Niels Henrik Abel-The Greatest Norwegian Mathematician

Niels Henrik Abel was one of the most prominent mathematicians of the world in the first half of the 19th century. He is probably the most well-known Norwegian mathematician ever. Abel founded the theory of groups. He showed that the general fifth-degree equation is not solvable algebraically. Abel’s theorem and Abelian functions and equations were all valuable additions to the science of mathematics. He revolutionized the important area of elliptic integrals with his theory of elliptic and transcendental functions. He also contributed to the theory of infinite series. Abel on realising that much of the previous mathematical work was unproved, took it as his own responsibility to fill these gaps in mathematics by providing the proofs that had been left out. His most significant work was the first proof of the general binomial theorem, which had been stated by Newton and Euler. The adjective “abelian”, derived from his name, has become a commonplace in mathematical writing. Abel’s work in mathematics was so revolutionary that one mathematician stated: “He has left mathematicians something to keep them busy for five hundred years.” Abel’s life story is one of the most tragic in the history of science.

Niels Henrik Abel was born on August 05, 1802 in Finnøy, an island near the Norwegian town of Stavanger. His family moved to Gjerstad shortly after his birth. His father Soren Georg Abel was a Lutheran minister. Soren Abel studied at the University of Copenhagen and he had a degree in theology. He was a prominent Norwegian nationalist who was active politically in the movement to make Norway independent. Abel’s mother Anne Marie (nee Simonson) was the daughter of a wealthy merchant. Abel was brought up at Gjerstad, where his father was appointed as minister to succeed his father-in-law. Abel was taught by his father in the vicarage until he reached 13 years of age. Abel’s father was a member of the session of the Norwegian Parliament (Storting) that was specially convened in 1814 with a specific purpose—rewriting the Norwegian constitution reflecting union with Sweden in place of Denmark. He again tried to enter the Parliament in 1816 but he failed to be elected. In 1818, he was re-elected but his political career ended in disgrace by making false charges against his colleagues in the Storting.

Abel was growing in a period when Norway was passing through a difficult period. At the end of the 18th century Norway was part of Denmark. During the Napoleonic wars Denmark decided to remain neutral. Accordingly they signed a neutrality treaty in 1794. However, in 1801 England considered this neutrality treaty as an aggressive act. The English fleet destroyed most of the Danish fleet in a battle in the harbour at Copenhagen. Denmark avoided wars until 1807. But then England feared that the French may use the Danish fleet to invade and they thought it will be in their own interest to attack Denmark. They captured the whole Danish fleet in October 1807. In this way Denmark was compelled to join the alliance against England. The war led to an economic crisis in Norway. Due to war restrictions they could neither export timber (which was largely to England) nor import food grains from Denmark. There were wide spread poverty and suffering among the people. In 1813 Denmark was attacked by Sweden from the south. Following a treaty between the two countries, Denmark handed over Norway to Sweden in 1814. After a few months there was an attempt by Norway to gain independence. This prompted Sweden to attack Norway. Sweden after gaining control of Norway set up a complete internal self-government for Norway. The seat of the government was at Christiania.

At the age of 13, Abel entered the Cathedral School of Christiania (today’s Oslo). At the time when Abel joined the school it was in a bad state. This is because most of the good teachers had left the school in 1813 to join the newly established University of Christiania. The environment of the school failed to inspire Abel and he was nothing but an ordinary student with some talent for mathematics and physics. Though he had developed some liking for mathematics but his mathematics teacher was very cruel. The teacher hardly cared for the students. One day he hit a student so badly that he died a few days later. This incident proved to be a turning-point for Abel. The teacher was suspended and the Bernt Michael Holmboe replaced him in 1817. He was an inspiring and caring teacher. Holmboe saw that Abel had special skills in mathematics and he helped and supported Abel as long as he lived. After recognizing the exceptional mathematical talent of Able, Holmboe persuaded Abel to study the works of great mathematicians like Leonhard Euler (1707-1783), Comte Joseph Louis Lagrange (1736-1813), and Pierre-Simon Laplace (1749-1827).Abel borrowed books and studied on his own. He went above the usual level and soon he attacked problems that were unsolved at that time, such as fifth degree equations. He went deep into the mathematics but he did not do very well in the other subjects.

