ABEL, NIELS HENRIK

born Aug. 5, 1802, island of Finny, near Stavanger, Nor. died April 6, 1829, Froland Norwegian mathematician, a pioneer in the development of several branches of modern mathematics. Abel was the son of a poor Protestant minister. Soon after Abel was born, his family moved to the parish of Gjerstad, near the town of Risr (southeast Norway), where the boy grew up. In 1815, when he entered the cathedral school in Oslo, his mathematical talent was recognized by a teacher who introduced him to the classics in mathematical literature and proposed original problems for solution. Thoroughly challenged, Abel studied the works of the 17th-century English mathematician and physicist Isaac Newton and the contemporary mathematicians Leonhard Euler (German), Joseph-Louis Lagrange (French), and Carl Friedrich Gauss (German) and learned to detect gaps in their mathematical reasoning. Although when Abel's father died in 1820 the family was left in straitened circumstances, the boy was able to enter the University of Christiania (Oslo) in 1821 because his teacher contributed and raised funds. He obtained a preliminary degree from the university in 1822 and continued his studies independently with further subsidies obtained by his teacher. His first papers, published in 1823 in the new periodical Magazin for Naturvidenskaberne, were on functional equations and integrals, his solution of an integral equation being the first. Abel's friends urged the Norwegian government to grant him a fellowship for study in Germany and France. While waiting for the royal decree to be issued, in 1824 he published at his own expense his proof of the impossibility of solving algebraically the general equation of the fifth degree, which he hoped would bring him recognition. He sent the pamphlet to Gauss, who dismissed it, failing to recognize that the famous problem had indeed been settled. Abel spent the winter of 1825-26 with Norwegian friends in Berlin, where he met August Leopold Crelle, civil engineer and self-taught enthusiast of mathematics, who became his close friend and mentor. With Abel's warm encouragement, Crelle founded the Journal fr die reine und angewandte Mathematik ("Journal for Pure and Applied Mathematics"), the first volume of which (1826) contains papers by Abel, including a more elaborate version of his work on the quintic equation. Other papers dealt with equation theory, functional equations, integration in finite forms, and problems from theoretical mechanics. In Berlin new directions in mathematics stimulated him to do further independent work. Soon distracted socially, however, Abel traveled throughout Europe. Arriving in Paris in the summer of 1826, he called on the foremost mathematicians and completed a memoir on transcendental functions. This major work presented a theory of integrals of algebraic functions, in particular the result known as Abel's theorem: there is a finite number, or genus, of independent integrals of this nature. This theorem is the basis for the later theory of Abelian integrals and Abelian functions. Abel was accepted with restrained civility in Paris, for his work was still unknown. He submitted his memoir for presentation to the Academy of Sciences, hoping to establish his reputation, but he waited in vain. Before leaving Paris, thinking he had a persistent cold, Abel consulted a physician, who informed him he had tuberculosis. Abel returned to Norway heavily in debt. He subsisted by tutoring, by receiving a small grant from the university, and, in 1828, by accepting a substitute-teaching position. His poverty and ill health did not decrease his production; he wrote a great number of papers during this period, principally on equation theory and elliptic functions. Among them are the theory of the Abelian equations with Abelian groups. He rapidly developed the theory of elliptic functions in competition with Karl Gustav Jacobi. By this time Abel's fame had spread to all mathematical centres, and strong efforts were made to secure a suitable position for him by a group from the French Academy, who addressed Bernadotte, the king of Norway-Sweden; Crelle worked to secure a professorship for him in Berlin. In the fall of 1828, Abel became seriously ill, and his condition deteriorated on a sled trip at Christmas time to visit his fiance at Froland, where he died. The French Academy of Sciences published his memoir in 1841. Additional reading The Oeuvres compltes de N.H. Abel, new ed., 2 vol. (1881, reprinted 2 vol. in 1, 1973), includes a short biographical note. For further information, see ystein Ore, Niels Henrik Abel, Mathematician Extraordinary (1957, reprinted 1974); and Eric T. Bell, Men of Mathematics (1937, reprinted 1986), pp. 307-326.