Niels Henrik Abel

Norwegian mathematician

• Born: August 5, 1802
• Birthplace: Finnøy, Norway
• Died: April 6, 1829
• Place of death: Froland, Norway

Abel was a central figure in the evolution of modern mathematics, especially in the field of algebra. Regarded as one of the foremost analysts of his time, he insisted that a rigorous approach to mathematical proof was critical for the further development of abstract mathematics.

Early Life

Niels Henrik Abel (AH-behl) was the second child of S ren Georg Abel, a second-generation Lutheran minister, and Anne Marie Simonsen, a daughter of a successful merchant and shipowner. Soon after his birth, his father was transferred to the parish of Gjerstad, in southeastern Norway, about 150 miles from Oslo, where Abel spent his childhood with his five brothers and sisters. Abel was an attractive youth, with light ash-brown hair and blue eyes.

Although his father’s earnings were never adequate to provide for the large family, the emphasis on educational stimulation in the Abel household was an important formative influence on the young boy. Although his early education was conducted at home, it was sufficient to allow him to attend the Cathedral School at Oslo when he was thirteen years old. It was there that his talent in mathematics was discovered, although his initial efforts were somewhat unpromising.

The Cathedral School had once been quite good, but many positions had been filled by inexperienced or inadequate teachers because their predecessors had been recruited to join the faculty of the newly formed University of Oslo. Indeed, Abel’s first mathematics instructor was dismissed abruptly after beating a student to death. The replacement in that position was Bernt Michael Holmboe, who was the first to recognize Abel’s talent and who later edited the first edition of his work. Holmboe also assisted Christopher Hansteen, a professor at the university; this connection would prove valuable to Abel.

When Holmboe first arrived at the school, he noticed Abel’s ability in mathematics and suggested that the two of them study some of the contemporary mathematics works together. Abel soon outpaced Holmboe and began developing a general solution for the quintic equation, that is, an equation of the fifth degree (ax⁵ + bx⁴ + cx³ + dx² + ex + f = 0). When Abel believed that the work was complete, Holmboe and Hansteen sensed that no one in Norway, including themselves, could review the work capably. They forwarded the paper to Ferdinand Degen of the Danish Academy, who carefully reviewed the work. Before publication, Degen helped Abel discover that his solution was flawed, but he steered Abel into the field of elliptic functions, which Degen believed would be more fruitful.

At about that same time, Abel discovered that several of his predecessors, particularly Leonhard Euler and Joseph-Louis Lagrange, had not completed the reasoning required to prove some of their work. Abel diligently supplied rigorous proofs where they were missing; a noted case is his proof of the general binomial theorem, which had been stated previously in part by Sir Isaac Newton and Euler. The mathematics community later was to find his meticulous treatment of the works he studied invaluable. Unfortunately for his personal life and his financial situation, Abel’s father, who had served two terms in the Storting (congress), was impeached and disgraced. His father died in 1829, leaving his family in even more desperate financial straits than ever before.

Life’s Work

The nineteen-year-old Abel entered the University of Oslo in 1821. Although this entering age would not normally denote a prodigy, the fact that the university granted him a free room and that several professors donated funds for his support does. Abel completed the preliminary requirements for a degree in a single year. He was then free to study mathematics on his own, as he had no peers among the faculty. He developed a love for the theater at that time that lasted throughout his short life. A modest person, he made many lasting friendships.

In addition to studying all available work, he began writing papers, the first of which were published in the journal Magazin for Naturvidenskaberne begun by Hansteen. In 1823, Abel’s first important paper, “Opläsning afet Par Opgaver ved bjoelp af bestemte Integraler” (“Solution of Some Problems by Means of Definite Integrals”), was published, containing the first published solutions of integral equations. During 1822 and 1823, he also developed a longer paper discussing the integration of functions. This work is recognized as very significant in the evolution of that field of study.

At that time, Abel’s work was largely ignored by the international mathematics community because Abel was from Norway and wrote in Norwegian, and the focal point of the mathematics community of the day was Paris, with the language of the learned being French. By applying himself diligently, Abel learned French and began to publish work in that language. The quintic equation still held his attention, and, as he thought of possibilities for its solution, he also considered that there might be no solution that could be found for all such equations. In time, he was able to prove this result. Nevertheless, the mathematicians whose approval he desired so fervently, those in Paris, ignored his work.

Abel began to press for the opportunity to go to Paris, but penniless as he was he was forced to rely on grants. After his first application, it was decided that he needed to study more foreign languages before going abroad. Although it meant delaying his dream for nearly two years, Abel applied himself to learning various languages. Meanwhile, he became engaged to Christine (Krelly) Kemp before he finally received a royal grant to travel abroad in 1825.

This trip was unsuccessful in many ways. When he arrived at Copenhagen, he discovered that Degen had died. Instead of going on to Paris, Abel decided to go to Berlin because several of his friends were there. The time in Berlin was invaluable, for he met and befriended August Leopold Crelle, who became his strongest supporter and mentor. When Abel met him, Crelle was preparing to begin publication of a new journal, Journal für die reine und angewandte Mathematik (journal for pure and applied mathematics). Crelle was so taken by Abel’s ability that much of the first few issues was devoted to Abel’s work in an attempt to win recognition for the young mathematician.

