Niels Henrik Abel

Niels Henrik Abel was born on August 5, 1802, at Nedstrand in southwestern Norway, where his father, Søren Georg Abel, was serving as a pastor ^[1]. The family soon moved to Gjerstad, a parish in the south of the country, and it was there that Abel spent his childhood. Norway in these years was a poor nation caught in the turbulence of the Napoleonic Wars: a British blockade, food shortages, and the political separation from Denmark in 1814 all left their mark on the household. Søren Georg Abel was politically active, serving in the Storting, the new Norwegian parliament, but his career ended in scandal and drink, and he died in 1820 leaving his widow and children in serious poverty ^[2].

In 1815 the thirteen-year-old Abel was sent to the Cathedral School in Christiania, as Oslo was then called. His first years there were unremarkable, and the school itself was in decline, staffed by teachers of uneven quality. The turning point came in 1818, when a brutal mathematics teacher was dismissed after beating a pupil so badly that the boy later died, and a young instructor named Bernt Michael Holmboe took over the class ^[1]. Holmboe quickly saw what he had in front of him. He gave Abel private lessons, guided him to the works of Newton, Euler, Lagrange, and Gauss, and wrote in a school report that the boy could become the greatest mathematician in the world ^[3].

When his father died, Abel became partly responsible for his mother and six siblings. He had no money of his own, and his continued education depended on the generosity of Holmboe and other professors, who arranged small stipends and contributed from their own pockets. In 1821 he entered the Royal Frederick University in Christiania, where he had already outgrown the mathematics anyone there could teach him ^[2].

Anyone asking who was Niels Henrik Abel in the early 1820s would have found a shy, badly dressed student who was nonetheless already producing research no one in Norway could evaluate. While still at school he attacked one of the most famous open problems of the age: whether the general equation of the fifth degree, the quintic, could be solved by a formula built from radicals, as the quadratic, cubic, and quartic equations could. At first he believed he had found such a formula, and his professors, unable to find the error, sent the work toward the Danish mathematician Ferdinand Degen. Degen asked for a numerical example, and in working one out Abel discovered his own mistake ^[3].

That failure redirected him. By 1823 he had reversed his approach entirely, and in 1824 he published, at his own expense, a short pamphlet proving that no general solution in radicals exists for equations of degree five. To save printing costs he compressed the argument onto six pages, which made it nearly unreadable, and the copy he sent to Carl Friedrich Gauss in Göttingen was apparently set aside unread ^[1]. The result, now known as the Abel-Ruffini theorem because the Italian Paolo Ruffini had earlier published an incomplete proof, closed a problem that had stood open for more than two and a half centuries ^[4].

Abel understood that his future depended on contact with the mathematicians of Paris and the German states. After two years of preparation, and armed with a modest travel grant from the Norwegian government, he left Christiania in the autumn of 1825. In Berlin he met August Leopold Crelle, an engineer and enthusiast who was about to found the Journal für die reine und angewandte Mathematik, the first major German journal devoted purely to mathematics. Crelle became Abel's closest ally, and the journal's opening volumes in 1826 carried a string of Abel's papers, including an expanded and rigorous proof of the insolvability of the quintic ^[4].

Any account of Niels Henrik Abel achievements has to begin with the quintic, but his deepest work lay elsewhere. In studying which equations can be solved by radicals, he isolated a class where solvability holds: equations whose roots can all be expressed rationally in terms of one of them, with the relevant expressions commuting with one another. Such equations are today called abelian, and the commutative groups that grew out of this circle of ideas are known as abelian groups, a naming that reaches into nearly every branch of modern algebra ^[4].

His second great subject was the theory of elliptic integrals, expressions that arise when computing arc lengths of ellipses and related curves. Mathematicians since Euler and Legendre had tabulated and manipulated these integrals directly. Abel's insight was to invert them: instead of studying the integral as a function of its endpoint, he studied the endpoint as a function of the integral. The resulting elliptic functions are doubly periodic and far easier to handle, and the inversion opened an entire field ^[5]. He developed the theory in a long memoir published in Crelle's journal in 1827 and 1828, working in an intense and productive rivalry with the young German mathematician Carl Gustav Jacob Jacobi, who reached similar ideas independently at almost the same moment ^[1].

