Niels Abel's life was dominated by poverty and
we begin by putting this in context by looking briefly at the political
problems which led to economic problems in Norway. At the end of the 18th
century Norway was part of Denmark and the Danish tried to remain
neutral through the Napoleonic wars. However a neutrality treaty in 1794
was considered a aggressive act by Britain and, in 1801, the British
fleet destroyed most of the Danish fleet in a battle in the harbour at
Copenhagen. Despite this Denmark-Norway avoided wars until 1807 when
Britain feared that the Danish fleet might be used by the French to
invade. Using the philosophy that attack is the best form of defence,
the English attacked and captured the whole Danish fleet in October
1807.
Denmark then joined the alliance Britain Britain. The continental powers
blockaded Britain, and as a counter to this Britain blockaded Norway.
The twin blockade was a catastrophe to Norway preventing their timber
exports, which had been largely to Britain, and preventing their grain
imports from Denmark. An economic crisis in Norway followed with the
people suffering hunger and extreme poverty. In 1813 Sweden attacked
Denmark from the south and, at the treaty of Kiel in January 1814,
Denmark handed over Norway to Sweden. An attempt at independence by
Norway a few months later led to Sweden attacking Norway in July 1814.
Sweden gained control of Norway, setting up a complete internal
self-government for Norway with a government in Christiania (which is
called Oslo today). In this difficult time Abel was growing up in
Gjerstad in south-east Norway.
Abel's father, Sören Georg Abel, had a degree in theology and philology
and his father (Niels Abel's grandfather) was a Protestant minister at
Gjerstad near Risor. Sören Abel was a Norwegian nationalist who was
active politically in the movement to make Norway independent. Sören
Abel married Ane Marie Simonson, the daughter of a merchant and ship
owner, and was appointed as minister at Finnoy. Niels Abel, the second
of seven children, was one year old when his grandfather died and his
father was appointed to succeed him as the minister at Gjerstad. It was
in that town that Abel was brought up, taught by his father in the
vicarage until he reached 13 years of age. However, these were the 13
years of economic crisis for Norway described above and Abel's parents
would have not been able to feed their family that well. The problems
were not entirely political either for [
14]:-
[Abel's] father was probably a drunkard and his mother was accused of having lax morals.
Abel's father was, however, important in the politics of Norway and,
after Sweden gained control of Norway in 1814, he was involved in
writing a new constitution for Norway as a member of the Storting, the
Norwegian legislative body. In 1815 Abel and his older brother were sent
to the Cathedral School in Christiania. The founding of the University
of Christiania had taken away the good teachers from the Cathedral
School to staff the University when it opened for teaching in 1813. What
had been a good school was in a bad state when Abel arrived. Uninspired
by the poor school, he proved a rather ordinary pupil with some talent
for mathematics and physics.
When a new mathematics teacher
Bernt Holmboë
joined the school in 1817 things changed markedly for Abel. The
previous mathematics teacher had been dismissed for punishing a boy so
severely that he had died. Abel began to study university level
mathematics texts and, within a year of
Holmboë's arrival, Abel was reading the works of
Euler,
Newton,
Lalande and
d'Alembert.
Holmboë was convinced that Abel had great talent and encouraged him greatly taking him on to study the works of
Lagrange and
Laplace. However, in 1820 tragedy struck Abel's family when his father died.
Abel's father had ended his political career in disgrace by making false
charges against his colleagues in the Storting after he was elected to
the body again in 1818. His habits of drinking to excess also
contributed to his dismissal and the family was therefore in the deepest
trouble when he died. There was now no money to allow Abel to complete
his school education, nor money to allow him to study at university and,
in addition, Abel had the responsibility of supporting his mother and
family.
Holmboë
was able to help Abel gain a scholarship to remain at school and Abel
was able to enter the University of Christiania in 1821, ten years after
the university was founded.
Holmboë
had raised money from his colleagues to enable Abel to study at the
university and he graduated in 1822. While in his final year at school,
however, Abel had begun working on the solution of
quintic equations by
radicals.
He believed that he had solved the quintic in 1821 and submitted a
paper to the Danish mathematician Ferdinand Degen, for publication by
the Royal Society of Copenhagen. Degen asked Abel to give a numerical
example of his method and, while trying to provide an example, Abel
discovered the mistake in his paper. Degen had given Abel some important
advice that was to set him working on an area of mathematics (see [
2]):-
... whose development would have the greatest consequences for
analysis and mechanics. I refer to elliptic integrals. A serious
investigator with suitable qualifications for research of this kind
would by no means be restricted to the many beautiful properties of
these most remarkable functions, but could discover a Strait of Magellan
leading into wide expanses of a tremendous analytic ocean.
