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
product of two absolutely convergent series converges to the product of the sums of the two series was shown half a century later by F. Mertens of Graz to be still true if, of the two
convergent series to be multiplied together, only one is absolutely convergent.
The most outspoken critic of the old methods in series was Abel. His letter to his friend Holmboe (1826) contains
severe criticisms. It is very interesting reading, even to modern students. In his demonstration of the binomial theorem he established the theorem that if two series and their product series are all convergent, then the product series will converge towards the product of the sums of the two given series. This remarkable result would dispose of the whole problem of multiplication of series if we had a universal
practical criterion of convergency for semi-convergent series.
Since we do not possess such a criterion, theorems have been recently established by A. Pringsheim of Munich and A. Voss
of Würzburg which remove in certain cases the necessity of applying tests of convergency to the product series by the application of tests to easier related expressions. Pringsheim reaches the following interesting conclusions: The product of two semi-convergent series can never converge absolutely, but a semi-convergent series, or even a divergent series, multiplied by an absolutely convergent series, may yield an absolutely convergent product.
The researches of Abel and Cauchy caused a considerable stir. We are told that after a scientific meeting in which Cauchy had presented his first researches on series, Laplace
hastened home and remained there in seclusion until he had examined the series in his Mécanique Céleste. Luckily, every one was found to be convergent! We must not conclude, however, that the new ideas at once displaced the old. On the contrary, the new views were generally accepted only after a