The chief contribution by the French mathematician and physicist Jean le Rond d'Alembert (1717-1783) is D'Alembert's principle, in mechanics. He was also a pioneer in the study of partial differential equations.

# Jean le Rond d'Alembert

Jean le Rond d'Alembert was born on Nov. 16, 1717, and abandoned on the steps of the church of St-Jean-le-Rond in Paris. He was christened Jean Baptiste le Rond. The infant was given into the care of foster parents named Rousseau. Jean was the illegitimate son of Madame de Tencin, a famous salon hostess, and Chevalier Destouches, an artillery officer, who provided for his education. At the age of 12, Jean entered the Collège Mazarin and shortly afterward adopted the name D'Alembert. He became a barrister but was drawn irresistibly toward mathematics.

Two memoirs, one on the motion of solid bodies in a fluid and the other on integral calculus, secured D'Alembert's election in 1742 as a member of the Paris Academy of Sciences. A prize essay on the theory of winds in 1746 led to membership in the Berlin Academy of Sciences. D'Alembert wrote the introduction and a large number of the articles on mathematics and philosophy for Denis Diderot's *Encyclopédie.* He entered the Académie Française as secretary in 1755.

D'Alembert had a generous nature and performed many acts of charity. Two people especially claimed his affection; his foster mother, with whom he lived until he was 50, and the writer Julie de Lespinasse, whose friendship was terminated only by her death. D'Alembert died in Paris on Oct. 29, 1783.

## Rigid Body and Fluid Motion

D'Alembert's principle appeared in his *Traité de dynamique* (1743). It concerns the problem of the motion of a rigid body. Treating the body as a system of particles, D'Alembert resolved the impressed forces into a set of effective forces, which would produce the actual motion if the particles were not connected, and a second set. The principle states that, owing to the connections, this second set is in equilibrium. An outstanding result achieved by D'Alembert with the aid of his principle was the solution of the problem of the precession of the equinoxes, which he presented to the Berlin Academy in 1749. Another form of D'Alembert's principle states that the effective forces and the impressed forces are equivalent. In this form the principle
had been applied earlier to the problem of the compound pendulum, but these anticipations in no way approach the clarity and generality achieved by D'Alembert.

In his *Traité de l'équilibre et du mouvement des fluides* (1744), D'Alembert applied his principle to the problems of fluid motion, some of which had already been solved by Daniel Bernoulli. D'Alembert recognized that the principles of fluid motion were not well established, for although he regarded mechanics as purely rational, he supposed that the theory of fluid motion required an experimental basis. A good example of a theoretical result which did not seem to correspond with reality was that known as D'Alembert's paradox. Applying his principle, D'Alembert deduced that a fluid flowing past a solid obstacle exerted no resultant force on it. The paradox disappears when it is remembered that the inviscid fluid envisaged by D'Alembert was a pure fiction.

## Partial Differential Equations

Applying calculus to the problem of vibrating strings in a memoir presented to the Berlin Academy in 1747, he showed that the condition that the ends of the string were fixed reduced the solution to a single arbitrary function. D'Alembert also deserves credit for the derivation of what are now known as the Cauchy-Riemann equations, satisfied by any holomorphic function of a complex variable.

Research on vibrating strings reflected only one aspect of D'Alembert's interest in music. He wrote a few of the musical articles for the *Encyclopédie.*

He favored the views of the composer Jean Philippe Rameau and expounded them in his popular *Élemens de musique théorique et pratique* (1752).

## Further Reading on Jean le Rond d'Alembert

D'Alembert's more important mathematical works are available in English, as are his many contributions to the *Encyclopédie,* the most significant of which is his *Preliminary Discourse.* His contributions are discussed in Thomas L. Hankins, *Jean d'Alembert: Science and the Enlightenment* (1970; reprinted, 1990). Excellent studies on D'Alembert as a *philosophe* are Ronald Grimsley, *Jean D'Alembert* (1963), and John Nicholas Pappas, *Voltaire and D'Alembert* (1962). The standard biography, in French, is Joseph Bertrand, *D'Alembert* (1889). A full account of D'Alembert's work in dynamics appears in René Dugas, *A History of Mechanics* (1950; trans. 1955).