ALEMBERT, JEAN LE ROND D' (1717–1783)

ALEMBERT, JEAN LE ROND D' (1717–1783), French mathematician, scientist, philosopher, and writer. Born 17 November 1717, Jean Le Rond d'Alembert was the illegitimate son of the famous Claudine Alexandrine Guérin, marquise de Tencin, and an artillery officer, Louis-Camus Destouches. Abandoned on the steps of Saint-Jean-Le-Rond in Paris, he was taken to the Foundling Home and named after the church where he was discovered. Through his father's efforts he was placed with a foster mother, Mme. Rousseau, to whom he remained devoted. His father also saw to it that his son received a good education; he attended first a private school, then the Collège des Quatre-Nations. After three years studying law and medicine, it became clear to d'Alembert that mathematics was his true vocation. In 1741 he was named an adjoint (adjunct) at the Academy of Sciences, and in 1743 he published his most important mathematical work, the Traité de dynamique (Treatise on dynamics). In addition to six other major scientific treatises, his 1752 Éléments de musique, théorique et pratique, suivant les principes de Rameau (Elements of practical and theoretical music following Rameau's principles) is noteworthy as a lucid exposition of Rameau's hugely influential harmonic theory.

Today d'Alembert is somewhat undervalued, remembered mostly as coeditor of the Encyclopédie, although even in that enterprise he was eclipsed by Denis Diderot (1713–1784). In his day d'Alembert was esteemed second only to Voltaire (1694–1778) in leading the philosophe movement, the very core of Enlightenment ideology. Through his role in the French Academy, to which he was elected in 1754, and of which he became permanent secretary in 1772, the discreet and cautious d'Alembert was able to confer legitimacy on many of the philosophes' deepest concerns while remaining immune to the imprisonments and exiles that punctuated the lives of so many of his colleagues.

Largely because of his scientific reputation, but also because he was a popular, brilliant participant in Parisian salons, d'Alembert was asked as early as 1745 to participate in the production of the Encyclopédie; in 1747 he was named coeditor with Diderot and was charged primarily with the mathematical and scientific articles. His nonscientific entry, the infamous "Genève," created a controversy with Jean-Jacques Rousseau (1712–1778) and then with Genevan Protestants, leading d'Alembert to resign from his editorial post in 1758.

The desire to avoid scandal at all costs, which led to his resignation, was consistent with the public comportment d'Alembert adopted for the rest of his career. Although he shared many of the goals of the other philosophes, his correspondence (in particular with Voltaire) consistently shows not only a refusal to jeopardize his career and freedom to remain in Paris but also an unflinching conviction that enlightenment must be a gradual and tactful process of persuasion rather than a series of attacks, whether open or anonymous. He thought he could best serve that end by promoting the philosophe party at large and especially in the Academy, by mediating disputes within the group and by functioning as a de facto public relations manager as a foil to the polemical outpourings from Voltaire at Ferney and from numerous other quarters (most notably the baron Paul Thiry, baron d'Holbach; 1723–1789). Indeed, it had long been Voltaire's wish that when he died, d'Alembert would succeed him as leader of the philosophes. Much of d'Alembert's immense stature in the eighteenth century, then, came not from his writings but from his ceaseless efforts to unite and promote his colleagues and advance their mutual cause.

In 1759 he laid out his philosophical principles and methodology in his Essai sur les éléments de philosophie: ou sur les principes des connaissances humaines (Essay on the elements of philosophy, or on the principles of human knowledge). In this work he provides a synthesis of his prior thought in epistemology, metaphysics, language theory, science, and aesthetics. The Éclaircissements (Explanations), added in 1767, round out the Essay, forming a composite that represents the ambitious scope of d'Alembert's empiricist philosophy.

However, his most important work is without doubt the 1751 Preliminary Discourse to the Encyclopedia. In this concise and occasionally flawed but often brilliant document, d'Alembert seeks to justify the encyclopedic enterprise in a Lockean vein, by showing the unity of all thought from its sensorial origins (in "direct" and "reflected" ideas deriving from corporeal impressions). However, he also attempts to provide a rational, scientific method for the mapping of human knowledge as well as a historical account of the evolution of human thought. The result is not merely an apology for the ends as well as the means of the Encyclopédie, it is also a superb summation of Enlightenment empirical and sensualist thought, a forceful rejection of Cartesian metaphysics (if not Cartesian method, which d'Alembert admired), and a valorization of the scientific method of Francis Bacon (1561–1626) and (particularly) Isaac Newton (1642–1727). In the Discourse, d'Alembert succeeds in showing the intimate connection between the spirit of the Encyclopédie and the concerns of the Enlightenment generally, in a way that is not always obvious to the reader of the encyclopedia's articles themselves.

D'Alembert's last important work, the fifth volume of Mélanges de littérature, d'histoire, et de philosophie, was published in 1767. From that point on, his health became increasingly fragile. In his last years he wrote little, instead concentrating on his duties as permanent secretary of the French Academy. As the result of his refusal of an operation (without which his doctors informed him he would not survive) for a painful bladder ailment he had had for years, d'Alembert died on 29 October 1783.

See also Diderot, Denis ; Encyclopédie ; Enlightenment ; Mathematics ; Philosophes ; Voltaire.

BIBLIOGRAPHY

Primary Sources

Alembert, Jean Le Rond d'. Œuvres de d'Alembert. 5 vols, reprint of 1821–1822 Paris edition. Geneva, 1967.

——. Œuvres et correspondances inédites de d'Alembert. Edited by Charles Henry. Reprint. Geneva, 1967.

——. Preliminary Discourse to the Encyclopedia of Diderot. Edited by Walter E. Rex and Richard N. Schwab. Chicago, 1995.

——. Traité de dynamique. Sceaux, 1990.

Secondary Sources

Essar, Dennis F. The Language Theory, Epistemology, and Aesthetics of Jean Lerond d'Alembert. Oxford, 1976.

Grimsley, Ronald. Jean d'Alembert (1717–1783). Oxford, 1963.

Hankins, Thomas L. Jean d'Alembert: Science and the Enlightenment. Oxford, 1970.

Pappas, John N. Voltaire and D'Alembert. Bloomington, Ind., 1962.

Patrick Riley, Jr.

Alembert, Jean Le Rond D’

(b. Paris, France, 17 November 1717; d. Pairs, 29 October 1783)

physics, mathematics.

Jean Le Rond d’Alembert was the illegitimate child of Madame de Tencin, a famous salon hostess of the eighteenth century, and the Chevalier DestouchesCanon, a cavalry officer. His mother, who had renounced her nun’s vows, abandoned him, for she feared being returned to a convent. His father, however, located the baby and found him a home with a humble artisan named Rousseau and his wife. D’Alembert lived with them until he was forty-seven years old. Destouches-Canon also saw to the education of the child. D’Alembert attended the Collège de Quatre-Nations (sometimes called after Mazarin, its founder), a Jansenist school offering a curriculum in the classics and rhetoric–and also offering more than the average amount of mathematics. In spite of the efforts of his teachers, he turned against a religious career and began studies of law and medicine before he finally embarked on a career as a mathematician. In the 1740’s he became part of the philosophes, thus joining in the rising tide of criticism of the social and intellectual standards of the day. D’Alembert published many works on mathematics and mathematical physics, and was the scientific editor of the Encyclopédie.

D’Alembert never married, although he lived for a number of years with Julie de Lespinasse, the one love of his life. A slight man with an expressive face, a high-pitched voice, and a talent for mimicry, he was known for his wit, gaiety, and gift for conversation, although later in life he became bitter and morose. D’Alembert spent his time much as the other philosophes did: working during the morning and afternoon and spending the evening in the salons, particularly those of Mme. du Deffand and Mlle. de Lespinasse. He seldom traveled, leaving the country only once, for a visit to the court of Frederick the Great. D’Alembert was a member of the Académie des Sciences, the Académie Française, and most of the other scientific academies of Europe. He is best known for his work in mathematics and rational mechanics, and for his association with the Encyclopédie.

