Niels Henrik Abel (1802–1829) died at the modest age of 26 years old. Largely self-taught, in his short life the young Abel made pioneering contributions to variety of subjects in pure mathematics, including: algebraic equations, elliptic functions, elliptic integrals, functional equations, integral transforms and series representations.

Born on the small island of Finnøy in Rogaland, Norway, the early years of Abel’s short life was dominated by the instability of an alcoholic father who died when he was 16 years old. More or less self-taught, at 21, Abel provided the first complete proof demonstrating that there is no general algebraic solution for the roots of a quintic equation, or any general polynomial equation of degree greater than four. At that point, the problem had been unresolved for over 250 years. In the process of writing the proof, he laid the foundation — independently of Galois — for the branch of mathematics now known as group theory. At 22, he also wrote a fundamental work on elliptic integrals which helped lay the foundation for what would later be the theory of elliptic functions. Then, on April 6th 1829, at the age of 26, Abel died of tuberculosis. He had contracted the disease while in Paris and become ill while traveling (by sled) to the south of Norway to visit his fiancée.

Although he was never able to attain a permanent research or teaching position, in the ultimate cruel irony, two days after he passed a letter arrived from August Crelle (1780–1855) (of Crelle’s Journal) announcing that Abel been appointed Professor of Mathematics at the University of Berlin.

This is the story of the “Mozart of Mathematics”, Niels Henrik Abel.

Niels Henrik was either born on the island of Finnøy or in Nedstrand near Stavanger in (then) Denmark-Norway, on August 5th 1802. His father Søren Georg Abel was a pastor, as was his father Hans Mathias before him. Their family originated from Denmark. His mother Anne Marie Abel (born Simonsen) was the daughter of a wealthy shipping merchant. Niels Henrik had one older brother (Hans Mathias), three younger brothers (Thomas Hammond, Peder Tuxen and Thor Henrik) and a younger sister (Elisabeth). Their father Søren had earned a degree in theology and philosophy in Bergen, and was appointed pastor at Gjerstad Church in Risør when Abel’s grandfather Hans Mathias died. The family moved there in 1804.

When Norway adopted its constitution ten years later, in 1814, Abel’s father Søren became a member of Parliament in Oslo (then Christiania). He held the seat until 1818, when his political career abruptly ended. He was forced to step down after almost being impeached for making unfounded accusations against a cabinet member. Sick and worn down, Abel’s father returned to Gjerstad. An alcoholic, he died two years later at 48 years old.

Although he was home schooled at first, by the age of 13 Abel left home to join his older brother Hans at Christiania Cathedral School. The formal requirements for admission to the school were

Be at least ten years of ageBe able to demonstrate insight into history and geographyProficiency to read Danish and Latin, andBe acquainted with the four species of mathematics

Despite his rural upbringing, Abel qualified and left for the capital in 1815. His father Søren helped him find housing with a merchant. His quarters included a small room with a bed, table, stool and (perhaps) a window overlooking a stable yard where horses and wagons came and went with packing crates and other goods (Stubhaug, 1996). With the considerable time required for travel (by horse and buggy or sled, depending on the time of year), Abel likely did not return home to see his family for the duration of his first and second year. He was fourteen years old.

Being a disciple at the Cathedral School was a full-time occupation. School began at nine o'clock with four hours of morning lessons, and as a rule, three hours in the late afternoon. The day ended at six in the evening. In addition there was homework and preparation, day in and day out, six days a week without variation.- Excerpt, "Niels Henrik Abel and his Times: Called Too Soon by Flames Afar*" by Stubhaug (1996)

As his posthumous biographer Arild Stubhaug writes — somewhat fortunately for the burgeoning mathematical protege — more than half of Abel’s 40 hour school week as a teenager was devoted to languages. These included lessons in Danish as well as in modern foreign languages such as German and French, and of course, Latin. A proficiency in such languages was no doubt crucial for any aspiring scientist of the age, as the lingua franca of mathematics at the time was a mixture of French and increasingly, German. For the duration of Abel’s brief career, Gauss at Göttingen was by far the most influential mathematician in the world (see essay below).

Somewhat surprisingly, Abel did not fall in love with mathematics immediately. Indeed, he paid no special attention to the subject until he had attended the Cathedral School for nearly three years, in 1818. This because, for the first three years there his mathematics instructions “did not venture much beyond the demand that the pupils copy things down from the blackboard“ (Stubhaug, 1996).

