Jean Le Rond d'Alembert
Nasceu: 17 Nov 1717 em Paris, França
Faleceu: 29 Oct 1783 em Paris, França
Jean Le Rond D'Alembert — matemático, físico e filósofo francês, nasceu na cidade de Paris em novembro de 1717. Filho natural do general Destouches e de Mme. de Tencin da aristocracia parisiense e amante de Philippe d'Orléans e de La Fresnais. Abandonado pelos pais nas proximidades da igreja de Saint-Jean-Le-Rond, perto da Notre-Dame de Paris, foi criado pela mulher de um vidraceiro que após alguns anos baptizou-o com o nome da igreja o qual representa um santo a que ela se dedica. Iniciou seus estudos no Collége des Quatre Nations, sob a direcção dos jansenistas. Estes estudos foram custeados pela Madame de Tencin, que tentou reclamá-lo quando notou que seu filho era um génio. Ao tomar conhecimento da reivindicação de sua mãe, D'Alembert, respondeu-lhe directamente com a seguinte frase: " Minha mãe é a mulher do vidraceiro ". Aos dezoito anos, substitui o sobrenome de Daremberg, até então adoptado, por D'Alembert.
Precoce, autodidacta, dotado para a filosofia, as ciências e as línguas mortas, tentou o direito e a medicina antes de descobrir que a sua vocação estava voltada para as matemáticas.
Em 1738, formou-se em Direito e iniciou, no mesmo ano, seus estudos de medicina, renunciando, em seguida, aos referidos cursos pelo fato de que como já foi dito, sua vocação era mesmo as matemáticas.
Em 1739, aos 22 anos, publica " Mémoire sur le calcul intégral " ( Memória sobre o cálculo integral ) a qual lhe valeu, um ano após, seu ingresso na Académie des Sciences em Paris.
Em 1740, enunciou e demonstrou o Teorema Fundamental da Álgebra, também conhecido como teorema de d'Alembert, e apresentou-o à Academia de Ciências de Berlim com o seguinte enunciado: " Toda e qualquer equação algébrica que representa uma função racional inteira, admite sempre uma raiz ".
Em 1741, apresentou pela primeira vez, em sua publicação " Mémoire sur la réfraction des corps solides " ( Memória sobre a refracção dos corpos sólidos ), uma bela exposição teórica, explicando por que um corpo passa de um fluido para outro mais denso, seguindo uma direcção oblíqua em relação à superfície que separa os dois fluidos.
Até o século XVIII os cientistas não percebiam que a diferença entre energia cinética e momento linear é a relação entre estes conceitos e o de força. Eles indagavam qual das duas grandezas, energia cinética ou momento linear, representaria a ' verdadeira ' medida do efeito de uma força sobre um corpo. Descartes argumentava que, quando os corpos interagem, tudo o que pode acontecer é apenas transferência de momento de um corpo para outro, pois o momento linear total do universo permanece constante; então, a ' verdadeira ' medida de uma força é a variação do momento linear que ela produz em um dado intervalo de tempo. Leibniz atacava este ponto de vista e dizia que a ' verdadeira ' medida de uma força é a variação que ela produz na energia cinética ( chamada por ele de vis viva ou ' força viva ', tomada como sendo o dobro daquilo que chamamos de energia cinética ).
Em 1743, d'Alembert publicou o " Traité de dynamique " ( Tratado de dinâmica ) expondo o princípio fundamental que levou o seu nome, consolidando a mecânica em três conceitos básicos que são a inércia, o movimento composto e o equilíbrio entre dois corpos. Com esta publicação, d'Alembert encerrou a discussão, considerando-a sem sentido, uma vez que provinha de uma confusão de terminologia. O efeito cumulativo de uma força pode ser medido por seu efeito integrado no tempo que produz a variação no momento linear, ou por seu efeito integrado no espaço que cria variação de energia cinética. Ambos os conceitos são úteis e válidos, embora diferentes.