For a variety of reasons, Abel did not proceed to Paris until the spring of 1826. By this time, he had spent most of his grant and was physically tired, and the Parisian mathematicians he had hoped to convince were nearly all on holiday. However, his masterwork, Mémoire sur une propriété générale d’une classe très-étendue de fonctions transcendantes (memoir on a general property of a very extensive class of transcendental functions), was presented to the Academy of Sciences on October 30, 1826. The paper was left in the keeping of Augustin-Louis Cauchy, a prominent mathematician, and Cauchy and Adrien-Marie Legendre were to be the referees. Whether the paper was illegible, as Cauchy claimed, or was misplaced, as most historians believe, no judgment was issued until after Abel’s death.

Abel felt a great sense of failure, for many young mathematicians had been established by recognition from the academy. He returned first to Berlin and finally to Oslo in May, 1827. His prospects were bleak: He had contracted tuberculosis, there was no prospect for a mathematical position in Norway, and he was in debt. Abel began tutoring and lecturing at the university on a substitute basis in order to support himself.

Another young mathematician, Carl Gustav Jacob Jacobi, soon began publishing work in Abel’s foremost field, the theory of elliptic functions and integrals. The rivalry created between them dominated the rest of Abel’s life. He worked furiously to prove his ideas, and his efforts were spurred by his correspondence with Legendre. As he finally began to be recognized in Europe, many mathematicians, led by Crelle, attempted to secure a patronage for him. However, he succumbed to tuberculosis on April 6, 1829, two days before Crelle wrote to inform him that such financial support had been found. In June, 1830, he and Jacobi were awarded the Grand Prix of the French Academy of Sciences for their work in elliptic integrals. Abel’s original manuscript was found and finally published in 1841.

Significance

Although Niels Henrik Abel’s life was short and his work was unrecognized for most of his life, he has exercised a great influence on modern mathematics. His primary work with elliptic functions and integrals led to interest in what became one of the great research topics of his century. Without his preliminary findings, many of the developments in mathematics and, consequently, science, may not have been made. One example of this is his theory of elliptic functions, much of which was developed very quickly during his race with Jacobi. In addition, his proof that there is no general solution to the quintic equation is quite important, as are his other findings in equation theory.

Abel’s theory of solutions using definite integrals, including what is now called Abel’s theorem, is also widely used in engineering and the physical sciences and provided a foundation for the later work of others. Abelian (commutative) groups, Abelian functions, and Abelian equations are but three of the ideas that commonly carry his name. Given Abel’s short life span and his living in Norway, a definite academic backwater at the time, his prolific achievements are amazing.

Abel is also significant because his writing and mathematical styles, which were easily comprehended, made his discoveries available to his contemporaries and successors. Abel’s insistence that ideas should be demonstrated in such a way that the conclusions would be supported by clear and easily comprehended arguments, that is, proved rigorously, is the cornerstone of modern mathematics. It is in this regard that Abel is most often remembered.

Bibliography

Abel, Niels Henrik. “From a Memoir on Algebraic Equations, Proving the Impossibility of a Solution of the General Equation of the Fifth Degree.” In Classics of Mathematics, edited by Ronald Calinger. Oak Park, Ill.: Moore, 1982. This extract of Abel’s paper on the general quintic equation demonstrates Abel’s style. Although it is too technical for the casual reader, it is of interest to mathematicians and demonstrates Abel’s place in the development of mathematics. Also includes a brief biography.

Bell, Eric T. “Genius and Poverty: Abel.” In Men of Mathematics. New York: Simon & Schuster, 1937. This compilation of brief biographies of famous mathematicians includes a chapter on Abel, focusing more on the subject’s life than on his mathematical achievements.

Boyer, Carl B. A History of Mathematics. New York: John Wiley & Sons, 1968. This general history of mathematics will help the reader place Abel within the general development of mathematics.

Kline, Morris. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1972. Kline includes both a brief biography of Abel and discussions of his most important work in this history of mathematics.

Ore, ystein. Niels Henrik Abel: Mathematician Extraordinary. Minneapolis: University of Minnesota Press, 1957. This English-language biography gives a detailed account of Abel’s life without requiring a specialized knowledge of mathematics.

Pesic, Peter. Abel’s Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability. Cambridge, Mass.: MIT Press, 2003. Describes Abel’s life, focusing on his proof of the quintic equation. Pesic discusses why and how the proof changed the perception of mathematics. Includes a new annotated translation of Abel’s original proof.

Stubhaug, Arild. Niels Henrik Abel and His Times: Called Too Soon by Flames Afar. Translated by Richard H. Daly. Berlin: Springer, 2000. Stubhaug, a Norwegian mathematician and historian, recounts Abel’s brief life and achievements in this readable biography intended for a general audience. Includes illustrations.