In 1820 tragedy struck Abel’s family when his father died. Abel was still in School. His father’s death left the family in dire poverty. There was now no money to allow Abel to complete his school education. His mathematics teacher helped him to complete his school education. A small pension from the state allowed Abel to enter Christiania University in in 1821. But before entering the University Abel made a contribution to mathematics. For hundreds of years, mathematicians had searched in vain to discover the general solution for the quintic (the fifth power) equation, a x5 + b x4 + c x3 + d x2 + e x + f = 0. Abel developed what he thought was the formula to solve the fifth degree equation. To see whether the answer was correct or not, Abel’s paper containing the solution to the age-old problem was sent to the mathematician Ferdinand Degen in Denmark. However, before Degen could send his observations, Abel himself discovered a mistake in his figures and wondered whether there was really an answer to the problem. He eventually proved that an algebraic solution to the quintic equation was impossible. But his interaction with Degen proved to be useful in another way. Degen could not find anything wrong about it and he praised Abel’s work but he also recommended him to take up the subject of elliptic integrals. This became the focus of Abel’s subsequent work and the source of his fame.
At the Christiania University, Abel was patronized by Christopher Hansteen, professor of astronomy. Hansteen not only supported Abel financially but also encouraged him to continue his studies. Hansteen’s wife cared for Abel as her own son.

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. This paper contained the first ever solutions of an integral equation. He also produced another valuable work on the integration of functions. His papers, though they were very important, failed to bring him fame or an appointment. In fact his papers were not read by important mathematicians of Europe. This is because Abel wrote his papers in Norwegian while the leading mathematicians of Europe wrote in French and German.

In 1823, Abel visited Copenhagen. The purpose was to be familarised with the works of the Danish mathematicians. In those days Abel’s own country Norway had no good school of mathematics. His visit to Copenhagen was possible because he received financial support from Christopher Hansteen. It was at Copenhagen, Abel met Christine Kemp, with whom he became engaged. The authorities of the University of Christiania taking recognition of Abel’s mathematical talent provided necessary funds to Able for studying in Paris. As per the original plan Abel was to visit Gauss at Gottingen first and then go to Paris. However, this did not happen. The two great mathematicians never met. Gauss’ biographer G. Waldo Dunnington wrote: “When Niels Henrik Abel (1802-1829) of Norway, one of the most important mathematicians of the nineteenth century, went to Germany in 1825, he had originally intended to visit Karl Friedrich Gauss (1777-1855). Abel was not well known at the time. A copy of his proof of the impossibility of solving the general equation of the fifth degree had been sent to Gauss, who did not consider it very important. As he did not get any response from Gauss, Abel cancelled his planned visit to Gottingen. Abel thought Gauss did not do enough to put him before the public. After this incident he had no further interaction with Gauss and was exceedingly critical of him". It is very unfortunate that the two great mathematicians did not meet. Besides meeting Gauss, Abel had wanted to use the splendid university library in Gottingen. Gauss did realize his mistake. But then it was too late. After Abel’s death Gauss wrote to Schumacher on May 19, 1829: “Abel’s death, which I have not seen announced in any newspaper, is a very great loss for science. Should anything about life circumstances of his highly distinguished mind he printed, and come to your hands, I beg you to communicate it to me. I would also like to have his portrait if it were to be had anywhere.”

Abel, with some other students of the university went to Berlin before finally going to Paris. The year was 1826. It was not a good decision when we consider the fact that Abel spent most part of his grants for visiting Berlin. However, the most positive side of this visit was that Able met August Leopold Crelle (1780-1855), who had just founded the Journal fur die reine und angewandte Mathematik (Journal for Pure and Applied Mathematics). The journal was popularly called Crelle’s Journal. Crelle became Abel’s mentor. He encouraged Abel to publish his results in his Journal. The very first volume of the Journal had Abel’s seven papers. Abel published most of his major works in Crelle’s Journal. Abel’s association with Crelle was important because Abel could not persuade the French Academie des Sciences to publish his work.