D'Alembert combinou o conceito de movimento composto e o de equilíbrio entre dois corpos e estabeleceu o princípio que assim enunciou: " Num sistema as forças internas de inércia são iguais e opostas às forças que produzem a aceleração ". Este princípio, baseou-se numa observação bastante elementar de que a equação fundamental do movimento , ou seja , a força é igual ao produto da massa pela aceleração, expressa pela fórmula F = m.a ou F – m.a = 0; substituindo o produto ( –m.a ) por F*, a expressão assume a nova forma F + F* = 0 . Essa nova forma, que, para a dinâmica de um ponto livre, é de notória evidência, foi brilhantemente generalizada por d'Alembert para todo e qualquer sistema mecânico. Com isso, ele notabilizou-se pelo enunciado de um princípio básico de mecânica, conhecido como " Princípio de d'Alembert ".
Em 1744, publicou " Traité de l'équilibre et du mouvement des fluides " ( Tratado do equilíbrio e do movimento dos fluidos ) aplicando o seu princípio à solução de vários problemas relacionados com o movimento e o equilíbrio dos fluidos.
Em 1747, D'Alembert publicou " Réflexions sur la cause générale des vents " ( Reflexões sobre a causa geral dos ventos ) tendo sido conduzido, pelo fato de usar aplicações nesta e na obra supracitada princípios fundamentais estudados em 1742, ao estudo das derivadas parciais de segunda ordem, do tipo hiperbólico, as quais regem as pequenas oscilações transversais de uma corda homogénea, uniformemente distendida. A equação que rege tais oscilações é a seguinte:
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Equação das Ondas de D'Alembert Coube a ele estabelecer
um método |
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Estudou as equações diferenciais ordinárias, definiu a noção de limite e
inventou um critério de convergência das séries. Em 1748, iniciou seus trabalhos
com respeito a moderna e ampla teoria dos sistemas de equações diferenciais
lineares.
Em 1749, elabora trabalho sobre mecânica celeste sob o título " Recherches sur la précession des équinoxes et sur la mulation de l'arxe de la terre dans le système newtonien " ( Pesquisas sobre a precessão dos equinócios e sobre a nulação do eixo da terra no sistema newtoriano ) e as " Recherches sur différents points importants du système du monde " ( 1754 - 1756; Pesquisas sobre diferentes pontos importantes do sistema do mundo ), nos quais aplicou o seu princípio de dinâmica ao estudo do movimento da Terra em torno do Sol.
Em 1751 os livreiros Briasson, David, Le Breton e Durand publicam o primeiro tomo da " Encyclopédie ou Dictionnaire raisonné des sciences, des arts et des métiers " ( 1751 - 1780; Enciclopédia ou Dicionário racional das ciências, das artes e dos ofícios ), elaborado por d'Alembert e Diderot, tendo como parte inicial da referida enciclopédia o famoso " Discours préliminaire " da autoria de d'Alembert, uma peça magistral de grande conteúdo filosófico-literário, despertando o mundo científico para novos campos do conhecimento.
D'Alembert foi membro e secretário perpétuo da Academia Francesa. Recusou a presidência da Academia de Berlim e o lugar de preceptor do czarevich Paulo que Catarina II desejou confiar-lhe. Passou uma temporada em Genebra, na casa de Voltaire; visitou duas vezes Frederico II, o Grande, sendo que uma das vezes aconselha Euler a colocar o seu cargo à disposição, o qual foi assumido por Joseph-Louis Lagrange como matemático da corte, havendo, em seguida, o seguinte pronunciamento do rei: " Devo aos seus cuidados e recomendações por ter substituído um matemático meio cego por outro com ambos os olhos, o que vai satisfazer especialmente aos membros anatómicos da minha Academia ". D'Alembert, escreveu, também, verbetes sobre ciências e matemática para a Grande Enciclopédia, sendo o principal incentivador, com Diderot, da referida enciclopédia da qual redigiu e editou, em 1751, o Discurso preliminar.
D'Alembert, até 1765, viveu modestamente na casa de sua mãe de criação, morando, posteriormente, com Julie de Lespinasse, o grande amor de sua vida. Faleceu no dia 29 de outubro de 1783 em Louvre - Paris, para onde se transferira em 1776.
