Abel has left mathematicians enough to keep them busy for 500 years.
Niels Henrik Abel (1802—1829) was one of the foremost Norwegian mathematicians of the nineteenth Century. Along with his older contemporaries Gauss and Cauchy, Abel was one of the pioneers in the development of modern mathematics, which is characterized by its insistence on rigorous proof.
He was the second of six children. Abel and his brothers received their first education from their father. At the age of 13, Abel along with his older brother, was sent to the Cathedral school in Christiania (Oslo). In 1817, his mathematics teacher was Bernt Michael Holmbë recognized Abel’s talent and started giving him special problems and recommended special books outside the curriculum. Abel and Holmbë read the calculus text of Euler and the work of Lagrange and Laplace. Soon Abel became familiar with most of the important mathematical literature.
Abel’s father died when he was 18 years old and the responsibility of supporting the family fell on his shoulders. They subsisted with the aid of friends and neighbours, but somehow managed to enter the University of Oslo in 1821. His earliest researches were published in 1823, and included his solution of the classic Tautochrone problem (the problem of finding the curve down which a bead placed anywhere will fall to the bottom in the same amount of time) by means of an integral equation that is now known by his name.
Abel, in his last year of school, attacked the problem of the solvability of the quantic (fifth-degree) equation ax5 + bx4 + ex3 + dx2 + ex + f = 0, a problem that had been unsettled since the sixteenth century. Abel thought that he had solved the problem and submitted his work for publication. Unable to find an error and understand his arguments, he was asked by the editor to illustrate his method. In 1824, during the process of illustration he discovered an error. This discovery led Abel to a proof that no such solution exists. He also worked on elliptic functions and in essence revolutionized the theory of elliptic functions.
He travelled to Paris and Berlin in order to find a teaching position. Then poverty took its toll, and Abel died from tuberculosis on April 6, 1829. Two days later a letter from Crelle reached his address, conveying the news of his appointment to the professorship of mathematics at the University of Berlin.
Mathematicians, however, have their own ways of remembering their great men, and so we speak of Abel’s integral equation, Abelian integrals and functions, Abelian groups, Abel’s series, Abel’s partial summation formula, Abel’s limit theorem in the theory of power series, and Abel summability. Few have had their names linked to so many concepts and theorems in modern mathematics, and what he might have accomplished in a normal lifetime is beyond conjecture.