while the necessity for determining the convergence was generally lost sight of. Euler reached some very pretty results on infinite series, now well known, and also some very absurd results, now quite forgotten. The faults of his time found their culmination in the Combinatorial School in Germany, which has now passed into deserved oblivion. At the beginning of the period now under consideration, the doubtful, or plainly absurd, results obtained from infinite series stimulated profounder inquiries into the validity of operations with them. Their actual contents came to be the primary, form a secondary, consideration. The first important and strictly rigorous investigation of series was made by Gauss in connection with the hypergeometric series. The criterion developed by him settles the question of convergence in every case which it is intended to cover, and thus bears the stamp of generality so characteristic of Gauss's writings. Owing to the strangeness of treatment and unusual rigour. Gauss's paper excited little interest among the mathematicians of that time.
More fortunate in reaching the public was Cauchy, whose Analyse Algébrique of 1821 contains a rigorous treatment of series. All series whose sum does not approach a fixed limit as the number of terms increases indefinitely are called divergent. Like Gauss, he institutes comparisons with geometric series, and finds that series with positive terms are convergent or not, according as the th root of the th term, or the ratio of the th term and the th term, is ultimately less or greater than unity. To reach some of the cases where these expressions become ultimately unity and fail, Cauchy established two other tests. He showed that series with negative terms converge when the absolute values of the terms converge, and then deduces Leibniz's test for alternating series. The product of two convergent series was not found to be necessarily convergent. Cauchy's theorem that the