Jean d'Alembert's father was an artillery officer,
Louis-Camus Destouches and his mother was Mme de Tencin. She had been a nun but
had received a papal dispensation in 1714 which allowed her to begin [4]:-
... a brilliant social career in which political intrigues
and amorous liaisons contended for first place; a timely participation in the
famous John Law Scheme allowed her to pursue these activities in complete
financial security. [John Law was a Scottish monetary reformer who founded a bank
in Paris in 1716 with authority to issue notes. It was highly successful at
first, the time when Mme de Tencin made her money, but collapsed in 1720.]
D'Alembert was the illegitimate son from one of Mme de Tencin
'amorous liaisons'. His father, Louis-Camus Destouches, was out of the country
at the time of d'Alembert's birth and his mother left the newly born child on
the steps of the church of St Jean Le Rond. The child was quickly found and
taken to a home for homeless children. He was baptised Jean Le Rond, named after
the church on whose steps he had been found.
When his father returned to Paris he made contact with his
young son and arranged for him to be cared for by the wife of a glazier, Mme
Rousseau. She would always be d'Alembert's mother in his own eyes, particularly
since his real mother never recognised him as her son, and he lived in Mme
Rousseau's house until he was middle-aged.
The first school that d'Alembert attended was a private school,
his education being arranged by his father. His father died in 1726 when
d'Alembert was nine years old and he left him just enough money to give him
security. The Destouches family continued to look after d'Alembert's education
and they arranged for him to enter the Jansenist Collège des Quatre Nations. He
enrolled in the name of Jean-Baptiste Daremberg but soon changed his name to
Jean d'Alembert.
The Collège des Quatre Nations was an excellent place for
d'Alembert to study mathematics even though the course was elementary. The
mathematics course, given by Professor Carron, was based on Varignon's
lectures and d'Alembert was able to make use of the excellent mathematics
library at the Collège. As well as the mathematical training, he learnt about Descartes'
physical ideas at the Collège but, when he formed his own ideas later in his
life, he would have little respect for the views of Descartes.
The main aim of the Jansenist Collège des Quatre Nations was to
produce scholars who could become experts in theology and argue the Jansenist
case against the Jesuits. However, d'Alembert was turned off the study of
theology at the Collège. After graduating in 1735 he decided that he would make
a career in law but his real passion was for mathematics and he continued to
work in his spare time on that subject. In 1738 d'Alembert qualified as an
advocate but he seems to have decided that this was not the career for him. The
following year d'Alembert studied medicine but this was a topic that he found
even worse than theology. Of all the topics he had studied the one that he had
real enthusiasm for was mathematics and his progress in this was quite
remarkable, particularly given that he had studied almost exclusively on his own
and at a time when he was supposed to be studying foe other qualifications.
In July 1739 d'Alembert read his first paper to the Paris
Academy of Science on some errors he had found in Reyneau's
standard text Analyse démontrée which were not of great significance but
marked the start of his mathematical career. In 1740 he submitted a second work
on the mechanics of fluids which was praised by Clairaut.
In May 1741 d'Alembert was admitted to the Paris Academy of Science, on the
strength of these and papers on the integral calculus. It took some
determination on his part, submitting three unsuccessful applications in quick
succession, before his appointment.
Before discussing d'Alembert's contributions it is useful to
discuss his personality, which was to have a major effect on the way his
scientific work was to develop. In one sense d'Alembert's life was uneventful.
He travelled little and worked at the Paris Academy of Science and the French
Academy all his life. On another level his life was one of great drama as he
argued with almost everyone around him. As stated in [5]:-
D'Alembert was always surrounded by controversy. ... he
was a lightning rod which drew sparks from all the foes of the philosophes.