In 1826 Abel moved to Paris, where he stayed for about ten months. He met leading mathematicians of France. However, Abel’s work was poorly appreciated, as his work was scarcely known. Abel managed to present his “masterpiece,” a paper on elliptic functions and integrals which included Abel’s theorem to the French Academy of Sciences. The Academy referred the paper to Adrien-Marie Legendre (1752-1833) and Augustin Louis Cauchy (1789-1857). Legendre, who was in his seventies, claiming that he had difficulty in reading the handwriting Abel left the entire work to Cauchy. Cauchy brought the work home for reading but he promptly reported that the work was misplaced. It is said that he ‘misplaced’ it intentionally. This is because Cauchy was much more interested in his own work and he was a little jealous of Abel. The paper was given its due recognition in 1830, a year after Abel’s death. The French Academy awarded the grand prize. However, the paper was not published until 1841.

Commenting on his experience of the visit, Abel wrote: “Legendre is an exceedingly courteous man, but unfortunately as old as the stones. Cauchy is mad, and you cannot get anywhere with him, although he is the mathematician who knows at the moment how to treat mathematics. Cauchy is extremely Catholic and bigoted. A very strange thing in a mathematician…Poisson is a short man with a nice little belly. He carries himself with dignity. Likewise Fourier. Lacroix is terribly bald and extremely old. On Monday I am going to be introduced to several of these gentlemen by Hachette. Otherwise I do not like the Frenchman as much as the German, the Frenchman is uncommonly reserved towards foreigners. It is difficult to make his close acquaintance. And I dare not count on such a thing. Everyone wants to teach and nobody to learn. The most absolute egotism prevails everywhere. The only things that the Frenchman seeks from foreigners are the practical…He is the only one who can create something theoretical…you can imagine that it is difficult to become noticed, especially for a beginner.”

Because of financial compulsion Abel had to abandon his tour. After returning to Norway he taught for some time at Christiana. Abel failed to get the recognition that he rightly deserved. He had no appointment. A vacancy in mathematics department of the Christiana University arose but this was given to his teacher and mentor Holmboe. Holmboe wanted that the job should go to Abel. But when the university authorities threatened to give the job to a foreigner if he did not agree to take it, Holmboe accepted it. To increase his misery Abel was in debt and had contracted tuberculosis. Abel could manage to survive with meager grants and support from his friends. However, with all difficulties Abel continued to work. He produced several papers on the theory of equations, including sections that introduced a new class of equations, now known as the Abelian equations. In his study of elliptic functions and integrals Abel found a rival in Carl Gustav Jacob Jacobi (1804-1851) . He was also worried that his illness could end his life at any time. He was not deterred. He continued to work with a fervent zeal. His work laid the foundation of all further studies into the field. Eventually mathematicians had to take note of Abel’s work. Legendre started a correspondence with both Abel and Jacobi, praising them as two of “the foremost analysts of our times.” A demand for a professorship for Abel was raised by mathematicians all across Europe.

Niels Henrik Abel died April 6th in 1829 of tuberculosis. Two days later, Crelle sent him a letter informing him that Cerel had finally succeeded in getting a position for Abel at the University of Berlin. Abel’s works edited Holmboe were published in 1839 by the Swedish government. Later a more complete edition by Ludwig Sylow and Sophus Lie was brought out in 1881. After his death Abel became a national hero in Norway. His birth centenary (1902) was widely celebrated and a number of memorioals were erected—the most important among them was the monument by Vigeland which stands in the ‘Abel Garden’, the park of the Royal Palace.

We will end this write-up on Abel, one of the greatest mathematicians of all time by quoting August Leopold Crelle: “All of Abel’s works carry the imprint of an ingenuity and force of thought which is unusual and sometimes amazing, even if the youth of the author is not taken into consideration. One may say that he was able to penetrate all obstacles down to the very foundations of the problems, with a force which appeared irresistible; he attacked the problems with extraordinary energy; he regarded them from above and was able to soar so high over their present state that all difficulties seemed to vanish under the victorious onslaught of his genius…But it was not only his great talent which created the respect for Abel and made his loss infinitely regrettable. He distinguished himself equally by the purity and nobility of his character and by a rare modesty which made his person cherished to the same unusual degree as was his genius”.