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9.1. Abel, Niels (1802-1829)

Niels Abel was one of the innovators in the field of elliptic functions, discoverer of Abelian functions and one of the leaders in the use of rigor in mathematics. His work was so revolutionary that one mathematician stated: "He has left mathematicians something to keep them busy for five hundred years." However, his life did not mirror his mathematical success and his story is one of the most tragic in the sciences.

Niels Henrik Abel was born to a Lutheran minister in Finnoy, Norway, on August 5, 1802. His family, which moved to Gjerstad shortly after his birth, was poor, but somehow they managed to support seven children. His early education came at home and at the age of 13, he was admitted to the Cathedral School in Oslo. The school had recently lost most of its teachers to the new University of Oslo and was staffed with inexperienced and incompetent instructors. Under these circumstances, Abel's performance was rather unimpressive. However, this changed when his first mathematics professor was dismissed for beating a student to death while disciplining him. He was replaced by a young Bernt Michael Holmboe, an assistant to Christopher Hansteen at the university. Both of these men would become close friends and strong supporters of Abel. Holmboe saw Abel's ability in mathematics and encouraged him with books and problems. Soon, the student was teaching the teacher as Abel quickly outpaced his professor.

Abel's first main contribution to mathematics came before entering college. For hundreds of years, mathematicians had searched in vain to discover the general solution for the quintic equation a x5 + b x4 + c x3 + d x2 + e x + f = 0. Abel developed what he thought was the answer. Holmboe and Hansteen knew there was no one in Norway with the ability to understand if the answer was correct, so they sent the paper to the mathematician Ferdinand Degen in Denmark. Before receiving an answer, Abel discovered a mistake in his figures and questioned if there was an answer. Taking the tract that there was not, he eventually proved that an algebraic solution to the quintic equation was impossible. More important, however, Degen suggested that Abel take up the subject of elliptic integrals, which would become the focus of his work and the source of his fame.

Before entering the University of Oslo in 1821, Abel's father died, leaving his son to support his mother and six siblings. Unable to meet his financial needs, he relied on grants from the university, gifts from his math professors and tutoring positions to keep his family afloat. However, his mathematics flourished. After fulfilling the requirements for graduation in one year, he was left on his own to study. In 1823, he published his first important paper on definite integrals, which included the first ever solutions of an integral equation. He also produced another valuable work on the integration of functions. Both of these works would have brought him instant renown and a professorship -- if anyone would have read them. Unfortunately, the works were written in Norwegian while the leading mathematicians of Europe wrote in French and German. The papers were ignored.

Desperately trying to make a name for himself, Abel tried to get a royal grant to travel Europe, but was forced to wait two years to learn French and German. During this period, he was engaged to marry. Mathematically, he sent a French copy of his work on the quintic equation to Gauss, the time's leading mathematician who could have brought him instant fame. However, the German rejected it as "another of those monstrosities" without ever reading it. Finally, in 1825, he was given a grant to travel Europe for two years by the nearly bankrupt Norwegian government.

The trip would prove to be a near disaster. His first stop was Denmark to meet with Degen, only to discover that he had died. His next trip brought him to Berlin, the only successful leg of the journey. Here he met August Leopold Crelle, an influential engineer and amateur mathematician who was pondering the creation of a journal dedicated to new mathematical ideas. According to the story, Crelle originally thought Abel was a candidate for the trade school where he worked. After a long struggle to find a language both understood, Abel replied that he was interested in mathematics. They then discussed one of Crelle's papers and after praising it as interesting, Abel impolitely pointed out several mistakes. Fortunately, the German kept an open mind and listened. Despite not understanding most of what he was talking about, he understood he was in the presence of genius. Crelle started his journal with Abel's works as the features of his first few editions. The two would hold a close friendship and Crelle would use his influence to try to get Abel the recognition he deserved.

Together, the two planned to travel first to Gottingen to meet Gauss and then onto Paris, the hub of mathematics. Unfortunately, Crelle was unable to go and Abel went to Paris alone. He arrived just in time to catch almost every mathematician on vacation. When they returned, they were civil but uninterested in talking about anything but their own work. However, Abel did manage to have his "masterpiece," a paper on elliptic functions and integrals which included Abel's theorem, presented to the French Academy of Sciences. If the work was accepted, he would have been made. Unfortunately, the Academy picked Legendre and Cauchy as referees to judge it. The former, who was in his seventies, claimed that he could not read the handwriting and left all the work to the latter. The latter, who was much more interested in his own work and possibly just a bit jealous, brought the work home and promptly "misplaced" it. Not until 1830, a year after Abel's death, was the paper given the recognition it deserved and awarded the grand prize by the Academy. It was not published until 1841.

Abel returned to Norway in failure. Not only was he unable to get the recognition he deserved and the professorship he desperately needed, he was in debt and had contracted tuberculosis. To add insult to injury, he had been passed over to fill a vacancy in the mathematics department at the university. The position had instead been given to his friend Holmboe, who had accepted it only after they threatened to give the job to a foreigner if he did not agree to take it. Abel survived on grants and gifts from both the university and his friends.

However, his mathematics did not suffer. He produced several papers on the theory of equations, including sections that introduced a new class of equations, now known as the Abelian. Meanwhile, he gained a rival in his study of elliptic functions and integrals in Carl Jacobi. Spurred on by his competitor and also fearing his illness would soon finish him, Abel production on the subject increased at a blazing pace. His work laid the foundation of all further studies into the field. Finally, people began to notice. Legendre, who had failed to read his masterpiece, started a correspondence with both Abel and Jacobi, praising them as two of "the foremost analysts of our times." Slowly, mathematicians all across Europe were calling for a professorship for the Norwegian.

Unfortunately, this was all too late. In 1829, he suffered an attack from his tuberculosis that would slowly kill him. With his fiancee at his side, he lost his battle to the disease on April 6, 1829. Two days later, Crelle sent him notice that he had finally been able to secure a position for him at the University of Berlin.

This tragic life contributed much to the field of mathematics. His proof of no solution to the quintic equation, the solutions to definite integrals, Abel's theorem and Abelian functions and equations were all valuable additions to the science. His most important achievements were his discovery of elliptic functions and his use of rigor. On the first subject, mathematicians had been studying elliptic integrals with limited success over the past century. Abel inverted these integrals into elliptic functions which were much easier to manipulate. This is similar to inverting the complicated inverse trigonometric functions arcsin and arccos into the much simpler sin and cos. On the second subject, Abel quickly realized that much of the previous mathematical work was unproved. He saw it as his responsibility to fill these holes in mathematics and provide the proofs that had been left out. He most significant work was the first proof of the general binomial theorem, which had been stated by Newton and Euler.

For related information on Abel, see: Augustin Cauchy , Leonhard Euler , Abel's test for series.

Sources

Historical information compiled by Paul Golba
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