... Unfortunately he carried this... pugnacity into his scientific research
and once he had entered a controversy, he argued his cause with vigour and
stubbornness. He closed his mind to the possibility that he might be
wrong... Despite this tendency to quarrel with all around him, his
contributions were truly outstanding. D'Alembert helped to resolve the
controversy in mathematical physics over the conservation of kinetic energy by
improving Newton's
definition of force in his Traité de dynamique which he published in
1743. This also contains d'Alembert's principle of mechanics. This is an
important work and the preface contains a clear statement by d'Alembert of an
attempt to lay a firm foundation for mechanics. In [5] d'Alembert's ideas, as
presented in this preface, are described:-
... d'Alembert was a mathematician, not a physicist, and
he believed mechanics was just as much a part of mathematics as geometry or
algebra. Rational mechanics was a science based on simple necessary principles
from which all particular phenomenon could be deduced by rigorous mathematical
methods. ... d'Alembert thought mechanics should be made into a completely
rationalistic mathematical system. D'Alembert had begun to read parts of his Traité de
dynamique to the Academy in late 1742 but soon afterwards Clairaut
began to read his own work on dynamics to the Academy. Clearly a rivalry quickly
sprung up and d'Alembert stopped reading the work to the Academy and rushed into
print with the treatise. The two mathematicians had come up with similar ideas
and indeed the rivalry was to become considerably worse in the few years.
D'Alembert stated his position clearly that he believed
mechanics to be based on metaphysical principles and not on experimental
evidence. He seems not to have realised in his reading of Newton's
Principia how strongly Newton
based his laws of motion on experimental evidence. For d'Alembert these laws of
motion were logical necessities.
In 1744 d'Alembert applied his results to the equilibrium and
motion of fluids and published Traité de l'equilibre et du mouvement des
fluides. This work gave an alternative treatment of fluids to the one
published by Daniel
Bernoulli. D'Alembert thought it a better approach, of course, as one might
expect, Daniel
Bernoulli did not share this view.
D'Alembert became unhappy at the Paris Academy, almost
certainly because of his rivalry with Clairaut
and disagreements with others. His position became even less happy in 1745 when
Maupertuis
left Paris to take up the post of head of the Berlin Academy where, at that
time, Euler
was working.
In around 1746 d'Alembert's life took a rather sudden change.
This is described in [4] as follows:-
Until [1746] he had been satisfied to lead a
retired but mentally active existence at the house of his foster-mother. In
1746 he was introduced to Mme Geoffrin, the rich, imperious,
unintellectual but generous founder of a salon to which d'Alembert was
suddenly invited. He soon entered a social life in which, surprisingly enough,
he began to enjoy great success and popularity. Around the same time d'Alembert began to become involved in a
major project, namely editing the Encyclopédie with Diderot. He was
contracted as an editor to cover mathematics and physical astronomy but his work
covered a wider field. When the first volume appeared in 1751 it contained a
Preface written by d'Alembert which was widely acclaimed as a work of great
genius. Buffon said that:-
It is the quintessence of human knowledge...
D'Alembert worked on the Encyclopédie for many years. In
fact he wrote most of the mathematical articles in this 28 volume work. However,
he continued his mathematical work while working on the Encyclopédie. He
was a pioneer in the study of partial differential
equations and he pioneered their use in physics. His work on this
topic first appeared in an article which he submitted for the 1747 prize of the
Prussian Academy Réflexions sur la cause générale des vents which indeed
he won the prize. Euler,
however, saw the power of the methods introduced by d'Alembert and soon
developed these far further than had d'Alembert. In fact this work by d'Alembert
on the winds suffers from a defect which was typical of all of his work, namely
it was mathematically very sound but was based on rather poor physical evidence.
In this case, for example, d'Alembert assumed that the winds were generated by
tidal effects on the atmosphere and heating of the atmosphere played only a very
minor role. Clairaut
attacked d'Alembert's methods [5]:-
In order to avoid delicate experiments or long tedious
calculations, in order to substitute analytical methods which cost them less
trouble, they often make hypotheses which have no place in nature; they pursue
theories that are foreign to their object, whereas a little constancy in the
execution of a perfectly simple method would have surely brought them to their
goal. A heated argument between d'Alembert and Clairaut
resulted in the two fine mathematicians trading insults in the scientific
journals of the day.
The year 1747 was an important one for d'Alembert in that a
second important work of his appeared in that year, namely his article on
vibrating strings. The article contains the first appearance of the wave
equation in print but again suffers from the defect that he used mathematically
pleasing simplifications of certain boundary conditions which
led to results which were at odds with observation.
