Augustin Louis Cauchy
Born: 21 Aug 1789 in Paris, France Died: 23 May 1857 in Sceaux (near
Paris), France
Paris was a difficult place to live in when Augustin-Louis
Cauchy was a young child due to the political events surrounding the French
Revolution. When he was four years old his father, fearing for his life in
Paris, moved his family to Arcueil. There things were hard and he wrote in a
letter:-
We never have more than a half pound of bread - and
sometimes not even that. This we supplement with the little supply of hard
crackers and rice that we are allotted.
They soon returned to Paris and Cauchy's father was active in
the education of young Augustin-Louis. Laplace and Lagrange were visitors at the
Cauchy family home and Lagrange in particular seems to have taken an interest in
young Cauchy's mathematical education. Lagrange advised Cauchy's father that his
son should obtain a good grounding in languages before starting a serious study
of mathematics. In 1802 Augustin-Louis entered the École Centrale du Panthéon
where he spent two years studying classical languages.
From 1804 Cauchy attended classes in mathematics and he took
the entrance examination for the École Polytechnique in 1805. He was examined by
Biot and placed second. At the École Polytechnique he attended courses by
Lacroix, de Prony and Hachette while his analysis tutor was Ampère. In 1807 he
graduated from the École Polytechnique and entered the engineering school École
des Ponts et Chaussées. He was an outstanding student and for his practical work
he was assigned to the Ourcq Canal project where he worked under Pierre Girard.
In 1810 Cauchy took up his first job in Cherbourg to work on
port facilities for Napoleon's English invasion fleet. He took a copy of
Laplace's Mécanique Céleste and one of Lagrange's Théorie des
Fonctions with him. It was a busy time for Cauchy, writing home about his
daily duties he said:-
I get up at four o'clock each morning and I am busy from
then on. ... I do not get tired of working, on the contrary, it invigorates me
and I am in perfect health...
Cauchy was a devout Catholic and his attitude to his religion
was already causing problems for him. In a letter written to his mother in 1810
he says:-
So they are claiming that my devotion is causing me to
become proud, arrogant and self-infatuated. ... I am now left alone about
religion and nobody mentions it to me anymore...
In addition to his heavy workload Cauchy undertook mathematical
researches and he proved in 1811 that the angles of a convex polyhedron are
determined by its faces. He submitted his first paper on this topic then,
encouraged by Legendre and Malus, he submitted a further paper on polygons and
polyhedra in 1812. Cauchy felt that he had to return to Paris if he was to make
an impression with mathematical research. In September of 1812 he returned to
Paris after becoming ill. It appears that the illness was not a physical one and
was probably of a psychological nature resulting in severe depression.
Back in Paris Cauchy investigated symmetric functions and
submitted a memoir on this topic in November 1812. This was published in the
Journal of the École Polytechnique in 1815. However he was supposed to return to
Cherbourg in February 1813 when he had recovered his health and this did not fit
with his mathematical ambitions. His request to de Prony for an associate
professorship at the École des Ponts et Chaussées was turned down but he was
allowed to continue as an engineer on the Ourcq Canal project rather than return
to Cherbourg. Pierre Girard was clearly pleased with his previous work on this
project and supported the move.
An academic career was what Cauchy wanted and he applied for a
post in the Bureau des Longitudes. He failed to obtain this post, Legendre being
appointed. He also failed to be appointed to the geometry section of the
Institute, the position going to Poinsot. Cauchy obtained further sick leave,
having unpaid leave for nine months, then political events prevented work on the
Ourcq Canal so Cauchy was able to devote himself entirely to research for a
couple of years.
Other posts became vacant but one in 1814 went to Ampère and a
mechanics vacancy at the Institute, which had occurred when Napoleon Bonaparte
resigned, went to Molard. In this last election Cauchy did not receive a single
one of the 53 votes cast. His mathematical output remained strong and in 1814 he
published the memoir on definite integrals that later became the basis of his
theory of complex functions.