At the University of Christiania Abel found a supporter in the professor
of astronomy Christopher Hansteen, who provided both financial support
and encouragement. Hansteen's wife began to care for Abel as if he was
her own son. In 1823 Abel published papers on functional equations and
integrals in a new scientific journal started up by Hansteen. In Abel's
third paper, Solutions of some problems by means of definite integrals he gave the first solution of an integral equation.
Abel was given a small grant to visit Degen and other mathematicians in
Copenhagen. While there he met Christine Kemp who shortly afterwards
became his fiancée. Returning to Christiania, Abel tried to get the
University of Christiania to give him a larger grant to enable him to
visit the top mathematicians in Germany and France. He did not speak
French of German so, partly to save money, he was given funds to remain
in Christiania for two years to give him the chance to become fluent in
these languages before travelling. Abel began working again on quintic
equations and, in 1824, he proved the impossibility of solving the
general equation of the fifth degree in radicals. He published the work
in French and at his own expense since he wanted an impressive piece of
work to take with him when he was on his travels. As Ayoub writes in [
6]:-
He chose a pamphlet as the quickest way to get it into print, and in
order to save on the printing costs, he reduced the proof to fit on half
a folio sheet [six pages].
By this time Abel seems to have known something of
Ruffini's work for he had studied
Cauchy's work of 1815 while he was an undergraduate and in this paper there is a reference to
Ruffini's work. Abel's 1824 paper begins ([
6]):-
Geometers have occupied themselves a great deal with the general
solution of algebraic equations and several among them have sought to
prove the impossibility. But, if I am not mistaken, they have not
succeeded up to the present.
Abel sent this pamphlet to several mathematicians including
Gauss,
who he intended to visit in Göttingen while on his travels. In August
1825 Abel was given a scholarship from the Norwegian government to allow
him to travel abroad and, after taking a month to settle his affairs,
he set out for the Continent with four friends, first visiting
mathematicians in Norway and Denmark. On reaching Copenhagen, Abel found
that Degen had died and he changed his mind about taking Hansteen's
advice to go directly to Paris, preferring not to travel alone and stay
with his friends who were going to Berlin. As he wrote in a later letter
([
7]):-
Now I am so constituted that I cannot endure solitude. Alone, I am
depressed, I get cantankerous, and I have little inclination to work.
In Copenhagen Abel was given a letter of introduction to
Crelle by one of the mathematicians there. Abel met
Crelle in Berlin and the two became firm friends. This proved the most useful part of Abel's whole trip, particularly as
Crelle was about to begin publishing a journal devoted to mathematical research. Abel was encouraged by
Crelle to write a clearer version of his work on the insolubility of the quintic and this resulted in
Recherches sur les fonctions elliptiques which was published in 1827 in the first volume of
Crelle's Journal,
along with six other papers by Abel. While in Berlin, Abel learnt that
the position of professor of mathematics at the University of
Christiania, the only university in Norway, had been given to
Holmboë. With no prospects of a university post in Norway, Abel began to worry about his future.
Crelle's Journal continued to be a source for Abel's papers and
Abel began to work to establish mathematical analysis on a rigorous
basis. He wrote to
Holmboë from Berlin [
2]:-
My eyes have been opened in the most surprising manner. If you
disregard the very simplest cases, there is in all of mathematics not a
single infinite series whose sum had been rigorously determined. In
other words, the most important parts of mathematics stand without
foundation. It is true that most of it is valid, but that is very
surprising. I struggle to find a reason for it, an exceedingly
interesting problem.
It had been Abel's intention to travel with
Crelle to Paris and to visit
Gauss in Göttingen on the way. However, news got back to Abel that
Gauss
was not pleased to receive his work on the insolubility of the quintic,
so Abel decided that he would be better not to go to Göttingen. It is
uncertain why
Gauss took this attitude towards Abel's work since he certainly never read it - the paper was found unopened after
Gauss's death. Ayoub gives two possible reasons [
6]:-
... the first possibility is that Gauss
had proved the result himself and was willing to let Abel take the
credit. ... The other explanation is that he did not attach very much
importance to solvability by radicals...
The second of these explanations does seem the more likely, especially since
Gauss
had written in his thesis of 1801 that the algebraic solution of an
equation was no better than devising a symbol for the root of the
equation and then saying that the equation had a root equal to the
symbol.
Crelle
was detained in Berlin and could not travel with Abel to Paris. Abel
therefore did not go directly to Paris, but chose to travel again with
his Norwegian friends to northern Italy before crossing the Alps to
France. In Paris Abel was disappointed to find there was little interest
in his work. He wrote back to
Holmboë ([
7]):-
The French are much more reserved with strangers than the Germans. It
is extremely difficult to gain their intimacy, and I do not dare to
urge my pretensions as far as that; finally every beginner had a great
deal of difficulty getting noticed here. I have just finished an
extensive treatise on a certain class of transcendental functions to
present it to the Institute which will be done next Monday. I showed it
to Mr Cauchy, but he scarcely deigned to glance at it.