Euler
had learnt of d'Alembert's work in around 1743 through letters from Daniel
Bernoulli. However, Daniel
Bernoulli became highly critical of d'Alembert after reading his Traité
de l'equilibre et du mouvement des fluides for reasons we noted above. When
d'Alembert won the prize of the Prussian Academy with his essay on winds he
produced a work which Euler
considered superior to that of Daniel
Bernoulli. Certainly at this time Euler
and d'Alembert were on very good terms with Euler
having high respect for d'Alembert's work and the two corresponded on many
topics of mutual interest.
However relations between Euler
and d'Alembert soon took a turn for the worse after the dispute in the Berlin
Academy involving Samuel
König which began in 1751. The situation became more relevant to d'Alembert
in 1752 when he was invited to became President of the Berlin Academy. Another
reason for d'Alembert to feel angry with Euler
was that he felt that Euler
was stealing his ideas and not giving him due credit. In one sense d'Alembert
was justified but on the other hand his work was usually so muddled that Euler
could not follow it and resorted to starting from scratch to clarify the problem
being solved.
The Paris Academy had not been a place for d'Alembert to
publish after he fell out with colleagues there and he was sending his
mathematical papers to the Berlin Academy during the 1750s. However Euler
was unhappy to publish these works and d'Alembert stopped publishing his
mathematical articles, collecting them together and publishing them as
Opuscules mathématique which appeared in eight volumes between 1761 and
1780.
Again Frederick II, the King of Prussia, tried to persuade
d'Alembert to accept the presidency of the Berlin Academy. Euler
was strongly opposed to this and wrote to Lagrange (see [5]):-
... d'Alembert has tried to undermine [my solution
to the vibrating strings problem] by various cavils, and that for the
sole reason that he did not get it himself. ... He thinks he can deceive the
semi-learned by his eloquence. ... He wished to publish in our journal not a
proof, but a bare statement that my solution is defective. ... From this you
can judge what an uproar he would let loose if he were to become our
president. Euler
need not have feared however, for d'Alembert visited Frederick II for three
months in 1764, turned down the offer of the presidency again, and tried to
persuade Frederick II to made Euler
president. This was not the only offer d'Alembert turned down. He also turned
down an invitation from Catherine II to go to Russia as a tutor for her son.
D'Alembert made other important contributions to mathematics
which we have not yet mentioned. In an article entitled Différentiel in
volume 4 of Encyclopédie written in 1754, he suggested that the theory of
limits be put on a firm foundation. He was one of the first to understand the
importance of functions and, in this article, he defined the derivative of a
function as the limit of a quotient of increments. His ideas on limits led him
to the test for convergence, known today as d'Alembert's ratio test, which
appears in Volume 5 of Opuscules mathématique. D'Alembert was elected to the French Academy on 28 November
1754. In 1772 he was elected perpetual secretary of the French Academy and spent
much time writing obituaries for the academy [1]:-
He became the academy's most influential member, but, in
spite of his efforts, that body failed to produce anything noteworthy in the
way of literature during his pre-eminence. D'Alembert complained from 1765, after a bout of illness, that
his mind was no longer able to concentrate on mathematics. In 1777, in a letter
to Lagrange,
he showed how much he regretted this:-
What annoys me the most is the fact that geometry, which
is the only occupation that truly interests me, is the one thing that I cannot
do. All that I do in literature, although very well received in our public
sessions of the French Academy, is for me only a way to fill the time for lack
of anything better to do. He suffered bad health for many years and his death was as the
result of a bladder illness. As a known unbeliever, d'Alembert was buried in a
common unmarked grave.
In the latter part
of his life d'Alembert turned more towards literature and philosophy.
D'Alembert's philosophical works appear mainly in the five volume work
Mélanges de littérature et de philosophie which appeared between 1753 and
1767. In this work he sets out his skepticism concerning metaphysical problems.
He accepts the argument in favour of the existence of God, based on the belief
that intelligence cannot be a product of matter alone. However, although he took
this public view in his books, evidence from his friends showed that he was
persuaded by Diderot towards materialism before 1770.