Niels Abel
1802-1829
Norwegian Mathematician
Niels Abel, in his tragically short life, made fundamental contributions to the study of mathematics. He may be best known for his work on elliptic functions and definite integrals, but he also proved the insolvability of quintic equations (equations in which a factor is raised to the fifth power) and a class of functions named for him, the Abelian functions. He also established new standards for mathematical rigor in his work and developed a general proof for Leonhard Euler's binomial theorem.
Abel was born the son of a Lutheran minister in a small town in Norway. He did not attend formal schooling until he was 13, and that school was inadequately staffed, including the Mathematics department. However, shortly after starting, the mathematics teacher was dismissed and replaced by a junior professor from a nearby university, Bernt Holmboe. Under Holmboe's tutelage and encouragement Abel's mathematical skills blossomed. Before long, Abel's skills overtook his teacher's.
Before entering university studies, Abel made his first major contribution to mathematics, trying to develop a general solution to the quintic equation, a problem that had vexed mathematicians for over 200 years. Thinking he had found a solution, he wrote a proof and sent it to the Danish mathematician Degen. Quickly finding a flaw in his work, Abel continued to work and eventually was able to prove that there was no general solution for problems of this sort. He sent this off to Degen, too (there were no Norwegian mathematicians at that time who were qualified to critique his work). Degen was impressed and suggested that Abel study elliptic integrals, another particularly difficult problem for mathematicians.
Abel's father died in 1821, just as Abel was beginning his studies. Abel supported himself and helped support his family by tutoring and with grants from the university and his professors. Nevertheless, he completed his graduation requirements in one year and continued to work on his research at the same time. In 1823 he published his next major work, which included the first solution to an integral equation. Thiswas followed by an important work on the integration of functions.
Niels Abel. (The Library of Congress. Reproduced by permission.)Unfortunately, Abel's work was published in Norwegian, a language not read by most mathematicians, who worked in French or German. Because of this, the general mathematical community ignored his early papers and he was not offered the professorship his work had already earned him. In fact, papers he sent to Carl Gauss (1777-1855), Adrien-Marie Legendre (1752-1833), and Augustin Cauchy (1789-1857) were completely ignored.
In 1825 the Norwegian government gave Abel a grant to visit mathematicians elsewhere in Europe. Unfortunately, Degen, his first stop, had died and he managed to visit Paris when virtually all mathematicians were on vacation. He did manage to meet with August Crelle (1780-1855) who, in spite of language difficulties, recognized Abel's genius and later published some of his work.
Returning to Norway, Abel was dejected. He had received virtually no recognition for his work, had not been offered a professorship that he desperately needed, and had contracted tuberculosis during his trip. On top of that,upon his return he found that a Norwegian teaching position had been offered to someone else, leaving him without a job. As before, he survived on grants and generosity while continuing to work on his mathematics.
During the next few years, understanding that he was ill, Abel worked at a feverish pace. He finally began to receive some degree of recognition from mainstream European mathematicians, including Legendre and Karl Jacobi (1804-1851). He was finally offered a much-deserved professorship at the University of Berlin in 1829. Unfortunately, he died shortly before the offer was sent. After Abel's death, a colleague commented, "he has left mathematicians something to keep them busy for five hundred years."
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