D’Alembert appeared on the scientific scene in July 1739, when he sent his first communication to the Acdémie des Sciences. It was a critique of a mathematical text by Father Charles Reyneau. During the next two years he sent the academy five more mémoires dealing with methods of integrating differential equations and with the motion of bodies in resisting media. Although d’Alembert had received almost no formal scientific training (at school he had studied Varignon’s work), it is clear that on his own he had become familiar not only with Newton’s work, but also with that of L’Hospital, the Bernoullis, and the other mathematicians of his day. His communications to the academy were answered by Clairaut, who although only four years older than d’Alembert was already a member.

After several attempts to join the academy, d’Alembert was finally successful. He was made adjoint in astronomy in May 1741, and received the title of associé géometre in 1746. From 1741 through 1743 he worked on various problems in rational mechanics and in the latter year published his famous Traité de dynamique. He published rather hastily (a pattern he was to follow all of his life) in order to forestall the loss of priority; Clairaut was working along similar lines. His rivalry with Clairaut, which continued until Clairaut’s death, was only one of several in which he was involved over the years.

The Traité de dynamique, which has become the most famous of his scientific works, is significant in many ways. First, it is clear that d’Alembert recognized that a scientific revolution had occurred, and he thought that he was doing the job of formalizing the new science of mechanics. That accomplishment is often attributed to Newton, but in fact it was done over a long period of time by a number of men. If d’Alembert was overly proud of his share, he was at least clearly aware of what was happening in science. The Traité also contained the first statement of what is now known as d’Alembert’s principle. D’Alembert was, furthermore, in the tradition that attempted to develop mechanics without using the notion of force. Finally, it was long afterward said (rather simplistically) that in this work he resolved the famous vis viva controversy, a statement with just enough truth in it to be plausible. In terms of his own development, it can be said that he set the style he was to follow for the rest of his life.

As was customary at the time, d’Alembert opened his book with a lengthy philosophical preface. It is true that he was not always faithful to the principles he set down in the preface, but it is astonishing that he could carry his arguments as far as he did and remain faithful to them. D’Alembert fully accepted the prevailing epistemology of sensationalism. Taken from John Locke and expanded by such men as Condillac, sensationalism was to be d’Alembert’s metaphysical basis of science. The main tenet of this epistemology was that all knowledge was derived, not from innate ideas, but from sense perception. In many ways, however, d’Alembert remained Cartesian. The criterion of the truth, for example, was still the clear and simple idea, although that idea now had a different origin. In science, therefore, the basic concepts had to conform to this ideal.

In developing his philosophy of mechanics, d’Alembert analyzed the ideas available to him until he came to those that could be analyzed no further; these were to be his starting points. Space and time were such. So simple and clear that they could not even be defined, they were the only fundamental ideas he could locate. Motion was a combination of the ideas of space and time, and so a definition of it was necessary. The word “force” was so unclear and confusing that it was rejected as a conceptual building block of mechanics and was used merely as a convenient shorthand when it was properly and arbitrarily defined. D’Alembert defined matter as impenetrable extension, which took account of the fact that two objects could not pass through one another. The concept of mass, which he defined, as Newton had done, as quantity of matter, had to be smuggled into the treatise in a mathematical sense later on.

In the first part of the Traité, d’Alembert developed his own three laws of motion. It should be remembered that Newton had stated his laws verbally in the Principia, and that expressing them in algebraic form was a task taken up by the mathematicians of the eighteenth century. D’Alembert’s first law was, as Newton’s had been, the law of inertia. D’Alembert, however, tried to give an a priori proof for the law, indicating that however sensationalistic his thought might be he still clung to the notion that the mind could arrive at truth by its own processes. His proof was based on the simple ideas of space and time; and the reasoning was geometric, not physical, in nature. His second law, also proved as a problem in geometry, was that of the parallelogram of motion. It was not until he arrived at the third law that physical assumptions were involved.

The third law dealt with equilibrium, and amounted to the principle of the conservation of momentum in impact situations. In fact, d’Alembert was inclined to reduce every mechanical situation to one of impact rather than resort to the effects of continual forces; this again showed an inheritance from Descartes. D’Alembert’s proof rested on the clear and simple case of two equal masses approaching each other with equal but opposite speeds. They will clearly balance one another, he declared, for there is no reason why one should overcome the other. Other impact situations were reduced to this one; in cases where the masses or velocities were unequal, the object with the greater quantity of motion (defined as mv) would prevail. In fact, d’Alembert’s mathematical definition of mass was introduced implicitly here; he actually assumed the conservation of momentum and defined mass accordingly. This fact was what made his work a mathematical physics rather than simply mathematics.

The principle that bears d’Alembert’s name was introduced in the next part of the Traité. It was not so much a principle as it was a rule for using the previously stated laws of motion. It can be summarized as follows: In any situation where an object is constrained from following its normal inertial motion, the resulting motion can be analyzed into two components. One of these is the motion the object actually takes, and the other is the motion “destroyed” by the constraints. The lost motion is balanced against either a fictional force or a motion lost by the constraining object. The latter case is the case of impact, and the result is the conservation of momentum (in some cases, the conservation of vis viva as well). In the former case, an infinite force must be assumed. Such, for example, would be the case of an object on an inclined plane. The normal motion would be vertically downward; this motion car. be resolved into two others. One would be a component down the plane (the motion actually taken) and the other would be normal to the surface of the plane (the motion destroyed by the infinite resisting force of the plane). Then one can easily describe the situation (in this case, a trivial problem).

It is clear that the use of d’Alembert’s principle requires some knowledge beyond that of his laws. One must have the conditions of constraint. or the law of falling bodies, or some information derived either empirically or hypothetically about the particular situation. It was for this reason that Ernst Mach could refer to d’Alembert’s principle as a routine form for the solution of problems, and not a principle at all. D’Alembert’s principle actually rests on his assumptions of what constitutes equilibrium, and it is in his third law of motion that those assumptions appear. Indeed, in discussing his third law (in the second edition of his book, published in 1758) d’Alembert arrived at the equation φ = dv/dt, which is similar to the standard expression for Newton’s second law, but which lacks the crucial parameter of mass. The function φ was to contain the parameters for specific problems. For example (and this is d’Alembert’s example), should the assumption be made that a given deceleration is proportional to the square of the velocity of an object, then the equation becomes –gv²=dv/dt. The minus sign indicates deceleration, and the constant g packs in the other factors involved, such as mass. In this fashion d’Alembert was able to avoid dealing with forces.

It has often been said that d’Alembert settled the vis viva controversy in this treatise, but such a view must be qualified. In the preface d’Alembert did discuss the issue, pointing out that in a given deceleration the change in velocity was proportional to the time. One could therefore define force in terms of the velocity of an object. On the other hand, if one were concerned with the number of “obstacles” that had to be overcome to stop a moving body (here he probably had in mind ’sGravesande’s experiments with objects stopped by springs), then it was clear that such a definition of force depended on the square of the velocity and that the related metric was distance, not time. D’Alembert pointed out that these were two different ways of looking at the same problem, that both methods worked and were used with success by different scientists. To use the word “force” to describe either mv or mv² was therefore a quarrel of words; the metaphysical notion of force as a universal causal agent was not clarified by such an argument. In this way d’Alembert solved the controversy by declaring it a false one. It involved convention, not reality, for universal causes (the metaphysical meaning of the idea of force) were not known, and possibly not even knowable. It was for this reason that d’Alembert refused to entertain the possibility of talking of forces in mechanics. He did not throw the word away, but used it only when he could give it what today would be called an operational definition. He simply refused to give the notion of force any metaphysical validity and, thus, any ontological reality.

In this way d’Alembert was clearly a precursor of positivistic science. He employed mathematical abstractions and hypothetical or idealized models of physical phenomena and was careful to indicate the shortcomings of his results when they did not closely match the actual events of the world. The metaphysician, he warned in a later treatise, too often built systems that might or might not reflect reality, while the mathematician too often trusted his calculations, thinking they represented the whole truth. But just as metaphysics was suspect because of its unjustified claim to knowledge, so mathematics was suspect in its similar claim. Not everything could be reduced to calculation.