In 1815 Cauchy lost out to Binet for a mechanics chair at the
École Polytechnique, but then was appointed assistant professor of analysis
there. He was responsible for the second year course. In 1816 he won the Grand
Prix of the French Academy of Sciences for a work on waves. He achieved real
fame however when he submitted a paper to the Institute solving one of Fermat's
claims on polygonal numbers made to Mersenne. Politics now helped Cauchy into
the Academy of Sciences when Carnot and Monge fell from political favour and
were dismissed and Cauchy filled one of the two places.
In 1817 when Biot left Paris for an expedition to the Shetland
Islands in Scotland Cauchy filled his post at the Collège de France. There he
lectured on methods of integration which he had discovered, but not published,
earlier. Cauchy was the first to make a rigorous study of the conditions for
convergence of infinite series in addition to his rigorous definition of an
integral. His text Cours d'analyse in 1821 was designed for students at
École Polytechnique and was concerned with developing the basic theorems of the
calculus as rigorously as possible. He began a study of the calculus of residues
in 1826 in Sur un nouveau genre de calcul analogue au calcul
infinitésimal while in 1829 in Leçons sur le Calcul Différentiel he
defined for the first time a complex function of a complex variable.
Cauchy did not have particularly good relations with other
scientists. His staunchly Catholic views had him involved on the side of the
Jesuits against the Académie des Sciences. He would bring religion into his
scientific work as for example he did on giving a report on the theory of light
in 1824 when he attacked the author for his view that Newton had not believed
that people had souls. He was described by a journalist who said:-
... it is certain a curious thing to see an academician who
seemed to fulfil the respectable functions of a missionary preaching to the
heathens.
An example of how Cauchy treated colleagues is given by
Poncelet whose work on projective geometry had, in 1820, been criticised by
Cauchy:-
... I managed to approach my too rigid judge at his
residence ... just as he was leaving ... During this very short and very rapid
walk, I quickly perceived that I had in no way earned his regards or his respect
as a scientist ... without allowing me to say anything else, he abruptly walked
off, referring me to the forthcoming publication of his Leçons à 'École
Polytechnique where, according to him, 'the question would be very properly
explored'.
Again his treatment of Galois and Abel during this period was
unfortunate. Abel, who visited the Institute in 1826, wrote of him:-
Cauchy is mad and there is nothing that can be done about
him, although, right now, he is the only one who knows how mathematics should be
done.
Belhoste in [4] says:-
When Abel's untimely death occurred on April 6, 1829,
Cauchy still had not given a report on the 1826 paper, in spite of
several protests from Legendre. The report he finally did give, on June 29,
1829, was hasty, nasty, and superficial, unworthy of both his own brilliance
and the real importance of the study he had judged.
By 1830 the political events in Paris and the years of hard
work had taken their toll and Cauchy decided to take a break. He left Paris in
September 1830, after the revolution of July, and spent a short time in
Switzerland. There he was an enthusiastic helper in setting up the Académie
Helvétique but this project collapsed as it became caught up in political
events.
Political events in France meant that Cauchy was now required
to swear an oath of allegiance to the new regime and when he failed to return to
Paris to do so he lost all his positions there. In 1831 Cauchy went to Turin and
after some time there he accepted an offer from the King of Piedmont of a chair
of theoretical physics. He taught in Turin from 1832. Menabrea attended these
courses in Turin and wrote that the courses:-
were very confused, skipping suddenly from one idea to
another, from one formula to the next, with no attempt to give a connection
between them. His presentations were obscure clouds, illuminated from time to
time by flashes of pure genius. ... of the thirty who enrolled with me, I was
the only one to see it through.
In 1833 Cauchy went from Turin to Prague in order to follow
Charles X and to tutor his grandson. However he was not very successful in
teaching the prince as this description shows:-
... exams .. were given each Saturday. ... When
questioned by Cauchy on a problem in descriptive geometry, the prince was
confused and hesitant. ... There was also material on physics and chemistry. As
with mathematics, the prince showed very little interest in these subjects.
Cauchy became annoyed and screamed and yelled. The queen sometimes said to him,
soothingly, smilingly, 'too loud, not so loud'.
