Niels Henrik Abel

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Lore & Background

Niels Henrik Abel was born prematurely in Nedstrand, Norway, as the second child of the pastor Søren Georg Abel and Anne Marie Simonsen. When Abel was born, the family was living at a rectory on Finnøy. Much suggests that Niels Henrik was born in the neighboring parish, as his parents were guests of the bailiff in Nedstrand in July and August of his year of birth. His father had a degree in theology and philosophy and served as pastor at Hesby Church on Finnøy. After his grandfather's death in 1804, Søren was appointed pastor at Gjerstad and the family moved there. The Abel family originated in Schleswig and came to Norway in the 17th century. Abel's mother, from Risør, was said to be early on an alcoholic and took little interest in the upbringing of the children. Niels and his brothers were given their schooling by their father, with handwritten books to read. An addition table in a book of mathematics reads: 1+0=0. In 1815, at age 13, Abel entered the Cathedral School. A new mathematics teacher, Bernt Michael Holmboe, was appointed in 1818. He saw Niels Henrik's talent in mathematics, and encouraged him to study the subject to an advanced level. He even gave Niels private lessons after school. In 1818, Søren Abel had a public theological argument with the theologian Stener Johannes Stenersen, which was well covered in the press. Søren was given the nickname 'Abel Treating' (Norwegian: 'Abel Spandabel'). Niels' reaction to the quarrel was said to have been 'excessive gaiety'. At the same time, Søren also almost faced impeachment after insulting Carsten Anker, and in September 1818 he returned to Gjerstad with his political career in ruins. He began drinking heavily and died only two years later, in 1820, aged 48. Bernt Michael Holmboe supported Niels Henrik Abel with a scholarship to remain at the school and raised money from his friends to enable him to study at the Royal Frederick University. When Abel entered the university in 1821, he was already the most knowledgeable mathematician in Norway. Holmboe had nothing more he could teach him and Abel had studied all the latest mathematical literature in the university library. During that time, Abel started working on the quintic equation in radicals. In 1821, Abel thought he had found the solution. The two professors of mathematics in Christiania, Søren Rasmussen and Christopher Hansteen, found no errors in Abel's formulas, and sent the work on to Carl Ferdinand Degen in Copenhagen. He too found no faults but still doubted that the solution could really have been found by an unknown student in far-off Christiania. Degen noted Abel's unusually sharp mind, and believed that such a talented young man should not waste his abilities on such a 'sterile object' as the fifth degree equation, but rather on elliptic functions and transcendence; for then, wrote Degen, he would 'discover Magellanian thoroughfares to large portions of a vast analytical ocean'. Degen asked Abel to give a numerical example of his method. While trying to provide an example, Abel found a mistake in his paper. This led to a discovery in 1823 that a solution by formula to a fifth- or higher-degree equation was not necessarily possible. Abel graduated in 1822. His performance was exceptionally high in mathematics and average in other matters. After he graduated, professors from university supported Abel financially, and Professor Christopher Hansteen let him live in a room in the attic of his home. Abel would later view Ms. Hansteen as his second mother. While living here, Abel helped his younger brother, Peder Abel, through examen artium. He also helped his sister Elisabeth to find work in the town. In early 1823, Niels Abel published his first article in 'Magazin for Naturvidenskaberne', Norway's first scientific journal, which had been co-founded by Professor Hansteen. Abel published several articles, but the journal soon realized that this was not material for the common reader. In 1823, Abel also wrote a paper in French. It was 'a general representation of the possibility to integrate all differential formulas' (Norwegian: en alminnelig Fremstilling af Muligheten at integrere alle mulige Differential-Formler). He applied for funds at the university to publish it. However, the work was lost while being reviewed, never to be found thereafter. In mid-1823, Professor Rasmussen gave Abel a gift of 100 speciedaler so he could travel to Copenhagen and visit Ferdinand Degen and other mathematicians there. While in Copenhagen, Abel did some work on Fermat's Last Theorem. Abel's uncle, Peder Mandrup Tuxen, lived at the naval base in Christianshavn, Copenhagen, and at a ball there Niels Abel met Christine Kemp, his future fiancée. In 1824, Christine moved to Son, Norway, to work as a governess and the couple got engaged over Christmas. After returning from Copenhagen, Abel applied for a government scholarship in order to visit top mathematicians in Germany and France, but he was instead granted 200 speciedaler yearly for two years, to stay in Christiania and study German and French. In the next two years, he was promised a scholarship of 600 speciedaler yearly and he would then be permitted to travel abroad. While studying these languages, Abel published his first notable work in 1824, Mémoire sur les équations algébriques où on démontre l'impossibilité de la résolution de l'équation générale du cinquième degré (Memoir on algebraic equations, in which the impossibility of solving the general equation of the fifth degree is proven). By 1823, Abel had at last proved the impossibility of solving the quintic equation in radicals (now referred to as the Abel–Ruffini theorem). However, this paper was in an abstruse and difficult form, in part because he had restricted himself to only six pages in order to save money on printing. A more detailed proof was published in 1826 in the first volume of Crelle's Journal. In 1825, Abel wrote a personal letter to King Carl Johan of Norway/Sweden requesting permission to travel abroad. He was granted this permission, and in September 1825 he left Christiania together with four friends from university (Christian P.B Boeck, Balthazar M. Keilhau, Nicolay B. Møller and Otto Tank). These four friends of Abel were traveling to Berlin and to the Alps to study geology. Abel wanted to follow them to Copenhagen and from there make his way to Göttingen. The terms for his scholarship stipulated that he was to visit Gauss in Göttingen and then continue to Paris. However, when he got as far as Copenhagen, he changed his plans. He wanted to follow his friends to Berlin instead, intending to visit Göttingen and Paris afterwards. On the way, he visited the astronomer Heinrich Christian Schumacher in Altona, now a district of Hamburg. He then spent four months in Berlin, where he became well acquainted with August Leopold Crelle, who was then about to publish his mathematical journal, Journal für die reine und angewandte Mathematik. This project was warmly encouraged by Abel, who contributed much to the success of the venture. Abel contributed seven articles to it in its first year. From Berlin Abel also followed his friends to the Alps. He went to Leipzig and Freiberg to visit Georg Amadeus Carl Friedrich Naumann.

Reader's Guide

Niels Henrik Abel's significance lies in his resolution of a problem that had baffled mathematicians for over 250 years: the impossibility of solving the general quintic equation in radicals. This proof, along with his innovations in elliptic functions and his discovery of Abelian functions, established him as a foundational figure in modern algebra and analysis. Most of his work was done in six or seven years of his working life, and he made his discoveries while living in poverty. He died at age 26 from tuberculosis. The French mathematician Charles Hermite remarked: 'Abel has left mathematicians enough to keep them busy for five hundred years.' Another French mathematician, Adrien-Marie Legendre, said: 'What a head the young Norwegian has!' Abel's legacy endures through the concepts named after him, including Abelian functions, which remain central to mathematics.

Did You Know?

• Abel was born prematurely in Nedstrand, Norway, and much suggests he was born in the neighboring parish while his parents were guests of the bailiff.
• His first notable paper on the quintic equation was restricted to only six pages in order to save money on printing.
• Abel's mathematics teacher Bernt Michael Holmboe raised money from his friends to enable Abel to study at the Royal Frederick University.
• While trying to provide a numerical example of his supposed solution to the quintic equation, Abel found a mistake that led to his discovery that a solution by formula was not necessarily possible.
• Abel contributed seven articles to the first year of Crelle's Journal, which he warmly encouraged.