Geometry owes its certainty to the simplicity of the things it deals with; as the phenomena become more complicated, the results become less certain. It is necessary to know when to stop, when one is ignorant of the thing being studied, and one must not believe that the words theorem and corollary have some secret virtue so that by writing QED at the end of a proposition one proves something that is not true [Essai d’une nouvelle thèorie de la rèsistance des fluides. pp. xlii-xliii]. D’Alembert’s instincts were good. Unfortunately, in this case they diverted him from the path that was eventually to produce the principle of the conservation of energy.

A major question that beset all philosophers of the Enlightenment was that of the nature of matter. While d’Alembert’s primary concern was mathematical physics, his epistemology of sensationalism led him to speculate on matter theory. Here again, he was frustrated, repeating time after time that we simply do not know what matter is like in its essence. He tended to accept the corpuscular theory of matter, and in Newton’s style; that is, he conceived of the ideal atom as perfectly hard. Since this kind of atom could not show the characteristic of elasticity, much less of other chemical or physical phenomena, he was sorely perplexed. In his Traité de dynamique, however, he evolved a model of the atom as a hard particle connected to its neighbors by springs. In this way, he could explain elasticity, but he never confused the model with reality. Possibly he sensed that his model actually begged the question, for the springs became more important that the atom itself, and resembled nothing more than a clumsy ether, the carrier of an active principle. Instead of belaboring the point, however, d’Alembert soon returned to mathematical abstraction, where one dealt with functional relations and did not have to agonize over ontology.

In 1744 d’Alembert published a companion volume to his first work, the Traité de l’équilibre et du mouvement des fluides. In this work d’Alembert used his principle to describe fluid motion, treating the major problems of fluid mechanics that were current. The sources of his interest in fluids were many. First, Newton had attempted a treatment of fluid motion in his Principia, primarily to refute Descartes’s tourbillon theory of planetary motion. Second, there was a lively interest in fluids by the experimental physicists in the eighteenth century, for fluids were most frequently invoked to give physical explanations for a variety of phenomena, such as electricity, magnetism, and heat. There was also the problem of the shape of the earth; What shape would it be expected to take if it were thought of as a rotating fluid body? Clairaut published a work in 1744 which treated the earth as such, a treatise that was a landmark in fluid mechanics. Furthermore, the vis viva controversy was often centered on fluid flow, since the quantity of vis viva was used almost exclusively by the Bernoullis in their work on such problems. Finally, of course, there was the inherent interest in fluids themselves. D’Alembert’s first treatise had been devoted to the study of rigid bodies; now he was giving attention to the other class of matter, the fluids. He was actually giving an alternative treatment to one already published by Daniel Bernoulli, and he commented that both he and Bernoulli usually arrived at the same conclusions. He felt that his own method was superior. Bernoulli did not agree.

In 1747 d’Alembert published two more important works, one of which, the Réflexions sur la cause générale des vents, won a prize from the Prussian Academy. In it appeared the first general use of partial differential equations in mathematical physics. Euler later perfected the techniques of using these equations. The pattern was to become a familiar one: d’Alembert, Daniel Bernoulli, or Clairaut would pioneer a technique, and Euler would take it far beyond their capacity to develop it. D’Alembert’s treatise on winds was the only one of his works honored by a prize and, ironically, was later shown to be based on insufficient assumptions. D’Alembert assumed that wind patterns were the result of tidal effects on the atmosphere, and he relegated the influence of heat to a minor role, one that caused only local variations from the general circulation. Still, as a work on atmospheric tides it was successful, and Lagrange continued to praise d’Alembert’s efforts many years later.

D’Alembert’s other important publication of 1747 was an article in the Mémoirs of the Prussian Academy dealing with the motion of vibrating strings, another problem that taxed the minds of the major mathematicians of the day. Here the wave equation made its first appearance in physics. D’Alembert’s mathematical instincts led him to simplify the boundary conditions, however, to the point where his solution, while correct, did not match well the observed phenomenon. Euler subsequently treated the same problem more generally; and although he was no more correct than d’Alembert, his work was more useful.

During the late 1740’s, d’Alembert, Clairaut, and Euler were all working on the famous three-body problem, with varying success. D’Alembert’s interest in celestial mechanics thus led him, in 1749, to publish a masterly work, the Recherches sur la précession des équinoxes et sur la nutation de la terre. The precession of the equinoxes, a problem previously attacked by Clairaut, was very difficult. D’Alembert’s method was similar to Clairaut’s, but he employed more terms in his integration of the equation of motion and arrived at a solution more in accord with the observed motion of the earth. He was rightly proud of his book.

D’Alembert then applied himself to further studies in fluid mechanics, entering a competition announced by the Prussian Academy. He was not awarded the prize; indeed, it was not given to anybody. The Prussian Academy took this action on the ground that nobody had submitted experimental proof of the theoretical work. There has been considerable dispute over this action. The claim has been made that d’Alembert’s work, although the best entered, was marred by many errors. D’Alembert himself viewed his denial as the result of Euler’s influence, and the relations between the two men deteriorated furhter. Whatever the case, the disgruntled d’Alembert published his work in 1752 as the Essai d’une nouvelle théorie de la résistance des fluides. It was in this essay that the differential hydrodynamic equations were first expressed in terms of a field and the hydrodynamic paradox was put forth.

In studying the flow lines of a fluid around an object (in this case, an elliptical object), d’Alembert could find no reason for assuming that the flow pattern was any different behind the object than in front of it. This implied that whatever the forces exerted on the front of the object might be, they would be counteracted by similar forces on the back, and the result would be no resistance to the flow whatever. The paradox was left for his readers to solve. D’Alembert had other difficulties as well. He found himself forced to assume, in order to avoid the necessity of allowing an instantaneous change in the velocity of parts of the fluid moving around the object, that a small portion of the fluid remained stagnant in front of the object, an assumption required to prevent breaking the law of continuity.

In spite of these problems, the essay was an important contribution. Hunter Rouse and Simon Ince have said that d’Alembert was the first “to introduce such concepts as the components of fluid velocity and acceleration, the differential requirements of continuity, and even the complex numbers essential to modern analysis of the same problem.” Clifford Truesdell, on the other hand, thinks that most of the credit for the development of fluid mechanics must be granted to Euler; thus historians have continued the disputes that originated among the scientists themselves. But it is often difficult to tell where the original idea came from and who should receive primary recognition. It is certain, however, that d’Alembert, Clairaut, Bernoulli, and Euler were all active in pursuing these problems, all influenced one another, and all deserve to be remembered, although Euler was no doubt the most able of the group. But they all sought claims to priority, and they guarded their claims with passion.

D’Alembert wrote one other scientific work in the 1750’s, the Recherches sur différens points importants du systême du monde. It appeared in three volumes, two of them published in 1754 and the third in 1756. Devoted primarily to the motion of the moon (Volume III included a new set of lunar tables), it was written at least partially to guard d’Alembert’s claims to originality against those of Clairaut. As was so often the case, d’Alembert’s method was mathematically more sound, but Clairaut’s method was more easily used by astronomers.

The 1750’s were more noteworthy in d’Alembert’s life for the development of interests outside the realm of mathematics and physics. Those interests came as a result of his involvement with the Encyclopédie. Denis Diderot was the principal editor of the enterprise, and d’Alembert was chosen as the science editor. His efforts did not remain limited to purely scientific concerns, however. His first literary task was that of writing the Discours préliminaire of the Encyclopédie, a task that he accomplished with such success that its publication was largely the reason for his acceptance into the Académie Française in 1754.

The Discours préliminaire, written in two parts, has rightly been recognized as a cardinal document of the Enlightenment. The first part is devoted to the work as an encyclopédie, that is, as a collection of the knowledge of mankind. The second part is devoted to the work as a dictionnaire raisonnèe, or critical dictionary. Actually, the first part is an exposition of the epistemology of sensationalism, and owes a great deal to both John Locke and Condillac. All kinds of human knowledge are discussed, from scientific to moral. The sciences are to be based on physical perception, and morality is to be based on the perception of those emotions, feelings, and inclinations that men can sense within themselves. Although d’Alembert gives lip service to the truths of religion, they are clearly irrelevant and are acknowledged only for the sake of the censors. For this reason, the Discours prèliminaire came under frequent attack; nevertheless, it was generally well received and applauded. It formed, so to say, the manifesto of the now coalescing party of philosophes; the body of the Encyclopédie was to be the expression of their program.

