Theory of Functions

integrals can be expressed by a definite number $p$ of similar integrals, where $p$ depends merely on the properties of the equation $F(x,y)=0$. It was shown later that $p$ is the deficiency of the curve $F(x,y)=0$. The addition theorems of elliptic integrals are deducible from Abel's theorem. The

hyperelliptic integrals introduced by Abel, and proved by him to possess multiple periodicity, are special cases of Abelian

integrals whenever $p=\text{or}>3$. The reduction of Abelian to elliptic integrals has been studied mainly by Jacobi, Hermite,

Königsberger, Brioschi, Goursat, E. Picard, and O. Bolza of

the University of Chicago.

Two editions of Abel's works have been published: the first by Holmboe in 1839, and the second by Sylow and Lie in

Abel's theorem was pronounced by Jacobi the greatest discovery of our century on the integral calculus. The aged

Legendre, who greatly admired Abel's genius, called it "monumentum

aere perennius." During the few years of work allotted to the young Norwegian, he penetrated new fields of research, the development of which has kept mathematicians busy for over half a century.

Some of the discoveries of Abel and Jacobi were anticipated by Gauss. In the Disquisitiones Arithmeticæ he observed

that the principles which he used in the division of the circle were applicable to many other functions, besides the circular, and particularly to the transcendents dependent on the integral ${\displaystyle \int \frac{dx}{\sqrt{1-x^4}}}$. From this Jacobi[]83 concluded that Gauss had

thirty years earlier considered the nature and properties of elliptic functions and had discovered their double periodicity. The papers in the collected works of Gauss confirm this conclusion.

Carl Gustav Jacob Jacobi[]84 (1804–1851) was born of Jewish parents at Potsdam. Like many other mathematicians he was initiated into mathematics by reading Euler. At the University of Berlin, where he pursued his mathematical studies independently of the lecture courses, he took the degree of Ph.D. in 1825. After giving lectures in Berlin for two years, he was elected extraordinary professor at Königsberg, and two years later to the ordinary professorship there. After the publication of his Fundamenta Nova he spent some time in travel, meeting Gauss in Göttingen, and Legendre, Fourier,

Poisson, in Paris. In 1842 he and his colleague, Bessel, attended

the meetings of the British Association, where they made the acquaintance of English mathematicians.

His early researches were on Gauss' approximation to the value of definite integrals, partial differential equations, Legendre's