The contents and importance of this treatise by Abel is described in [
2]:-
It dealt with the sum of integrals of a given algebraic function.
Abel's theorem states that any such sum can be expressed as a fixed
number p of these integrals, with integration arguments that are
algebraic functions of the original arguments. The minimal number p is
the genus of the algebraic function, and this is the first occurrence of
this fundamental quantity. Abel's theorem is a vast generalisation of Euler's relation for elliptic integrals.
Two referees,
Cauchy and
Legendre, were appointed to referee the paper and Abel remained in Paris for a few months [
14]:-
... emaciated, gloomy, weary and constantly worried. He ... could only afford to eat one meal a day.
He published some articles, mainly on the results he had already written for
Crelle's Journal,
then with no money left and his health in a very poor state, he
returned to Berlin at the end of 1826. In Berlin, Abel borrowed some
money and continued working on elliptic functions. He wrote a paper in
which [
2]:-
... he radically transformed the theory of elliptic integrals to the
theory of elliptic functions by using their inverse functions ...
Crelle
tried to persuade Abel to remain in Berlin until he could find an
academic post for him and he even offered Abel the editorship of
Crelle's Journal.
However, Abel wanted to get home and by this time he was heavily in
debt. He reached Christiania in May 1827 and was awarded a small amount
of money by the university although they made sure they had the right to
deduct a corresponding amount from any future salary he earned. To make
a little more money Abel tutored schoolchildren and his fiancée was
employed as a governess to friends of Abel's family in Froland.
Hansteen received a major grant to investigate the Earth's magnetic
field in Siberia and a replacement was needed to teach for him at the
University and also at the Military Academy. Abel was appointed to this
post which improved his position a little.
In 1828 Abel was shown a paper by
Jacobi on transformations of elliptic integrals. Abel quickly showed that
Jacobi's
results were consequences of his own and added a note to this effect to
the second part of his major work on elliptic functions. He had been
working again on the algebraic solution of equations, with the aim of
solving the problem of which equations were
soluble by radicals (the problem which
Galois solved a few years later). He put this to one side to compete with
Jacobi in the theory of elliptic functions, quickly writing several papers on the topic.
Legendre saw the new ideas in the papers which Abel and
Jacobi were writing and said ([
2]):-
Through these works you two will be placed in the class of the foremost analysts of our times.
Abel continued to pour out high quality mathematics as his health
continued to deteriorate. He spent the summer vacation of 1828 with his
fiancée in Froland. The masterpiece which he had submitted to the
Paris Academy seemed to have been lost and so he wrote the main result down again [
3]:-
The paper was only two brief pages, but of all his many works perhaps
the most poignant. He called it only "A theorem": it had no
introduction, contained no superfluous remarks, no applications. It was a
monument resplendent in its simple lines - the main theorem from his
Paris memoir, formulated in few words.
Abel travelled by sled to visit his fiancée again in Froland for
Christmas 1828. He became seriously ill on the sled journey and despite
an improvement which allowed them to enjoy Christmas, he soon became
very seriously ill again.
Crelle
was told and he redoubled his efforts to obtain an appointment for Abel
in Berlin. He succeeded and wrote to Abel on the 8 April 1829 to tell
him the good news. It was too late, Abel had already died. Ore [
3] describes his last few days:-
... the weakness and cough increased and he could remain out of bed
only the few minutes while it was being made. Occasionally he would
attempt to work on his mathematics, but he could no longer write.
Sometimes he lived in the past, talking about his poverty and about Fru
Hansteen's goodness. Always he was kind and patient. ...
He endured his worst agony during the night of April 5. Towards morning he became more quiet and in the forenoon, at eleven o'clock, he expired his last sigh.
After Abel's death his Paris memoir was found by
Cauchy
in 1830 after much searching. It was printed in 1841 but rather
remarkably vanished again and was not found until 1952 when it turned up
in Florence. Also after Abel's death unpublished work on the algebraic
solution of equations was found. In fact in a letter Abel had written to
Crelle on 18 October 1828 he gave the theorem [
13]:-
If every three roots of an irreducible equation of prime
degree are related to one another in such a way that one of them may be
expressed rationally in terms of the other two, then the equation is
soluble in radicals.
This result is essentially identical to one given by
Galois in his famous memoir of 1830. In this same year 1830 the
Paris Academy awarded Abel and
Jacobi the Grand Prix for their outstanding work.
Article by: J J O'Connor and E F Robertson
