## Jean-Baptiste d'Alembert

mathematician
philosopher
physicist
scientist

November 16, 1717
(age 65)
Paris, Ile-de-France, France

D’Alembert appeared on the scientific scene in July 1739, when he sent his first communication to the Académie 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. 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 géomètre 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 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.
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 im plicitly 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

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.
In 1744 d’Alembert published a companion volume to his first work, the Traité de equilibre 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. 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 further. 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 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 dijférens points importants du système 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 elforts 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 raisonnée, or critical dictionary
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 Èlémens 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.
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 Encyclopédie. 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.
Most of 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 expériences 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 Française, 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 those 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.”