There were, of course, no textbooks at the time. Indeed, had there been textbooks, Abel would not have been able to afford them. Writing home to his superior in Gjesdal, his father Søren reported in 1818 that the tuition at his son’s school was 90 speciedaler per year (about $450), which with the addition of costs for books and clothes meant that “There is nothing left but Milk and Gruel, Gruel and Milk for my remaining family” (Stubhaug, 1996). He ended his letter:

“I will pay with the shirt off my back. My children are my all” — Søren Abel

By 1818, a new, twenty-three year old teacher by the name of Bernt Michael Holmboe (1795–1850) had assumed the teaching of mathematics at Abel’s school, which would change 16-year-old Abel’s life forever. Inspired by the self-taught Joseph-Louis Lagrange (1736–1813), Holmboe was cautious about claiming to know the best way for students to learn, instead emphasizing that his students read the works of Euler (which Holmboe’s hero Lagrange had done).

As Stubhaug writes, in 1818 Abel “could hardly prove that there were infinitely many prime numbers”. By 1819 however, inspired by his new teacher, Abel had rapidly completed the required instruction in elementary mathematics. At 17 years old, he moved on to receive private lessons in higher mathematics from Holmboe, who helped him through the classics of Euler, Poisson and of course, his tutor’s idol Lagrange (Stubhaug, 1996). Before long, “Holmboe unashamedly boasted about his disciple, and with the conclusion of the school year in the summer of 1819, he wrote in the school register about Niels Henrik” :

“A splendid mathematical genius”

At the time of Abel’s graduation in 1820, Holmboe had run out of things to teach him as Abel had already ventured on and began studying the latest mathematical literature in the Royal Frederick University library. As his posthumous autobiographer Stubhaug later wrote:

“By 18, Abel was likely the most knowledgeable mathematician in Norway”

Abel entered the Royal Frederick University (now the University of Oslo) in 1821 on a scholarship raised for him by his patron and teacher Holmboe and his friends. He remained there for less than a year before earning a Bachelor of Arts degree in mathematics. His report card included “exceptionally high” marks in mathematics and mediocre grades in everything else.

Following his graduation, without any prospects of work or other income, Abel was on the verge of homelessness until he was taken in by his former professor — the leading natural scientist in Norway at the time — Christopher Hansteen (1784–1873) and his wife Johanne Borch Hansteen. Abel would later refer to Johanne as his “second mother”. Indeed, she “seems to have been the person who gained the deepest glimpse into Niels Henrik’s feelings and sensibilities” (Stubhaug, 1996). The couple provide for the young Abel, who by this point had lost his father and whose mother lived far away in rural Norway and knew nothing about her son’s life in the capital. In addition to providing him with a place to live and food, Hansteen also did his best to promote Abel’s name in the European mathematical community.

Indeed, throughout his brief life, the “unsophisticated” Abel was helped by many friends and patrons who — although most did not grasp the importance or implications of his work — supported his aspiring mathematical career. In addition to Holmboe and Hansteen, the following in particular stand out:

The “other” Professor of Mathematics at the University of Christiania (now the University of Oslo), Søren Rasmussen (1768–1850) who in 1823 gave Abel a gift of 100 speciedaler (about $580) so that he could travel to Copenhagen to visit mathematician Carl Ferdinand Degen (1766–1825).Abel’s principal at Christiania Cathedral School, Niels Treschow (1751–1833), without whom’s considerable effort in shaping school policy, Abel most likely would not have been exposed to Holmboe. When Abel applied for a two-year travel grant of 600 speciedaler (about $5500) in 1825, Treschow supported his request. When he returned from abroad, he stayed in Treschow’s home which also served as his sister Elisabeth’s home and place of work for years.The Professor and Doctor Michael Skjelderup (1769–1852) who let Abel live in his home when he needed it.

Abel’s research career began when he published his first proofs in the newly founded Magazin for Naturvitenskaberne (“Journal of Natural Scientists”) in 1823:

The paper regarded functional equations. In particular, Abel considered a very general type of functional equation (Houzel, 2002):

Where φ, f, F, … are unknown functions of one variable and α, β, γ, . . . are known functions of independent variables x and y. His investigation was in a method of eliminating the unknown equations of one variable until he could arrive at a differential equation with only one unknown function and one variable. In the paper he used the method to derive functional equations for various classical equations in mechanics and physics.

Following this publication, Abel also wrote a French paper (likely) titled:

Abel, N. H. (1823). Une représentation générale de la possibilité d’intégrer toutes les formules différentielles (“A General Representation of the Possibility to Integrate all Differential Formulas”)

He applied to the Royal Frederick University for funds to publish it, but despite the support of his patrons Professor Rasmussen and Hansteen, the committee did not immediately approve the grant and the paper was eventually lost.

