**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 for 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 next 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'équilibre 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. In1746he 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'équilibre et du mouvement des fluides* for reasons we noted above. When d'Alembert won the prize of the Prussian Academy of Sciences 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 mathématiques* 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 mathématiques.*

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.

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.

**Article by:** *J J O'Connor* and *E F Robertson*

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