Augustin Louis Cauchy
Born: 21 Aug 1789 in Paris, France
Died: 23 May 1857 in Sceaux (near Paris), France
Augustin Louis Cauchy - matemático e físico-matemático de origem Parisiense - França, nasceu em 21 de agosto de 1789. Filho de um homem de muita cultura, teve uma educação muito esmerada. Ademais, Laplace e Lagrange muito contribuíram para a sua formação, inclusive, influenciando-o nos estudos. Aos dezasseis anos, ingressou na École des Ponts et Chanssées, tendo concluído o curso de engenharia em 1809 como primeiro aluno da turma. Trabalhou até 1813 no cargo de engenheiro e colaborador no projecto do canal de Ourcq e na construção do porto militar de Cherbourg.
Em 1811, provou que os ângulos de um poliedro convexo são determinados por suas faces ( as superfícies planas que formam um sólido geométrico ). Em 1812, publicou uma memória onde lançou a ideia, posteriormente desenvolvida por Èvariste Galois com respeito a Teoria das Substituições. Neste mesmo ano, lançou seu primeiro trabalho sobre determinantes, com oitenta e quatro páginas, passando a aplicá-los nas mais diversas situações como, por exemplo, na propagação de ondas.
A memória de 1814, acerca da integral definida com um número complexo como limite, inicia sua carreira de criador de gênio e de reformador da teoria das funções de variável complexa.
Em 1815, com apenas vinte e seis anos passou a leccionar na École Polytechnique, assumindo a cadeira de análise e mecânica em face de a ele ser atribuída em virtude da quantidade e do valor de suas descobertas, principalmente nos domínios da geometria e da análise matemática, como também, estudou os grupos de permutação, criando praticamente a teoria moderna dos grupos. Uma longa lista de trabalhos nesse sector culmina com a teoria dos grupos finitos. Logo depois, foi nomeado membro do Collège de France e da Sorbonne. Em 1816, é nomeado para a secção de mecânica da Académie des Sciences, ocupando uma das vagas deixadas por Sadi Carnot e Gaspard Monge. Também neste mesmo ano, suas descobertas foram, por ele mesmo, aplicadas à mecânica celeste e à física, tendo obtido, o grande prémio do Instituto de Física, por essa contribuição ao estudos dos fluidos.
Em 1819, publica um tratado com o título " Traité du calcul differentiel et du calcul intégral " ( Tratado de cálculo diferencial e de cálculo integral ). Entre 1821 e 1830, publicou três obras, intituladas " Cours d'analyse de l'école royale polytechnique " ( Curso de análise da escola real politécnica ), " Le Calcul Infinitésimal "( 1823; O Cálculo Infinitesimal ) e " Leçons sur les applications du calcul infinitésimal à la géométrie " ( 1826 - 1828, 2 t. em1 v.; Lições sobre as aplicações do cálculo infinitesimal à geometria ) que deram ao cálculo elementar o carácter que tem hoje, definindo precisamente limite, derivada e integral; os conceitos de funções e de limites de funções eram de fundamental importância. Estas obras de Cauchy foram desenvolvidas quase ao mesmo tempo e com ideias semelhantes às de Bolzano, um padre Tcheco.
Em suas notas, Cauchy demonstrou teoremas de existência, relativos às equações diferenciais, adoptando um critério construtivo para a pesquisa da solução, método que seria adoptado por Lipschitz, um matemático alemão, em suas aproximações sucessivas.
Seus trabalhos sobre mecânica e física-matemática, especialmente sua teoria de ondas, são notáveis pela precisão e clareza das demonstrações. Na álgebra contribuiu decisivamente para o progresso da teoria dos determinantes, desenvolvendo os princípios fundamentais estabelecidos por Vandermonde e Laplace. Substituiu o termo " resultante ", adoptado pelos dois ilustres matemáticos, por " determinante " o qual embora já empregado por Gauss, um matemático alemão, só foi definitivamente introduzido na matemática graças aos trabalhos e ao prestígio de Cauchy.