Abel’s second journal paper was published in the same issue of Magazin for Naturvitenskaberne (“Journal of Natural Scientists”) as his first:

The paper regarded integral transforms and likely contains the first ever case of an integral equation (an equation in which an unknown function appears under the integral sign). In the paper, Abel first studies the integral equation

where ψ is a given function, s is an unknown function of variables x and n < 1. Abel next investigates the particular case where n = 1/2 by developing s as a power series and using the Euler function Γ, citing Adrien-Marie Legendre (1752–1833). He later published a version of the same paper in German in “Crelle’s Journal”, the Journal für die reine und angewandte Mathematik in 1826. In the second part of the 1823 paper, he also proved and the integral formula

In the spring of 1824, following the publication of his first two journal papers Abel’s fortunes were looking up. He was granted a scholarship for 200 speciedaler (about $1500) for the next two years and a promise for an additional 600 speciedaler for two years of travel after that. His most well-known result came around that time, when he wrote a booklet in French demonstrating (with proof) why it is impossible to solve the general quintic equation by radicals:

The general quintic equation is of the form:

As the story goes, already in 1821 Abel believed that he had discovered a method of solving the equation before realizing that there was an error in his solution—an error which he would later employ to prove that any such solution is impossible. He first showed his (unbeknownst to him, faulty) solution to professors Rasmussen and Hansteen, neither of whom found any errors. They next forwarded the solution to the leading mathematician in the Nordic countries, Carl Ferdinand Degen (1766–1825) who also did not find any mistakes but still doubted the solution, noting however that

“[The work] exhibits, even if the goal has not been proven, an uncommon mind, and uncommon insights, particularly for his age.” — Degen (1821)

Although he hesitated to forward the solution in its current form to the Royal Academy of the Sciences, Degen suggested Abel send a more detailed treatment and a numerical example, to strengthen his argument.

What Abel surely didn’t know at the time, was that supposed proofs of the impossibility of solving quintic equations had already been published by an Italian mathematician named Paulo Ruffini (1765–1822) in 1799, 1802 and 1813. Ruffini’s proofs were however incomplete, because he had assumed (without proof) that the radicals of a hypothetical solution were rational functions of the roots of the equation (Houzel, 2002). Abel’s proof (which he came to after failing to provide a numerical example to Degen) indeed begins with a proof of this supposition, namely that the root of the equation:

is of the form

In the interest of completeness, Abel’s proof of the impossibility of solving the quintic equation in radicals, in its entirety, is provided below:

Due to financial reasons it is likely that very few copies of the paper were sold. Abel sent a few copies around to colleagues and through Hansteen, to Heinrich Christian Schumacher (1780–1850) in Hamburg, who showed the paper to Carl Friedrich Gauss (1777–1855) in July 1824 (Stubhaug, 1996). Reportedly, Gauss’ initial reaction was negative, as he believed he could prove the opposite. However, as time moved on, Gauss too came to believe Abel’s proof to be correct.

Abel’s sole effort in applied mathematics was his 1824 paper:

At the encouragement of Professor Hansteen, as the story goes, Abel was given the task of calculating the moon’s gravitational pull using a pendulum on which a measuring apparatus was attached. As he was the editor of Magazin for Naturvidenskaberne, Hansteen published Abel’s paper reporting his findings during the spring of 1824. Hoping that the result would serve as Abel’s introduction to the academic circles of Europe, Hansteen even sent it to Schumacher who served as the editor of Astronomische Nachrichten. As it turned out, Abel’s calculations of the gravitational pull of the moon were very wrong (about six orders of magnitude) and so Schumacher refused to publish the paper. Worried that he might mention the blunder to his friend Gauss, Abel was embarrassed and never again ventured into topics outside of pure mathematics.

Abel’s fourth article published in the Norwegian journal Magazin for Natuvidenskaberne appeared the following year:

Building on the findings of his second paper in the same journal two years prior, this paper too regarded integral equations, in particular a derivation and extension of the formula:

Derivations of the formula had already been published five years earlier by Giovanni Antonio Amadeo Plana (1781–1864) in Memoirs of the Turin Academy (Houzel, 2002), but Abel was likely unaware of this work. It would not be until 1848 before Schaar established the result rigorously based on Cauchy’s residue theorem (Houzel, 2002).

