the theory of biquadratic residues (1825 and 1831), the second of which contains a theorem of biquadratic reciprocity.

Gauss was led to astronomy by the discovery of the planet Ceres at Palermo in 1801. His determination of the elements of its orbit with sufficient accuracy to enable Olbers to rediscover it, made the name of Gauss generally known. In 1809 he published the Theoria motus corporum coelestium, which contains a discussion of the problems arising in the determination of the movements of planets and comets from observations made on them under any circumstances. In it are found four formulæ in spherical trigonometry, now usually called "Gauss' Analogies," but which were published somewhat earlier by Karl Brandon Mollweide of Leipzig (17741825), and earlier still by Jean Baptiste Joseph Delambre (1749-1822). Many years of hard work were spent in the astronomical and magnetic observatory. He founded the German Magnetic Union, with the object of securing continuous observations at fixed times. He took part in geodetic observations, and in 1843 and 1846 wrote two memoirs, Ueber Gegenstände der höheren Geodesie. He wrote on the attraction of homogeneous ellipsoids, 1813. In a memoir on capillary attraction, 1833, he solves a problem in the calculus of variations involving the variation of a certain double integral, the limits of integration being also variable; it is the earliest example of the solution of such a problem. He discussed the problem of rays of light passing through a system of lenses.

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Among Gauss' pupils were Christian Heinrich Schumacher, Christian Gerling, Friedrich Nicolai, August Ferdinand Möbius, Georg Wilhelm Struve, Johann Frantz Encke.

Gauss' researches on the theory of numbers were the starting-point for a school of writers, among the earliest of whom was Jacobi. The latter contributed to Crelle's Journal an article on cubic residues, giving theorems without proofs. After the

publication of Gauss' paper on biquadratic residues, giving the law of biquadratic reciprocity, and his treatment of complex numbers, Jacobi found a similar law for cubic residues. By the theory of elliptical functions, he was led to beautiful theorems on the representation of numbers by 2, 4, 6, and 8 squares. Next come the researches of Dirichlet, the expounder of Gauss, and a contributor of rich results of his own.

Peter Gustav Lejeune Dirichlet 88 (1805-1859) was born in Düren, attended the gymnasium in Bonn, and then the Jesuit gymnasium in Cologne. In 1822 he was attracted to Paris by the names of Laplace, Legendre, Fourier, Poisson, Cauchy. The facilities for a mathematical education there were far better than in Germany, where Gauss was the only great figure. He read in Paris Gauss' Disquisitiones Arithmeticæ, a work which he never ceased to admire and study. Much in it was simplified by Dirichlet, and thereby placed within easier reach of mathematicians. His first memoir on the impossibility of certain indeterminate equations of the fifth degree was presented to the French Academy in 1825. He showed that Fermat's equation, "+y" 2", cannot exist when n= 5. Some parts of the analysis are, however, Legendre's. Euler and Lagrange had proved this when n is 3 and 4, and Lamé proved it when n = 7. Dirichlet's acquaintance with Fourier led him to investigate Fourier's series. He became docent in Breslau in 1827. In 1828 he accepted a position in Berlin, and finally succeeded Gauss at Göttingen in 1855. The general principles on which depends the average number of classes of binary quadratic forms of positive and negative determinant (a subject first investigated by Gauss) were given by Dirichlet in a memoir, Ueber die Bestimmung der mittleren Werthe in der Zahlentheorie, 1849. More recently F. Mertens of Graz has determined the asymptotic values of several numerical functions. Dirichlet gave some

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 P, inferior to a given number x, must be comprised. 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,

the modulus is a divisor of their resultant. Joseph Liouville (1809-1882), professor at the Collége de France, investigated mainly questions on the theory of quadratic forms of two, and of a greater number of variables. Profound researches were instituted by Ferdinand Gotthold Eisenstein (1823-1852), of Berlin. Ternary quadratic forms had been studied somewhat by Gauss, but the extension from two to three indeterminates was the work of Eisenstein who, in his memoir, Neue Theoreme der höheren Arithmetik, defined the ordinal and generic characters of ternary quadratic forms of uneven determinant; and, in case of definite forms, assigned the weight of any order or genus. But he did not publish demonstrations of his results. In inspecting the theory of binary cubic forms, he was led to the discovery of the first covariant ever considered in analysis. He showed that the series of theorems, relating to the presentation of numbers by sums of squares, ceases when the number of squares surpasses eight. Many of the proofs omitted by Eisenstein were supplied by Henry Smith, who was one of the few Englishmen who devoted themselves to the study of higher arithmetic.

Henry John Stephen Smith" (1826-1883) was born in London, and educated at Rugby and at Balliol College, Oxford. Before 1847 he travelled much in Europe for his health, and at one time attended lectures of Arago in Paris, but after that year he was never absent from Oxford for a single term. In 1861 he was elected Savilian professor of geometry. His first paper on the theory of numbers appeared in 1855. The results of ten years' study of everything published on the theory of numbers are contained in his Reports which appeared in the British Association volumes from 1859 to 1865. These reports are a model of clear and precise exposition and perfection of form. They contain much original matter, but the chief results of his own discoveries were

printed in the Philosophical Transactions for 1861 and 1867. They treat of linear indeterminate equations and congruences, and of the orders and genera of ternary quadratic forms. He established the principles on which the extension to the general case of n indeterminates of quadratic forms depends. He contributed also two memoirs to the Proceedings of the Royal Society of 1864 and 1868, in the second of which he remarks that the theorems of Jacobi, Eisenstein, and Liouville, relating to the representation of numbers by 4, 6, 8 squares, and other simple quadratic forms are deducible by a uniform method from the principles indicated in his paper. Theorems relating to the case of 5 squares were given by Eisenstein, but Smith completed the enunciation of them, and added the corresponding theorems for 7 squares. The solution of the cases of 2, 4, 6 squares may be obtained by elliptic functions, but when the number of squares is odd, it involves processes peculiar to the theory of numbers. This class of theorems is limited to 8 squares, and Smith completed the group. In ignorance of Smith's investigations, the French Academy offered a prize for the demonstration and completion of Eisenstein's theorems for 5 squares. This Smith had accomplished fifteen years earlier. He sent in a dissertation. in 1882, and next year, a month after his death, the prize was awarded to him, another prize being also awarded to H. Minkowsky of Bonn. The theory of numbers led Smith to the study of elliptic functions. He wrote also on modern geometry. His successor at Oxford was J. J. Sylvester.

Ernst Eduard Kummer (1810-1893), professor in the University of Berlin, is closely identified with the theory of numbers. Dirichlet's work on complex numbers of the form a+ib, introduced by Gauss, was extended by him, by Eisenstein, and Dedekind. Instead of the equation x-1= 0, the roots of which yield Gauss' units, Eisenstein used the equation