The second part of the Discours préliminaire is in fact a history of science and philosophy, and clearly shows the penchant of the philosophes for the notion of progress through the increased use of reason. As a history, it has often quite properly been attacked for its extreme bias against the medieval period and any form of thought developed within the framework of theology, but this bias was, of course, intentional. At the end of this history, the philosophes’ debt to Francis Bacon is clearly acknowledged in the outline of the organization of knowledge. A modified version of Bacon’s tree of knowledge is included and briefly explained. All knowledge is related to three functions of the mind: memory, reason, and imagination. Reason is clearly the most important of the three. Bacon’s emphasis on utility was also reflected in the Encyclopédie, although more by Diderot than by d’Alembert. D’Alembert’s concept of utility was far wider than that of most people. To him, the things used by philosophers–even mathematical equations–were very useful, even though the bulk of the public might find them mysterious and esoteric.

In the midst of this activity, d’Alembert found time to write a book on what must be called a psychophysical subject, that of music. In 1752 he published his Élémens de musique théorique et pratique suivant les principes de M. Rameau. This work has often been neglected by historians, save those of music, for it was not particularly mathematical and acted as a popularization of Rameau’s new scheme of musical structure. Yet it was more than simply a popularization. Music was still emerging from the mixture of Pythagorean numerical mysticism and theological principles that had marked its rationale during the late medieval period. D’Alembert understood Rameau’s innovations as a liberation; music could finally be given a secular rationale, and his work was important in spreading Rameau’s ideas throughout Europe.

As time went on, d’Alembert’s pen was increasingly devoted to nonscientific subjects. His articles in the Encyclopédie reached far beyond mathematics. He wrote and read many essays before the Académie Française; these began to appear in print as early as 1753. In that year he published two volumes of his Mélanges de littérature et de philosophie. The first two were reprinted along with two more in 1759; a fifth and last volume was published in 1767. The word mélanges was apt, for in these volumes were essays on music, law, and religion, his treatise on the Élémens de philosophie, translations of portions of Tacitus, and other assorted literary efforts. They make an odd mixture, for some are important in their exposition of Enlightenment ideals, while others are mere polemics or even trivial essays.

In 1757 d’Alembert visited Voltaire at Ferney, and an important result of the visit was the article on Geneva, which appeared in the seventh volume of the Encycloédie. It was clearly an article meant to be propaganda, for the space devoted to the city was quite out of keeping with the general editorial policy. In essence, d’Alembert damned the city by praising it. The furor that resulted was the immediate cause of the suspension of the license for the Encyclopédie. D’Alembert resigned as an editor, convinced that the enterprise must founder, and left Diderot to finish the task by himself. Diderot thought that d’Alembert had deserted him, and the relations between the men became strained. Rousseau also attacked d’Alembert for his view that Geneva should allow a theater, thus touching off another of the famous controversies that showed that the philosophes were by no means a totally unified group of thinkers.

D’Alembert’s chief scientific output after 1760 was his Opuscules mathématiques, eight volumes of which appeared from 1761 to 1780. These collections of mathematical essays were a mixed bag, ranging from theories of achromatic lenses to purely mathematical manipulations and theorems. Included were many new solutions to problems he had previously attacked–including a new proof of the law of inertia. Although the mathematical articles in the Encyclopédie had aired many of his notions, these volumes provide the closest thing to a collection of them that exists.

As Carl Boyer has pointed out, d’Alembert was almost alone in his day in regarding the differential as the limit of a function, the key concept around which the calculus was eventually rationalized. Unfortunately, d’Alembert could never escape the tradition that had made geometry preeminent among the sciences, and he was therefore unable to put the idea of the limit into purely algorithmic form. His concept of the limit did not seem to be any more clear to his contemporaries than other schemes invented to explain the nature of the differential.

It has often been said that d’Alembert was always primarily a mathematician and secondarily a physicist. This evaluation must be qualified. No doubt he sensed the power of mathematics. But, as he once said, “Mathematics owes its certainty to the simplicity of the things with which it deals.” In other words, d’Alembert was never able to remove himself to a world of pure mathematics. He was rather in the tradition of Descartes. Space was the realization of geometry (although, unlike Descartes. d’Alembert drew his evidence fró;m sense perception). It was for this reason that he could never reduce mathematics to pure algorithms, and it is also the reason for his concern about the law of continuity. In mathematics as well as physics, discontinuities seemed improper to d’Alembert; equations that had discontinuities in them gave solutions that he called “impossible,” and he wasted no time on them. It was for this reason that the notion of perfectly hard matter was so difficult for him to comprehend, for two such particles colliding would necessarily undergo sudden changes in velocity, something he could not allow as possible.

It was probably the requirement of continuity that led d’Alembert to his idea of the limit, and it also led him to consider the techniques of handling series. In Volume V of the Opuscules he published a test for convergence that is still called d’Alembert’s theorem. The mathematical statement is:

If and r < 1, the series converges. If r > 1, the series diverges; if r = 1, the test fails.

But in spite of such original contributions to mathematical manipulation, d’Alembert’s chief concern was in making this language not merely descriptive of the world, but congruent to it. The application of mathematics was a matter of considering physical situations, developing differential equations to express them, and then integrating those equations. Mathematical physicists had to invent much of their procedure as they went along. Thus, in the course of his work, d’Alembert was able to give the first formulation of the wave equation, to express the first partial differential equation, and to be the first to solve a partial differential equation by the technique of the separation of variables. But probably the assignment of “firsts” in this way is not the best manner of evaluating the development of mathematics or of mathematical physics. For every such first, one can find other men who had alternative suggestions or different ways of expressing themselves, and who often wrote down similar but less satisfactory expressions.

More important, possibly, is the way in which these ideas reflect the mathematicians’ view of nature, a view that was changing and was then very different from that of a mathematical physicist today. D’Alembert’s very language gives a clue. He used, for example, the word fausse to describe a divergent series. The word to him was not a bare descriptive term. There was no match, or no useful match, for divergence in the physical world. Convergence leads to the notion of the limit; divergence leads nowhere–or everywhere.

D’Alembert has often been cited as being oddly ineffective when he considered probability theory. Here again his view of nature, not his mathematical capabilities, blocked him. He considered, for example, a game of chance in which Pierre and Jacques take part. Pierre is to flip a coin. If heads turns up on the first toss, he is to pay Jacques one écu. If it does not turn up until the second toss, he is to pay two écus, If it does not turn up until the third toss, he is to pay four écus, and so on, the payments mounting in geometric progression. The problem is to determine how many écus Jacques should give to Pierre before the game begins in order that the two men have equal chances at breaking even. The solution seemed to be that since the probability on each toss was one-half, and since the number of tosses was unlimited, then Jacques would have to give an infinite number of écus to Pierre before the game began, clearly a paradoxical situation.

D’Alembert rebelled against this solution, but had no satisfactory alternative. He considered the possibility of tossing tails one hundred times in a row. Metaphysically, he declared, one could imagine that such a thing could happen; but one could not realistically imagine it happening. He went further: heads, he declared, must necessarily arise after a finite number of tosses. In other words, any given toss is influenced by previous tosses, an assumption firmly denied by modern probability theory. D’Alembert also said that if the probability of an event were very small, it could be treated as nothing, and therefore would have no relevance to physical events. Jacques and Pierre could forget the mathematics; it was not applicable to their game.

It is no wonder that such theorizing caused d’Alembert to have quarrels and arguments with others. Moreover, there were reasons for interest in probability outside games of chance. It had been known for some time that if a person were inoculated with a fluid taken from a person having smallpox, the result would usually be a mild case of the disease, followed by immunity afterward. Unfortunately, a person so inoculated occasionally would develop a more serious case and die. The question was posed: Is one more likely to live longer with or without inoculation? There were many variables, of course. For example, should a forty-year-old, who was already past the average life expectancy, be inoculated? What, in fact, was a life expectancy? How many years could one hope to live, from any given age, both with and without inoculation? D’Alembert and Daniel Bernoulli carried on extensive arguments about this problem. What is significant about d’Alembert’s way of thinking is that he expressed the feeling that the laws of probability were faint comfort to the man who had his child inoculated and lost the gamble. To d’Alembert, that factor was as important as any mathematical ratio. It was not, as far as he was concerned, irrelevant to the problem.