In the summer of 1825, Abel set out on a European tour with four other young Norwegian scientists. With him, Abel had geologists Balthazar Mathias Keilhau(1797–1858), Nikolaj Benjamin Møller (1802–1860, Nils Otto Tank (1800–1864) as well as doctor/veterinarian Christian Peter Boeck (1798–1877). Abel was intending to visit the mathematical centers in Göttingen and Paris. Most notably was Gauss at Göttingen, of whose style Abel once wrote:

“He is like the fox, who effaces his tracks in the sand with his tail.”

To which Gauss is reported to have replied “No self respecting architect leaves the scaffolding in place after completing his building.”

Despite his goal of going to Göttingen, while in Copenhagen Abel was convinced by his fellow travelers to instead travel to Berlin. There, he met August Crelle for the first time. Crelle introduced him to the mathematical milieu in Berlin. Following a successful stay from October 1825 to March 1826, the young Norwegians continued on to Dresden, Prague, Vienna, Graz and Trieste. Regarding his aborted plan to visit Gauss in Göttingen, Abel later stated that he had chosen to instead follow his friends to avoid growing homesick. Indeed, homesickness may have been an issue for Abel during his travels, as he in a letter dated January 1826 wrote to Holmboe:

Excerpt, Letter from Abel to Holmboe (Berlin, 1826)I wish I were home, which I miss terribly. Write me now at least a long letter, telling me every kind of thing. Sit down and do this as soon as you get my letter. Tomorrow I am going to the comedy theatre to see Die schöne Müllerinn. Goodbye, and greet my friends and acquaintances.Your Friend, N. H. Abel

Having made the acquaintance of Crelle, Abel wrote and submitted three papers to his newly established journal. The first was published as:

In the paper, Abel gives a general and (almost) rigorous proof of Newton’s binomial formula (Houzel, 2002):

Employing ideas by Euler, Cauchy and Lagrange Abel goes on to (almost) show continuity for the sum of the series of continued fractions.

When Abel in 1821 believed that he had found a solution to the quintic equation, Danish Professor Degen (who reviewed his proof) added the following comment in his review letter:

Excerpt, Letter from Carl Ferdinand Degen to Abel (1821)I can scarcely repress the wish, that the time and the mental powers of the head which Mr. A. blesses us with, not be utilized, from my perspective, on sterile subject matter, but ought rather be applied to a theme, whose edification would have the most important consequences for the whole analysis and its application to dynamic explorations: I am thinking of the elliptical transcendents. With the proper approach to investigations of this type, serious scrutiny by no means becomes static, nor do the highest and more remarkable functions, with their many and handsome properties, become something in and for themselves, but rather, it is going to reveal the Magellan-Voyages to great regions of one and the same immense analytic ocean.

By elliptical transcendents, Degen was referring to what was later named elliptic functions and integrals. Abel’s first contribution to this theory came while he was in Paris, by the publication of the paper:

This paper (in addition to two more, which he wrote in Berlin), as Houzel (2002) writes, grants Abel claim to the title of “the founder of the theory of elliptic functions”.

His most important work appeared in the last two papers he published while still alive, both in Crelle’s Journal, in its second (1827) and fourth edition (1829).

Abel, N. H. (1827). Recherches sur les fonctions elliptiques (“Research on Elliptic Functions”). Journal für die reine und angevandte Mathematik 2, pp. 101–181.Abel, N. H. (1829). Précis d’une théorie des fonctions elliptiques (“A Precise Theory of Elliptic Functions”). Journal für die reine und angevandte Mathematik 4, pp. 309–348.

In the 1827 paper, Abel sets out by recalling the main results of Euler, Lagrange and Legendre on elliptic integrals and defines the function φα = x by the relation

where c and e are real numbers. In the 1829 paper, Abel describes a method for expressing elliptic functions as quotients of functions of a similar type as Weierstrass functions.

Abel did write another paper while in Paris. It was presented to Augustin-Louis Cauchy (1789–1857) in October of the same year, who described it as “a monument more lasting than bronze”. However, the paper was lost by the French Academy of Sciences until 1841, when it was finally published as:

Abel, N. H. (1841). Mémoire sur une Propriété Générale d’une Classe Trés Étendue de Fonctions Transcendantes (“Dissertation on a General Property of a Very Extended Class of Transcendent Functions”). Mémoires présentés par divers savants à l’Académie Royale des Sciences de l’Institut de France, t.VII. pp. 176–264. “No one in Norway had sufficient mathematical knowledge to understand the significance of his stringent proofs”

Abel’s working career was remarkably short, lasting a mere seven years from his entering university in 1821 to his death in 1829. Although in his short life he published nine journal papers, most of his work remained unpublished until his collected works were gathered by fellow Norwegian mathematicians Peter Ludwig M. Sylow (1832–1918) and Sophus Lie (1842–1899) and published in two volumes under the title:

