We begin our sketch of the vast progress in the theory of functions by considering the special class called elliptic functions.
These were richly developed by Abel and Jacobi.
Niels Henrick Abel (1802–1829) was born at Findoë in Norway, and was prepared for the university at the cathedral school in Christiania. He exhibited no interest in mathematics until 1818, when B. Holmboe became lecturer there,
and aroused Abel's interest by assigning original problems to the class. Like Jacobi and many other young men who became eminent mathematicians, Abel found the first exercise of his talent in the attempt to solve by algebra the general equation of the fifth degree. In 1821 he entered the University in Christiania. The works of Euler, Lagrange, and Legendre were closely studied by him. The idea of the inversion of elliptic functions dates back to this time. His extraordinary success in mathematical study led to the offer of a stipend by the government, that he might continue his studies
in Germany and France. Leaving Norway in 1825, Abel visited
the astronomer, Schumacher, in Hamburg, and spent six
months in Berlin, where he became intimate with August Leopold Crelle (1780–1855), and met Steiner. Encouraged by
Abel and Steiner, Crelle started his journal in 1826. Abel began to put some of his work in shape for print. His proof of the impossibility of solving the general equation of the fifth degree by radicals,–-first printed in 1824 in a very concise form, and difficult of apprehension,–-was elaborated in greater detail, and published in the first volume. He entered also upon the subject of infinite series (particularly the binomial
theorem, of which he gave in Crelle's Journal a rigid general investigation), the study of functions, and of the integral calculus. The obscurities everywhere encountered by him owing to the prevailing loose methods of analysis he endeavoured to clear up. For a short time he left Berlin for Freiberg, where he had fewer interruptions to work, and it was there that he made researches on hyperelliptic and Abelian
functions. In July, 1826, Abel left Germany for Paris without having met Gauss! Abel had sent to Gauss his proof of
1824 of the impossibility of solving equations of the fifth
degree, to which Gauss never paid any attention. This slight, and a haughtiness of spirit which he associated with Gauss, prevented the genial Abel from going to Göttingen. A similar feeling was entertained by him later against Cauchy. Abel
remained ten months in Paris. He met there Dirichlet,