Ele definiu funções analíticas da seguinte maneira: uma função w da variável complexa z = (x + yi) é expressa da forma u + iv, onde u e v são funções reais de x e de y; diz-se que w é analítica se u, v e as derivadas dessas funções, relativamente a x e y, são contínuas e satisfazem as relações, chamadas de Cauchy-Riemann.
Vale salientar, também, o importante teorema de Cauchy que diz o seguinte:
Se f(z) é analítica em todos os pontos de uma curva fechada K e no interior de K, então a integral, sobre K, de f(z) é igual a zero ( percorrendo-se K no sentido positivo). Cauchy deduz ainda a integral ( dita de Cauchy ) que fornece o valor de f(z) em um ponto a do interior de K e são notáveis as fórmulas das derivadas sucessivas de f(z), calculadas no ponto a. Ele desenvolve, enfim, o cálculo dos resíduos, para aplicá-lo na avaliação de integrais definidas reais, tornando-se, com o desenvolvimento da tecnologia, uma verdadeira rotina.
A instabilidade na política fizeram-no interromper a carreira no magistério e, em 1830, após a deposição do rei Carlos X e o advento da nova monarquia, Cauchy recusou-se a prestar juramento de fidelidade ao novo regime de Luís Felipe de Orléans. Abandonou a França, exilando-se voluntariamente em Turim. Valendo-se de uma vasta cultura clássica, Cauchy propôs-se a leccionar em latim, mas seus alunos preferiram que ministrasse as aulas em italiano. Com sua presença, a universidade de Turim readquiriu o prestígio de que gozava ao tempo de Lagrange.
As firmes convicções políticas de Cauchy e sua fidelidade à dinastia dos Bourbons fizeram-no abandonar Turim, em 1833, para cuidar da educação do duque de Bordeaux, filho de Carlos X e herdeiro presuntivo do trono da França. Partiu então para Praga, onde o rei se havia exilado, permanecendo até 1838.
Voltou a Paris, onde passou a dedicar-se apenas à ciência, recusando todos os cargos políticos que lhe foram oferecidos. Quando Luís Napoleão assumiu a presidência da II República de 1848, aboliu a obrigatoriedade de juramento de fidelidade ao regime, que só foi restabelecida por Napoleão II, em 1852, o qual dispensou do juramento dois dos seus mais célebres homens da ciência: Cauchy e Arago, sendo que o primeiro, assumiu a cadeira de matemática da Faculté des Sciences, que ocupou até às vésperas da morte.
Cauchy foi o primeiro dos grandes matemáticos franceses cujo pensamento pertence claramente à Idade Moderna. Sua produção é imensa, comparável apenas à de Euler e à de Cayley. A fecundidade de Cauchy era tão prodigiosa , que sentiu necessidade de redigir entre 1826 e 1830 um espécie de diário, o qual denominou-se "Exercises de mathématique " ( Exercícios de matemática ) que se prolongou na série denominada " Exercises d'analyse mathématique et de physique " ( 1840-1847; Exercícios de análise matemática e de física ). Também, se deve a Cauchy a publicação em 1835, das atas " Comptes Rendus da Académie des Sciences " que inundou com suas produções, algumas muito extensas, com duzentas ou trezentas páginas. Ressalte-se que a direcção da revista, assombrada com seu elevado custo, deliberou aceitar apenas pequenos resumos para publicação - medida que ainda hoje está em vigor.
Cauchy escreveu mais de setecentas memórias, abarcando quase todos os ramos da matemática. Em todas elas sobressai o rigor com que os temas são tratados.
Mesmo após sua morte, ocorrida na cidade de Sceaux, França, em 23 de maio de 1857, foram publicados em 1882, sob os auspícios da Académie des Sciences sua obra com vinte e sete volumes intitulada " Euvres Complètes " ( Obras completas ).
-------------------------------------------------------------------------------------------------------------------------------------- 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
Méchanique Céleste and one of Lagrange's
Thèorie 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 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 Science 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 infinétesimal while in 1829 in
Leçons sur le Calcul Différential 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 , 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 [17] and [19] there are discussions on how much Cauchy's
definition of continuity is due to Bolzano,
Freudenthal's
view in [19] 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 Exercises d'analyse et de physique mathematique 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.