Most of these humanitarian concerns crept into d’Alembert’s work in his later years. Aside from the Opuscules, there was only one other scientific publication after 1760 that carried his name: the Nouvelles experiences sur la résistance des fluides (published in 1777). Listed as coauthors were the Abbé Bossut and Condorcet. The last two actually did all of the work; d’Alembert merely lent his name.

In 1764 d’Alembert spent three months at the court of Frederick the Great. Although frequently asked by Frederick, d’Alembert refused to move to Potsdam as president of the Prussian Academy. Indeed, he urged Frederick to appoint Euler, and the rift that had grown between d’Alembert and Euler was at last repaired. Unfortunately, Euler was never trusted by Frederick, and he left soon afterward for St. Petersburg, where he spent the rest of his life.

In 1765 d’Alembert published his Histoire de la destruction des Jésuites. The work was seen through the press by Voltaire in Geneva, and although it was published anonymously, everyone knew who wrote it. A part of Voltaire’s plan écraser l’infâme, this work is not one of d’Alembert’s best.

In the same year, d’Alembert fell gravely ill, and moved to the house of Mlle. de Lespinasse, who nursed him back to health. He continued to live with her until her death in 1776. In 1772 he was elected perpetual secretary of the Académie Françise, and undertook the task of writing the eulogies for the deceased members of the academy. 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 preeminence. D’Alembert sensed his failure. His later life was filled with frustration and despair, particularly after the death of Mlle. de Lespinasse.

Possibly d’Alembert lived too long. Many of the philosophes passed away before he did, and these who remained alive in the 1780’s were old and clearly not the vibrant young revolutionaries they had once been. What political success they had tasted they had not been able to develop. But, to a large degree, they had, in Diderot’s phrase, “changed the general way of thinking.”

BIBLIOGRAPHY

I. Original Works. There have been no collections made of d’Alembert’s scientific works, although reprints of the original editions of his scientific books (except the Opuscules mathématiques) have recently been issued by Éditions Culture et Civilisation, Brussels. There are two collection of d’Alembert’s Oeuvres which contain his literary pieces: the Bélin ed., 18 vols. (Paris, 1805); and the Bastien ed., 5 vols. (Paris, 1821). The most recent and complete bibliographies are in Grimsley and Hankins (see below).

II. Secondary Literature. The following works are devoted primarily to d’Alembert or accord him a prominent role: Joseph Bertrand, D’Alembert (Paris, 1889); Carl Boyer, The History of the Calculus and Its Conceptual Development (New York, 1949), ch. 4; René Dugas, A History of Mechanics (Neuchâtel, 1955), pp. 244–251, 290–299; Ronald Grimsley, Jean d’Alembert (Oxford, 1963); Maurice Müller, Essai sur la philosophie de Jean d’Alembert (Paris, 1926); Hunter Rouse and Simon Ince, A History of Hydraulics (New York, 1963), pp. 100–107; Clifford Truesdell, Continuum Mechanics, 4 vols. (New York, 1963–1964); and Arthur Wilson, Diderot: The Testing Years (New York, 1957). Of the above, Boyer, Dugas, Rouse and Ince, and particularly Truesdell, deal specifically and in detail with d’Alembert’s science.

Three recent doctoral dissertations on d’Alembert are J. Morton Briggs, D’Alembert: Mechanics, Matter, and Morals (New York, 1962): Thomas Hankins, Jean d Alembert, Scientist and Philosopher (Cornell University, 1964); and Harold Jarrett, D’Alembert and the Encyclopédie (Durham, N.C., 1962).

J. Morton Briggs

D'ALEMBERT (DALEMBERT), JEAN LE ROND

(b. Paris, France, 16 November 1717;

d. Paris, France, 29 October 1783), mathematics, mechanics, astronomy, physics, philosophy. For the original article on d’Alembert see DSB, vol. 1.

Since the first edition of DSB, d’Alembert has been the object of several studies, giving a better knowledge of his scientific works, especially in mathematics. Publication of the first two volumes of his œuvres complètes, concerning celestial mechanics, has put into evidence some of his unpublished early texts about lunar theory and it has given rise to critical studies. His correspondence with Euler, published in Euler’s Opera Omnia, has been also an important source of information.

As a scientist, d’Alembert made decisive contributions in mathematics and in various fields of mechanics. His leading scientific publications consist of six treatises (published from 1743 to 1756, two of them being reedited later); twenty-two memoirs in the collections of the Berlin Academy, the Paris Academy of Sciences, and the Turin Academy (from 1748 to 1774); and the fifty-eight memoirs gathered in the eight volumes of his Opuscules mathématiques (from 1761 to 1780). Other scientific writings appeared in the form of letters to Joseph-Louis Lagrange in the Memoirs of the Turin Academy and in those of the Berlin Academy (between 1766 and 1782). And d’Alembert also took the opportunity in some of the numerous entries he contributed to the Encyclopédie to clarify or develop scientific concepts. In addition, he left several unpublished works: early memoirs of mathematics and celestial mechanics, plis cachetés deposited at the Paris Academy, and a ninth volume of his Opuscules.

Having entered the Paris Academy of Sciences in 1741, d’Alembert only reached the highest rank of pensionnaire on 10 November 1765, but he enjoyed some of its prerogatives from 1756, when he became pensionnaire surnuméraire. He held the positions of sous-directeur and directeur in 1768 and 1769 respectively. As an academician, he was in charge of reporting on a large number of works submitted to the Academy, and he sat on many prize juries. In particular, one may believe that he had a decisive voice concerning the choice of works about lunar motion, libration, and comets for the astronomy prizes awarded to Leonhard Euler, Lagrange, and Nikolai Fuss between 1764 and 1780.

D’Alembert exchanged scientific correspondences with Euler (mainly between 1746 and 1751), Gabriel Cramer (between 1748 and 1751), and Lagrange (from 1759). He helped the latter to enter the Berlin Academy as a director, he protected Pierre Simon Laplace’s early career, and he was a friend of Marie-Jean-Antoine-Nicolas de Caritat, Marquis de Condorcet.

Mathematics . For imaginary (that is, complex) numbers, d’Alembert demonstrated the stability of the symbol (p and q real) with respect to elementary algebraic or analytical operations, in an unpublished memoir of 1745 and in his Réflexions sur la cause générale des vents(1747). Before the end of 1746, he demonstrated that any polynomial with degree n≥351 and real coefficients has at least one root either real or of the form (q≠ 0), and that nonreal roots can be associated in pairs (namely and ). Later (1772), he extended the former property to polynomials with complex coefficients. These results induce that any polynomial of the nth degree with complex coefficients has n complex roots separate or not, and also that any polynomial with real coefficients can be put in the form of a product of binomials of the first degree and trinomials of the second degree with real coefficients. Though d’Alembert’s analytical demonstration presents a few gaps, it stands as the first attempt to establish the fundamental theorem of algebra on rigorous bases, other demonstrations being given later by Carl Friedrich Gauss.

The study concerning polynomials with real coefficients was involved in the first of three memoirs devoted to integral calculus (published in 1748, 1750, 1752), in connection with the reduction of integrals of rational fractions to the quadrature of circle or hyperbola. In the same memoirs, d’Alembert began to deal with integrals that can reduce to the rectification of ellipse or hyperbola, in the continuation of Colin Maclaurin’s works, but using merely algebraic methods. Furthermore he considered another class of integrals, which included, where P is a polynomial of the third degree, an early approach to elliptic integrals whose theory was later started by Adrien-Marie Legendre. Works on the same topic can also be found in the Opuscules mathématiques and in other late memoirs.