While in Prague Cauchy had one meeting with Bolzano, at
Bolzano's request, in 1834. In [16] and [18] there are discussions on how much
Cauchy's definition of continuity is due to Bolzano, Freudenthal's view in [18]
that Cauchy's definition was formed before Bolzano's seems the more convincing.
Cauchy returned to Paris in 1838 and regained his position at
the Academy but not his teaching positions because he had refused to take an
oath of allegiance. De Prony died in 1839 and his position at the Bureau des
Longitudes became vacant. Cauchy was strongly supported by Biot and Arago but
Poisson strongly opposed him. Cauchy was elected but, after refusing to swear
the oath, was not appointed and could not attend meetings or receive a salary.
In 1843 Lacroix died and Cauchy became a candidate for his
mathematics chair at the Collège de France. Liouville and Libri were also
candidates. Cauchy should have easily been appointed on his mathematical
abilities but his political and religious activities, such as support for the
Jesuits, became crucial factors. Libri was chosen, clearly by far the weakest of
the three mathematically, and Liouville wrote the following day that he was:-
deeply humiliated as a man and as a mathematician by what
took place yesterday at the Collège de France.
During this period Cauchy's mathematical output was less than
in the period before his self-imposed exile. He did important work on
differential equations and applications to mathematical physics. He also wrote
on mathematical astronomy, mainly because of his candidacy for positions at the
Bureau des Longitudes. The 4-volume text Exercices d'analyse et de physique
mathématique published between 1840 and 1847 proved extremely important.
When Louis Philippe was overthrown in 1848 Cauchy regained his
university positions. However he did not change his views and continued to give
his colleagues problems. Libri, who had been appointed in the political way
described above, resigned his chair and fled from France. Partly this must have
been because he was about to be prosecuted for stealing valuable books.
Liouville and Cauchy were candidates for the chair again in 1850 as they had
been in 1843. After a close run election Liouville was appointed. Subsequent
attempts to reverse this decision led to very bad relations between Liouville
and Cauchy.
Another, rather silly, dispute this time with Duhamel clouded
the last few years of Cauchy's life. This dispute was over a priority claim
regarding a result on inelastic shocks. Duhamel argued with Cauchy's claim to
have been the first to give the results in 1832. Poncelet referred to his own
work of 1826 on the subject and Cauchy was shown to be wrong. However Cauchy was
never one to admit he was wrong. Valson writes in [7]:-
...the dispute gave the final days of his life a basic
sadness and bitterness that only his friends were aware of...
Also in [7] a letter by Cauchy's daughter describing his death
is given:-
Having remained fully alert, in complete control of his
mental powers, until 3.30 a.m.. my father suddenly uttered the blessed
names of Jesus, Mary and Joseph. For the first time, he seemed to be aware of
the gravity of his condition. At about four o'clock, his soul went to God. He
met his death with such calm that made us ashamed of our unhappiness.
Numerous terms in mathematics bear Cauchy's name:- the Cauchy
integral theorem, in the theory of complex functions, the Cauchy-Kovalevskaya
existence theorem for the solution of partial differential equations, the
Cauchy-Riemann equations and Cauchy sequences. He produced 789 mathematics
papers, an incredible achievement. This achievement is summed up in [4] as
follows:-
... such an enormous scientific creativity is nothing less
than staggering, for it presents research on all the then-known areas of
mathematics ... in spite of its vastness and rich multifaceted character,
Cauchy's scientific works possess a definite unifying theme, a secret wholeness.
... Cauchy's creative genius found broad expression not only in his work on the
foundations of real and complex analysis, areas to which his name is
inextricably linked, but also in many other fields. Specifically, in this
connection, we should mention his major contributions to the development of
mathematical physics and to theoretical mechanics... we mention ... his two
theories of elasticity and his investigations on the theory of light, research
which required that he develop whole new mathematical techniques such as Fourier
transforms, diagonalisation of matrices, and the calculus of residues.
His collected works, Oeuvres complètes d'Augustin Cauchy
(1882-1970), were published in 27 volumes.
Article by: J J O'Connor and E F Robertson
January 1997 |