Modern editions of this work is available on Amazon*. Among the most important findings of career, the following are considered major contributions:

Abel’s differential equation identity. In Abel’s 1829 paper in the Journal für die reine und angevandte Mathematik, he presented an equation that expresses the Wronskian determinant of two solutions of a homogeneous second-order linear ordinary differential equation in terms of a coefficient of the original differential equation.Algebraic solutions to higher order polynomials. Abel’s proof of the impossibility of solving polynomials of higher orders than four is now known as the Abel-Ruffini theorem for its two main contributors, Abel and Ruffini (1765–1822).Abelian integrals. However famous Abel’s result about the impossibility of solving quintic equations, the most important of his results among mathematicians likely still remains his extension of the Euler addition theorem for elliptic integrals, now known as Abelian integrals.The invention of group theory. In proving that there are no general algebraic solutions for the roots of quintic equations, Abel invented (independently of Galois) what later became known as group theory. In addition to Galois, the topic was also studied in the same period by Joseph-Louis Lagrange (1736–1813).

Abel’s former professor and patron Søren Rasmussen stepped down from his professorship in 1825, leaving a vacant position. Abel was (of course) nominated by Professor Hansteen for the position. In the end, two candidates were seriously considered for the position: Abel and his former mentor Holmboe. In December of the same year, Holmboe was awarded the position, primarily because Abel was considered “still young, and perhaps yet unable to teach elementary mathematics in an suitable manner”.

Abel returned from Paris in January 1827. While there, he had contracted tuberculosis. On his return back, he stopped in Berlin where he was offered a position as editor of Crelle’s Journal, which he turned down. He returned to Christiania in May. Despite his initial success in Berlin, he had not yet published in Paris and so was not able to secure an academic appointment. As such, he remained in poverty for the remainder of his life.

Abel had met the (only known) love of his life, Christine Kemp at a ballroom dance on the marine base Christianshavn in 1823. As the story goes, the two danced the new dance “the waltz” even though neither of them knew the steps. They were engaged during Christmas of the following year, in 1824. Having come back from his travels abroad, four years later, Abel traveled by sled to Froland where Christine was living. He became seriously ill on the journey and (despite a brief improvement) died in April of 1828 at 26 years old.

Two days after his death, a letter arrived from August Crelle in Paris announcing that he had managed to get him appointed as a Professor of Mathematics at the University of Berlin on the merits of his publications in Crelle’s Journal.

Although essentially unappreciated in his home country during his lifetime, Niels Henrik Abel is today widely considered to be the greatest Norwegian mathematician who has ever lived. Growing up in rural Norway at a time when the country was a deeply poor and uneducated, his brief but brilliant rise to prominence in the mathematical milieus of Paris and Berlin remains nothing short of unprecedented in a country of 5,5 million people as of 2024.

Monuments in his memory have been erected in the Royal Palace Park in Oslo, outside of his house on the campus of the University of Oslo, in his hometown of Gjerstad and on his birth island of Finnøy.. Abel’s likeness has also been featured on Norwegian banknotes, commemorative coins, stamps, and his name given to a crater on the moon, an astroid and an aircraft.

In memory of his short but eventful life, the Norwegian government in 2001 announced that it would award a yearly mathematics price in the memory of Niels Henrik Abel, beginning on the two-hundredth anniversary of his brith in 2002. Winners of the prize as of 2024 include:

Winners of the Abel Prize as of 20202002: Atle Selberg (memorial)
2003: Jean-Pierre Serre
2004: Michael Atiyah and Isadore Singer
2005: Peter Lax
2006: Lennard Carleson
2007: S.R.Srinivasa Varadhan
2008: John G. Thompson and Jacques Tits
2009: Mikhail Gromov
2010: John Tate
2011: John Milnor
2012: Endre Szemerédi
2013: Pierre Deligne
2014: Yakov Sinai
2015: John F. Nash Jr. and Louis Nirenberg
2016: Andrew Wiles
2017: Yves Meyer
2018: Robert Langlands
2019: Karen Uhlenbeck
2020: Hillel Furstenberg and Grigory Margulis
2021: László Lovász and Avi Wigderson
2022: Dennis Sullivan
2023: Luis Caffarelli
2024: Michel Talagrand

Those interested in reading more about the life and works of Niels Henrik Abel are encouraged to acquire the books Niels Henrik Abel and his Times* by Stubhaug (2002) and The Work of Niels Henrik Abel* by Houzel (2002).

I hope you enjoyed this essay,

Best

Jørgen

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