Saccheri was a Jesuit, and it was in 1690, while he was teaching grammar in Milan, that he first studied the _Elements_ of Euclid. He was a man of very great acumen, and when he, in turn, succumbed to the spell of the parallel postulate, he brought to bear on it a more subtle and rigorous logic than had yet been applied to it. Thirty-six years before he published his treatise on Euclid he had published a book on logic which gives him a high place as a logician. In it he is particularly concerned with investigating the compatibility of different assumptions or postulates. His method was to determine whether a member of a group of postulates is independent of the others by finding a particular case in which the postulate in question is not true while all the others remain true. If such a case can be found, it is obvious that the postulate in question cannot be deduced from the others, else it would be true whenever they were true. This was the method he applied to the parallel postulate of Euclid. He showed that the parallel postulate is equivalent to saying that the three interior angles of a triangle are equal to two right angles. He proceeds, therefore, in accordance with his method, to develop the consequences of supposing them less than, or greater than, two right angles. In the latter case he succeeds in showing that we are led to impossible conclusions, since he assumed, as everybody assumed for more than a century after, that the straight line is of infinite length. But in the former case, the hypothesis that the interior angles of a triangle are together less than two right angles, Saccheri, although he struggled very hard, did not succeed in falling into contradictions. He does not seem to have had the boldness necessary completely to trust his own logic, but the fact remains that, accepting the rest of Euclid’s axioms and denying the parallel axiom, he developed a logically consistent geometry.