The 1750 and 1752 memoirs additionally dealt with differential equations, a topic that appeared also in several of d’Alembert’s texts on mechanics and astronomy. In particular, he gave an original method, using multipliers, for solving systems of linear differential equations of the first order with constant coefficients, and he introduced the reduction of linear differential equations of any order to systems of equations of the first order.

D’Alembert was the first to solve partial differential equations, in hiséflexions sur la cause générale des vents. He considered a system of two differential expressions supposed to be exact differential forms in two independent variables, which should be equivalent to two independent linear partial differential equations of the second order with constant coefficients. He used the condition for exact differential forms and introduced multipliers leading to convenient changes of independent variables and unknown functions. His solution involved two arbitrary functions, to be determined by taking into account the boundary conditions of the physical problem. He applied this method to vibrating strings in two memoirs published in 1749, and to fluids in a 1749 manuscript at the origin of his d’une nouvelle théorie de la résistance des fluides (1752). That gave rise to a discussion with Euler about the nature of curves expressing boundary conditions. Another method, based on variable separation, appeared in a third memoir of d’Alembert about vibrating strings (1752), and a more general study of partial differential

equations, including equations of the first order— also studied by Euler—was undertaken in the twenty-sixth memoir of the Opuscules mathématiques (1768). These works were continued by Lagrange and Laplace.

Motion of a Solid Body around Its Mass Center and Astronomical Applications . In his first treatise, the Traité de dynamique published in 1743, d’Alembert introduced the principle that came to be called “d’Alembert’s principle,” which he next applied in several fields. One of them is the motion of a solid body around its center of mass.

D’Alembert started studying this problem in his fourth treatise, Recherches sur la précession des équinoxes, et sur la nutation de l’axe de la Terre, dans le système newtonien(1749). His purpose was to show that Newtonian attraction could explain a small motion of the Earth’s axis connected to the lunar node motion, which had just been discovered by James Bradley. First he separated the motion of the Earth attracted by the Sun and the Moon into two independent motions: the motion of the Earth mass center (relevant from the three-body problem) and the rotation of the Earth around its mass center, considered as a fixed point. Then applying his principle to the Earth, supposed to be a solid body of revolution about its polar axis (called axis of figure), he established two differential equations of the second order giving the motion of the figure axis in space and a third one expressing angular displacement around the figure axis. He also proved the existence of an instantaneous axis of rotation moving both in space and in the Earth, but close to the figure axis.

Using approximations in solving his differential equations, d’Alembert was the first to obtain analytical expressions for the two angles positioning the Earth’s polar axis with respect to the ecliptic plane. They accounted for the observed motions of the axis: precession known from antiquity and Bradley’s nutation.

Several common features exist between d’Alembert’s nutation theory and Edgar W. Woolard’s (1953), in the differential equations taken into account and in the solving methods. By comparing analytical amplitudes of precession and nutation to observed ones, d’Alembert gave values of two physical constants: the ratio of lunar and terrestrial masses, and a constant equivalent to modern dynamical flattening. Using the latter, he discussed the compatibility between the theory of the Earth’s figure, the observed motion of its polar axis, and the values of its geodetic flattening recently obtained by Pierre-Louis Moreau de Maupertuis and Pierre Bouguer.

In the third book of his Recherches sur différens points importans du systême du monde (1754), and in several later memoirs, d’Alembert attempted to apply his theory of precession-nutation to the lunar libration. But, though in a memoir published in 1759 he extended his differential equations to an ellipsoid with three unequal axes, he failed to account for the empirical laws found by Jean-Dominique Cassini. Finally, the complete theoretical explanation of Cassini’s laws was given by Lagrange in the 1780s, using differential equations similar to d’Alembert’s in the 1759 memoir, but a different method for solving them.

In the second memoir of the Opuscules mathématiques(1761), d’Alembert gave the way to obtain six differential equations of the second order representing the complete motion of any solid body in space. The position of the solid was defined by six functions of time: the coordinates of a particular point (for example, the mass center) in a system of rectangular axes fixed in space, and three angles (analogous to Euler’s angles) positioning the directions of body-fixed axes with respect to the space-fixed system. In the twenty-second memoir (1768), he simplified his equations by using what is called principal axes of inertia as body-fixed axes.

The problem of the motion of a solid around its mass center gave rise to a long polemic between d’Alembert and Euler. In 1751, Euler published his work about precession-nutation without referring to d’Alembert’s 1749 treatise, but he agreed to write in the next volume of the same Memoirs of the Berlin Academy that his French colleague had been the first to solve that particular problem. Nevertheless, he never recognized d’Alembert’s priority in solving the general problem of the motion of a solid around its mass center, as the latter required. In fact, Euler’s general solution, also published in the 1760s, is very different from d’Alembert’s and, as Wenceslaus Johann Gustav Karsten wrote in the preface of Euler’s Theoria Motus Corporum Solidorum seu Rigidorum, both scientists contributed equally to solve the problem.

Three-Body Problem . The three-body problem was the subject of an intense competition between Euler, Alexis-Claude Clairaut, and d’Alembert in the late 1740s and early 1750s. After preliminary attempts withdrawn from publication, d’Alembert published two memoirs about this topic in 1749, the second one (read at the Paris Academy of Sciences in February–March 1748) beginning the application of the first one (read in June 1747) to lunar theory. He achieved this lunar theory in August 1748, but like Clairaut’s and Euler’s at the same time, his theoretical calculations yielded only the half value of the observed mean motion of the lunar apsides. He did not take part in the controversy raised by Clairaut about the Newtonian formulation of universal gravitation, but he tried to account for the discrepancy between theory and observation by a force acting complementarily in the vicinity of the Earth. The unpublished manuscript of that 1748 lunar theory was deposited at the Paris Academy in May 1749, after Clairaut had stated his successful calculation of the apsidal mean motion.

Nevertheless, d’Alembert’s work was the first attempt at constructing a literal theory of the Moon (that is a theory involving explicitly four angles, functions of time, and constant mean orbital elements of the Moon and the Earth, kept under a literal form) and it presented a theoretical interest in the calculation of periodic inequalities. He resumed it from the end of 1749 on and then achieved an expression of the apsidal mean motion compatible with the observed value. His new theory was finished in January 1751, but he did not submit it to the St. Petersburg Academy of Sciences for the 1751 prize, because of the presence of Euler on the jury. The prize was awarded to Clairaut, and d’Alembert published his theory later in the first book of the Recherches sur différens points importans du systême du monde (1754).

In d’Alembert’s lunar theory, the geocentric motion of the Moon disturbed by the Sun is expressed by four differential equations: the first two expressing the motion of the body in projection on the ecliptic plane, the last two expressing the motion of the nodal line and the variations of the inclination of the instantaneous orbital plane on the ecliptic plane. Independent variable z is analogous to ecliptic longitude. The first equation is formulated as where unknown function t is simply connected to radius vector of the projection; N is a constant, 1 – N being proportional to the apsidal mean motion; and M depends on the position of the body through the disturbing forces. The whole system has to be solved by an iterative process; at each step, M is considered as a known function of z, and constant N is determined so that the differential equation in t could not have any solution increasing indefinitely with z. This determination of the apsidal mean motion at the differential equation level is one of the leading characteristics of d’Alembert’s method compared to Clairaut's. In the 1748 theory, only the first step of the iterative process was performed, whereas further steps are necessary to obtain a good value of N.

D’Alembert derived two sets of lunar tables from his theory: the first one was published in 1754 with the theory; the second one was published separately in 1756. A third set of tables, based on an empirical process, was published in the Opuscules mathématiques(1761).

The method introduced in the first memoir of 1749 was applied to planetary motions in the second and fifth books of Recherches sur différens points importans du systême du monde (1754, 1756), and to comets in the Opuscules mathématiques. These latter also contain interesting developments about lunar theory, some of them connected to the problem of the secular acceleration of the Moon.

SUPPLEMENTARY BIBLIOGRAPHY

A bibliography of d’Alembert has been published by Anne-Marie Chouillet in Analyse et dynamique: Études sur l’œuvre de d’Alembert, edited by Alain Michel and Michel Paty (see below). A critical complete edition of d’Alembert’s works is in progress in France (first volume published in 2002); information is available from http://dalembert.univ-lyon1.fr.

