Page:A History of Mathematics (1893).djvu/366

 exceptional phenomenon from the point of view of the differential equation.[89] A geometrical theory of singular solutions resembling the one used by Cayley was previously employed by W. W. Johnson of Annapolis.

An advanced Treatise on Linear Differential Equations (1889) was brought out by Thomas Craig of the Johns Hopkins University. He chose the algebraic method of presentation followed by Hermite and Poincaré, instead of the geometric method preferred by Klein and Schwarz. A notable work, the Traité d'Analyse, is now being published by Émile Picard of Paris, the interest of which is made to centre in the subject of differential equations. 

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