Fig. 1.
Saccheri attacked the problem of parallels in quite a new way. Examining a quadrilateral, ABCD, in which the angles A and B are right angles and the sides AC and BD are equal, he determined to show that the angles C and D are equal. He also sought to prove that they are either right angles, obtuse or acute. He undertook to prove the falsity of the latter two propositions (that they are either obtuse or acute), leaving as the only possibility that they must be right angles. In doing so, he found that his assumptions led him into contradictions which he experienced difficulty in explaining.
His labors in connection with the solution of the problems proposed by Count Ventimiglia, including his work on the question of parallels, led directly into the field of metageometrical researches, and perhaps to him as to no other who had preceded him, or at least to him in a larger degree, belongs the credit for a continued renewal of interest in that series of investigations which resulted in the formulation of the non-Euclidean geometry.
The last published work of Saccheri was a recital of his endeavors at demonstrating the parallel-postulate. This received the "Imprimatur" of the Inquisition, July 13, 1733; the Provincial Company of Jesus took possession of the book for perusal on August 16, 1733; but unfortunately within two months after it had been reviewed by these authorities, Saccheri passed away.
All efforts which had been made prior to the work of Saccheri were based upon the assumption that there must be an equivalent postulate which, if it could be demonstrated, would lead to a direct, positive proof of Euclid's proposition. Although these and all other attempts at reaching such a proof have signally failed and although it may correctly be said that the entire history of demonstrations aiming at the solution of the famous postulate has been one long series of utter failures, it can be asserted with equal certitude that it has proven to be one of the most fruitful problems in the history of mathematical thought. For out of these failures has been built a superstructure of analytical investigations which surpasses the most sanguine expectations of those who had labored and failed.
In 1766 John Lambert (1728-1777) wrote a paper upon the Theory of Parallels dated Sept. 5, 1766, first published in 1786, from the papers left by F. Bernoulli, which contained the following assertions:[2]
1. The parallel-axiom needs proof, since it does not hold for geometry on the surface of the sphere.
2. In order to make intuitive a geometry in which the triangle's sum is less than two right angles, we need an "imaginary" sphere (the pseudosphere).
3. In a space in which the triangle's sum is different from two right angles there is an absolute measure (a natural unit for length).