theory of Parallels, far from being a blot upon his work, is
one of his greatest triumphs. In the words of Heath: " When
we consider the countless successive attempts made through
more than twenty centuries to prove the Postulate, many of
them by geometers of ability, we cannot but admire the genius
of the man who concluded that such a hypothesis, which he
found necessary to the validity of his whole system of geometry,
was really indemonstrable." *
~ 6. The Work of Saccheri (1667-1733).
The history of these attempts to prove the Parallel Postulate
does not lie within the scope of this work.t But we must
refer to one or two of the most important contributions to that
discussion from their bearing on the rise and development of
the Non-Euclidean Geometries.
Saccheri, a Jesuit and Professor of Mathematics at the
University of Pavia, was the first to contemplate the possibility of hypotheses other than that of Euclid, and to work
out the consequences of these hypotheses. Indeed it required
only one forward step, at the critical stage of his memoir, and
the discovery of Lobatschewsky and Bolyai would have been
anticipated by one hundred years. Nor was that step taken
by his immediate successors. His work seems to have been
quickly forgotten. It had fallen completely into oblivion when
the attention of the distinguished Italian mathematician
Beltrami was called to it towards the end of the nineteenth
century. His Note entitled " un precursore italiano di Legendre
e di lobatschewsky " convinced the scientific world of the
importance of Saccheri's work, and of the fact that theorems,
which had been ascribed to Legendre, Lobatschewsky, and
Bolyai, had been discovered by him many years earlier.
Heath's Euclid, vol. i. p. 202.
Cf. Bonola, La geometria non-euclidea (Bologna, 1906); English
translation (Chicago, 1912). In quoting this work, we shall refer to
the English translation.
+ Rend. Ace. Lincei (4), t. v. pp. 441-448 (1889).