attention to prime numbers. Gauss and Legendre had given expressions denoting approximately the asymptotic value of the number of primes inferior to a given limit, but it remained for Riemann in his memoir, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, 1859, to give an investigation of the asymptotic frequency of primes which is rigorous. Approaching the problem from a different direction, Patnutij Tchebycheff, formerly professor in the University of St. Petersburg (born 1821), established, in a celebrated memoir, Sur les Nombres Premiers, 1850, the existence of limits within which the sum of the logarithms of the primes , inferior to a given number , must be comprised.[89] This paper depends on very elementary considerations, and, in that respect, contrasts strongly with Riemann's, which involves abstruse theorems of the integral calculus. Poincaré's papers, Sylvester's contraction of Tchebycheff's limits, with reference to the distribution of primes, and researches of J. Hadamard (awarded the Grand prix of 1892), are among the latest researches in this line. The enumeration of prime numbers has been undertaken at different times by various mathematicians. In 1877 the British Association began the preparation of factor-tables, under the direction of J. W. L. Glaisher. The printing, by the Association, of tables for the sixth million marked the completion of tables, to the preparation of which Germany, France, and England contributed, and which enable us to resolve into prime factors every composite number less than 9,000,000.
Miscellaneous contributions to the theory of numbers were made by Cauchy. He showed, for instance, how to find all the infinite solutions of a homogeneous indeterminate equation of the second degree in three variables when one solution is given. He established the theorem that if two congruences, which have the same modulus, admit of a common solution,
