Page:A History of Mathematics (1893).djvu/368

 as docent. Crelle secured at last an appointment for him at Berlin; but the news of it did not reach Norway until after the death of Abel at Froland.[82]

At nearly the same time with Abel, Jacobi published articles on elliptic functions. Legendre's favourite subject, so long neglected, was at last to be enriched by some extraordinary discoveries. The advantage to be derived by inverting the elliptic integral of the first kind and treating it as a function of its amplitude (now called elliptic function) was recognised by Abel, and a few months later also by Jacobi. A second fruitful idea, also arrived at independently by both, is the introduction of imaginaries leading to the observation that the new functions simulated at once trigonometric and exponential functions. For it was shown that while trigonometric functions had only a real period, and exponential only an imaginary, elliptic functions had both sorts of periods. These two discoveries were the foundations upon which Abel and Jacobi, each in his own way, erected beautiful new structures. Abel developed the curious expressions representing elliptic functions by infinite series or quotients of infinite products. Great as were the achievements of Abel in elliptic functions, they were eclipsed by his researches on what are now called Abelian functions. Abel's theorem on these functions was given by him in several forms, the most general of these being that in his Mémoire sur une propriété générale d'une classe très-étendue de fonctions transcendentes (1826). The history of this memoir is interesting. A few months after his arrival in Paris, Abel submitted it to the French Academy. Cauchy and Legendre were appointed to examine it; but said nothing about it until after Abel's death. In a brief statement of the discoveries in question, published by Abel in Crelle's Journal, 1829, reference is made to that memoir. This led Jacobi to inquire of Legendre what had become of it.