WORKS BY D’ALEMBERT

Traité de dynamique. Paris: David l’aîné, 1743.

Traité de l’équilibre et du mouvement des fluides. Paris: David l’aîné, 1744.

Réflexions sur la cause générale des vents. Paris: David l’aîné, 1747.

“Memoirs addressed to the Berlin Academy.” In Histoire de l’Académie royale des Sciences et Belles lettres. Berlin: Ambroise Haude, 1747–1771. For memoirs discussed in this article, see the volumes for the years 1746, 1747, 1748, 1750, 1763, 1765, and 1769.

“Memoirs addressed to the Paris Academy of Sciences (France).” Histoire de l’Académie Royale des Sciences. Paris: Jean Boudot, 1702–1797. For memoirs discussed in this article, see the volumes for the years 1745, 1754, 1757, 1764, 1765, 1767, 1768, and 1771.

Recherches sur la précession des équinoxes, et sur la nutation de l’axe de la Terre, dans le systême newtonien. Paris: David l’aîné, 1749.

Essai d’une nouvelle théorie de la résistance des fluides. Paris: David l’aîné, 1752.

Recherches sur différens points importans du systême du monde. 3 vols. Paris: David l’aîné, 1754–1756.

Opuscules mathématiques; ou, Mémoires sur différens sujets de geómétrie, de méchanique, d’optique, d’astronomie, &c. 8 vols. Paris: David, 1761–1780.

Œuvres complètes. Series 1, vol. 6. Premiers textes de mécanique céleste (1747–1749). Edited by Michelle Chapront-Touzé. Paris: CNRS Éditions, 2002. Contains his 1748 lunar theory and other early unpublished texts about the three-body problem .

———. Series 1, vol. 7. Précession et nutation (1749–1752). Edited by Michelle Chapront-Touzé and Jean Souchay. Paris: CNRS Éditions, 2006.

OTHER SOURCES

Auroux, Sylvain, and Anne-Marie Chouillet, eds. “D’Alembert (1717–1783).” Dix-huitième Siècle 16 (1984): 7–203. Special issue, with contributions from seventeen authors.

Chapront-Touzé, Michelle. “D’Alembert, Jean Le Rond.” In Biographical Encyclopedia of Astronomers, edited by Thomas A. Hockey. New York and London: Springer, 2007. Contains complementary information about d’Alembert’s works in astronomy and celestial mechanics.

De Gandt, François, Alain Firode, and Jeanne Peiffer, eds. “La formation de d’Alembert (1730–1738).” Recherches sur Diderot et sur l’Encyclopédie 38 (2005): 7–224. A special issue, with contributions from eleven authors.

Demidov, Serghei S. “Création et développement de la théorie des équations différentielles aux dérivées partielles dans les travaux de J. d’Alembert.” Revue d’histoire des sciences 35 (1982): 3–42.

Emery, Monique, and Pierre Monzani, eds. Jean d’Alembert, savant et philosophe: Portrait à plusieurs voix; actes du colloque. Paris: Editions des Archives Contemporaines, 1989.

Firode, Alain. La dynamique de d’Alembert. Montreal: Bellarmin; Paris: Vrin, 2001.

Fraser, Craig G. Calculus and Analytical Mechanics in the Age of Enlightenment. Aldershot, U.K.: Variorum, 1997.

Gilain, Christian. “Sur l’histoire du théorème fondamental de l’algèbre.” Archive for History of Exact Sciences 42 (1991): 91–136.

———. “D’Alembert et l’intégration des expressions différentielles à une variable.” In Analyse et dynamique: Études sur œuvre de d’Alembert, edited by Alain Michel and Michel Paty. Laval, Quebec: Les Presses de l’Université Laval, 2002.

———. “Équations différentielles et systèmes différentiels: De d’Alembert à Cauchy.” Oberwolfach Reports 1 (2004): 2741–2743.

Grimberg, Gérard. “D’Alembert et les équations aux dérivées partielles en hydrodynamique.” PhD diss., Université Denis Diderot, Paris, 1998.

———. “D’Alembert et les équations différentielles aux dérivées partielles en hydrodynamique.” In Analyse et dynamique:Études sur l’œuvre de d’Alembert, edited by Alain Michel and Michel Paty. Laval, Quebec: Les Presses de l’Université Laval, 2002.

Hankins, Thomas L. Jean d’Alembert: Science and the Enlightenment. Oxford: Clarendon Press, 1970.

Maheu, Gilles. “La vie et l’ œuvre de Jean d’Alembert: Étude bio-bibliographique.” PhD diss., École des Hautes Études en Sciences Sociales, Paris 1967.

Michel, Alain, and Michel Paty, eds. Analyse et dynamique: Études sur l’œuvre de d’Alembert. Laval, Quebec: Les Presses de l’Université Laval, 2002. With contributions from eleven authors.

Paty, Michel. D’Alembert; ou, La raison physico-mathématique au siècle des Lumières. Paris: Les Belles Lettres, 1998.

Viard, Jérôme. “Le principe de d’Alembert et la conservation du ‘moment cinétique’ d’un système de corps isolés dans le Traité de dynamique.” Physis 39 (2002): 1–40.

Wilson, Curtis. “D’Alembert versus Euler on the Precession of the Equinoxes and the Mechanics of Rigid Bodies.” Archive for History of Exact Sciences 37 (1987): 233–273.

Michelle Chapront-Touzé

ALEMBERT, JEAN LE ROND D'
(1717–1783)

The French mathematician and encyclopedist Jean Le Rond d'Alembert was the illegitimate son of Madame de Tencin and the artillery general Destouches-Canon. He was abandoned by his mother on the steps of the baptistry of Saint-Jean-Le-Rond in Paris, from which he received his name. Shortly afterward his father returned from the provinces, claimed the child, and placed him with Madame Rousseau, a glazier's wife, with whom d'Alembert remained until a severe illness in 1765 forced him to seek new quarters. Through the Destouches family, Jean Le Rond was placed in the exclusive Jansenist Collège de Mazarin and given the name of d'Aremberg, which he later changed to d'Alembert, no doubt for phonetic reasons. At the college an effort was made to win him over to the Jansenist cause, and he went so far as to write a commentary on St. Paul. The intense Jesuit-Jansenist controversy served only to disgust him with both sides, however, and he left the college with the degree of bachelor of arts and a profound distrust of, and aversion to, metaphysical disputes.

After attending law school for two years he changed to the study of medicine, which he soon abandoned for mathematics. His talent and fascination for mathematics were such that at an early age he had independently discovered many mathematical principles, only to find later that they were already known. In 1739 he submitted a mémoire on integral calculus to the Académie des Sciences, but it was his Traité de dynamique in 1743 that won him acclaim and paved the way for his entry into the academy that same year. The introduction to his treatise is significant as the first enunciation of d'Alembert's philosophy of science. He accepted the reality of truths rationally deduced from instinctive principles insofar as they are verifiable experimentally and therefore are not simply aprioristic deductions. Although admitting unproved axioms at the base of his principles of mechanics, thus revealing his debt to René Descartes, d'Alembert rejected metaphysical affirmations and the search for universals and expressed admiration for Bacon's experimental and inductive method.

The decade of the 1740s may be considered d'Alembert's mathematical period during which he made his most outstanding and fruitful contributions to that discipline. In addition to the Traité de dynamique he wrote Mémoire sur la réfraction des corps solides (1741); Théorie de l'équilibre du mouvement des fluides (1744 and 1751); Réflexions sur la cause générale des vents, which won him the prize of the Berlin Academy in 1746 as well as membership in that body; a mémoire on vibrating strings (Recherches sur les cordes vibrantes ), written in 1747 for the Berlin Academy; Recherches sur la précession des équinoxes et sur la nutation (1749); Réflexions sur la théorie de la résistance des fluides (1752); Recherches sur differents points importants du systéme du monde (1754–1756), plus eight volumes of Opuscules mathématiques (1761–1780).

