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”.
- James, Ioan. Remarkable Mathematicians: From Euler to von Neumann.
Cambridge: Cambridge University Press, 2002.
- Millar, David et al. The Cambridge Dictionary of Scientists
(Second Edition). Cambridge: Cambridge University Press, 2002.
- A Dictionary of Scientists. Oxford: Oxford University Press,
- Chambers Biographical Dictionary (Centenary Edition). New York:
Chambers Harrap Publishers Ltd, 1997.
- Various sources on the Internet.