The broadest of his results came in a manuscript he delivered to the French Academy of Sciences in Paris in October 1826. In it he proved a sweeping theorem on integrals of algebraic functions, generalizing the addition properties of elliptic integrals to a vastly wider class. Adrien-Marie Legendre later called it a monument more lasting than bronze, and the result is still taught as Abel's theorem; the associated abelian integrals and abelian functions occupied mathematicians for the rest of the nineteenth century ^[5]. The Academy handled the manuscript disastrously. Cauchy, appointed referee, took it home and mislaid it, and the memoir was not printed in full until 1841, long after its author's death ^[2]. Beyond these headline results, Abel also helped set a new standard of rigor in analysis, giving one of the first satisfactory studies of the convergence of the binomial series and insisting that infinite series be handled with proofs rather than habit ^[6].

Abel was engaged to Christine Kemp, called Crelly, a Danish governess he met at a ball in Christiania in 1823. The two became engaged around Christmas of 1824, but Abel's poverty made marriage impossible; he could not support himself, let alone a household. Crelly took positions as a governess in Norway to remain near him, including, in his final years, a post at Froland ironworks in the south of the country ^[3].

Those who knew Abel described a modest, cheerful, and unassuming man, devoted to his friends and fond of the theater and of card games. His letters show flashes of wit alongside constant anxiety about money and about his family in Gjerstad, for whom he felt responsible after his father's death. During his European journey of 1825 to 1827 he traveled with a circle of young Norwegian friends, and his choice to accompany them to Berlin and later through the Alps to Italy, rather than going straight to Paris and Gauss as his grant intended, drew criticism at home but gave him the Berlin connection with Crelle that carried his work into print ^[2].

Paris, when he finally reached it in the summer of 1826, disappointed him. The great French mathematicians were polite but distant, his memoir vanished into the Academy's files, and he lived cheaply and ate poorly. It was in Paris, in the winter of 1826 to 1827, that he first fell seriously ill; a doctor told him he had tuberculosis, a diagnosis Abel refused to accept ^[1].

Abel returned to Norway in May 1827 nearly penniless, having exhausted his grant. The university in Christiania had no chair for him; the only professorship in mathematics was held by Christopher Hansteen, and when Hansteen departed on an expedition to Siberia in 1828, Abel was given his teaching duties as a substitute, along with tutoring work, to scrape together a living ^[2]. Through all of this he kept publishing at an extraordinary rate, sending paper after paper to Crelle's journal as his health declined, aware that Jacobi was advancing quickly on elliptic functions and determined not to be overtaken ^[5].

His reputation abroad was finally catching up with his work. Legendre, then in his seventies, corresponded warmly with both Abel and Jacobi, and in 1828 a group of French academicians petitioned the Swedish-Norwegian king to find a suitable position for the young Norwegian. Crelle, meanwhile, lobbied persistently in Berlin to have a post created for Abel there ^[4].

At Christmas 1828 Abel traveled by sled to Froland to spend the holiday with his fiancée. The winter journey worsened his condition, and while there he suffered a violent hemorrhage. He never recovered. Confined to bed at the ironworks manor and nursed by Crelly, he died of tuberculosis on April 6, 1829, at the age of twenty-six ^[1]. Two days later, not yet knowing of the death, Crelle wrote in jubilation from Berlin: he had secured Abel an appointment there at last ^[3].

Recognition, so slow in life, came almost immediately after death. In June 1830 the French Academy of Sciences awarded its Grand Prix in mathematics jointly to Abel, posthumously, and to Jacobi for their work on elliptic functions ^[2]. Crelle's journal published the papers Abel left behind, and the lost Paris memoir was finally printed in 1841. A two-volume edition of his collected works, prepared by Ludvig Sylow and Sophus Lie, appeared in 1881 and fixed his place in the mathematical canon ^[6].

Few mathematicians have their name attached to so much. Abelian groups, abelian integrals, abelian functions, abelian varieties, and the Abel-Ruffini theorem are standard vocabulary, and the adjective is so common that it is conventionally written with a lowercase a, a quiet mark of how completely his ideas were absorbed. His insistence on proof over formal manipulation helped push analysis toward the rigor that defined the century after him, and his work on equations fed directly into the theory that Évariste Galois, another short-lived genius, completed a few years later ^[4].

Norway has honored him steadily. A monument by Gustav Vigeland was raised in the royal palace grounds in Oslo in 1908, his portrait appeared on banknotes and stamps, and a crater on the Moon carries his name. The most substantial tribute came in 2002, on the bicentennial of his birth, when the Norwegian government established the Abel Prize, an award for outstanding work in mathematics presented annually by the Norwegian Academy of Science and Letters and often described as the closest equivalent to a Nobel Prize in the field ^[7]. For readers of any Niels Henrik Abel biography, the sharpest of all Niels Henrik Abel facts remains the contrast at its center: a man who could not find a paying job in mathematics during his lifetime now names one of its highest honors.