D'Alembert's first philosophical work, the Discours préliminaire to the Encyclopédie, appeared in 1751. As early as 1746 he, with Denis Diderot, had been on the publisher's payroll as translator, in connection with the projected French version of Chambers's Cyclopaedia. We may suppose that, like Diderot, he had already worked for the publishers as a translator of English works for French consumption, thus exposing himself to the writings of the English empiricists and supplementing the meager pension left him by his father. In any event, d'Alembert had read Bacon as early as 1741; and his Discours préliminaire revealed not only his debt to the Descartes of the Regulae, shorn of metaphysics, but his admiration for, and indebtedness to, Bacon for his experimental method; Isaac Newton, whom he admired for proving gravitational force without trying to explain its first cause; and John Locke, whose metaphysical method he adopted. While paying lip service to the traditional religious concepts of his time, d'Alembert used Lockian sensationalist theory to arrive at a naturalistic interpretation of nature. It is not through vague and arbitrary hypotheses that nature can be known, he asserted, but through a careful study of physical phenomena. He discounted metaphysical truths as inaccessible through reason. In the Discours, d'Alembert began by affirming his faith in the reliability of the evidence for an external world derived from the senses and dismissed the Berkeleian objections as metaphysical subtleties that are contrary to good sense. Asserting that all knowledge is derived from the senses, he traced the development of knowledge from the sense impressions of primitive man to their elaboration into more complex forms of expression. Language, music, and the arts communicate emotions and concepts derived from the senses and, as such, are imitations of nature. For example, d'Alembert believed that music that is not descriptive is simply noise. Since all knowledge can be reduced to its origin in sensations, and since these are approximately the same in all men, it follows that even the most limited mind can be taught any art or science. This was the basis for d'Alembert's great faith in the power of education to spread the principles of the Enlightenment.

In his desire to examine all domains of the human intellect, d'Alembert was representative of the encyclopedic eighteenth-century mind. He believed not only that humanity's physical needs are the basis of scientific and aesthetic pursuits, but also that morality too is pragmatically evolved from social necessity. This would seem to anticipate the thought of Auguste Comte, who also placed morality on a sociological basis, but it would be a mistake to regard d'Alembert as a Positivist in the manner of Comte. If d'Alembert was a Positivist, he was so through temporary necessity, based on his conviction that since ultimate principles cannot be readily attained, one must reluctantly be limited to fragmentary truths attained through observation and experimentation. He was a rationalist, however, in that he did not doubt that these ultimate principles exist. In the Discours préliminaire he expressed the belief that everything could be reduced to one first principle, the universe being "one great truth" if we could only see it in a broader perspective. Similarly, in the realm of morality and aesthetics, he sought to reduce moral and aesthetic norms to dogmatic absolutes, and this would seem to be in conflict with the pragmatic approach of pure sensationalist theories. He was forced, in such cases, to appeal to a sort of intuition or good sense that was more Cartesian than Lockian, but he did not attempt to reconcile his inconsistencies and rather sought to remain within the basic premises of sensationalism. D'Alembert's tendency to go beyond the tenets of his own theories, as he did, for example, in admitting that mathematical realities are a creation of the human intellect and do not correspond to physical reality, has led Ernst Cassirer to conclude that d'Alembert, despite his commitment to sensationalist theory, had an insight into its limitations.

During the early 1750s d'Alembert engaged actively in the polemics of the time, particularly in the defense of the Encyclopédie and the party it represented. Many of the articles he wrote for that publication, as well as his preface to Volume III (1753), were aimed at the enemies of the Encyclopédie, notably the Jesuits, who were among the first to attack it for its antireligious and republican orientation. In addition, he took part in the controversy over French versus Italian music, which was inflamed by Jean-Jacques Rousseau's attack on French music in "Lettre sur la musique française" (1753). D'Alembert had already published his Éléments de musique (1752), based on Jean-Philippe Rameau's theories on harmonics, and in 1754 he published anonymously his Réflexions sur la musique en général et la musique française en particulier.

D'Alembert's chief preoccupation at this period, however, was with philosophy and literature. His Mélanges de littérature et de philosophie appeared in 1753 in two volumes (expanded to four volumes in 1759, with a fifth volume added in 1767), and it is here that his skepticism concerning metaphysical problems is delineated. Proceeding on the premise that certainty in this field cannot be reached through reason alone, he considered the arguments for and against the existence of God and cautiously concluded in the affirmative, on the grounds that intelligence cannot be the product of brute matter. Like Newton, d'Alembert viewed the universe as a clock, which necessarily implies a clockmaker, but his final attitude is that expressed by Montaigne's "Que sais-je? " Humankind's uncertainty before this enigmatic universe is the basis of d'Alembert's plea for religious tolerance. He maintained his skeptical deism as an official, public position throughout his life, but there is evidence for believing that in the late 1760s, under the influence of Diderot (whose Rêve de d'Alembert appeared in 1769), d'Alembert was converted to Diderot's materialism. In private correspondence with intimate friends, d'Alembert revealed his commitment to an atheistic interpretation of the universe. He accepted intelligence as simply the result of a complex development of matter and not as evidence for a divine intelligence.

Aside from the publication of a polemical brochure, Histoire de la destruction des Jésuites, in 1765 (with two additional Lettres on the subject in 1767), d'Alembert spent the last two decades of his life in furthering the cause of the philosophes in the Académie Française—by writing his Éloges, which were read in the Académie (and published in 1779), and by fostering the election of candidates of his own choice. Mademoiselle de Lespinasse's salon, where d'Alembert presided, became, in the words of Frédéric Masson, the "obligatory antechamber of the Académie." In this period he became influential with young aspiring men of letters, whom he recruited for his party and whose careers he fostered. The most notable of his disciples was the Marquis de Condorcet. After years of ill health, d'Alembert died of a bladder ailment and was buried as an unbeliever in a common, unmarked grave.

See also Bacon, Roger; Cassirer, Ernst; Comte, Auguste; Condorcet, Marquis de; Descartes, René Diderot, Denis; Locke, John; Music, Philosophy of; Newton, Isaac; Rousseau, Jean-Jacques.

Bibliography

works by d'alembert

Oeuvres philosophiques et littéraires. Edited by J. F. Bastien. 18 vols. Paris, 1805. Not so complete as the Belin edition but contains letters to d'Alembert not included elsewhere. Nouvelle édition augmentée. Edited by A. Belin. 5 vols. Paris, 1821. The most complete edition to date.

Oeuvres et correspondance inédites de d'Alembert. Edited by Charles Henry. Paris, 1887. Contains important supplements to above editions in the fields of philosophy, literature, and music, as well as additional correspondence.

Discours préliminaire de l'Encyclopédie. Edited by P. Picavet. Paris, 1912. Standard critical edition.

For a fuller listing, see A Critical Bibliography of French Literature. Edited by D. C. Cabeen. Syracuse, NY: Syracuse University Press, 1947–1951. Vol. IV, pp. 136–138.

works on d'alembert

Bertrand, Joseph. D'Alembert. Paris: Librarie Hochette, 1889. Despite shortcomings and reliance on Condorcet's Eloge de d'Alembert, the most complete biography to date.

Grimsley, Ronald. Jean d'Alembert. Oxford: Clarendon Press, 1963. A good, comprehensive treatment of d'Alembert's philosophy and ideas. Less concerned with biography.

Kunz, Ludwig. "Die Erkenntnistheorie d'Alemberts," in Archiv für Geschichte der Philosophie, Vol. 20 (1907), 96–126. Considers relation between d'Alembert's metaphysics and English empiricists. Presents him as a link between empiricists and Comte.

Misch, Georg. Zur Entstehung des französischen Positivismus. Berlin, 1900. Influence of d'Alembert's empiricism and materialistic viewpoint on Comte's Positivism.

Muller, Maurice. Essai sur la philosophie de Jean d'Alembert. Paris, 1926. Most important and complete study of d'Alembert's general philosophy.

Pappas, John N. Voltaire and d'Alembert. Bloomington: Indiana University Press, 1962. Considers d'Alembert's position and method in spreading the ideals of the Enlightenment and his influence on Voltaire.

John N. Pappas (1967)

Jean le Rond d'Alembert

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 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

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). □