## Traité de l'equilibre et du mouvement des fluides, pour servir de suite au Traité de dynamique.

Paris: David l’ainé, 1744.

First edition of this continuation of d’Alembert’s classic *Trait**é** de dynamique* published in the previous year. “The ‘Treatise on Dynamics’ was d’Alembert’s first major book and it is a landmark in the history of mechanics. It reduces the laws of the motion of bodies to a law of equilibrium. Its statement that ‘the internal forces of inertia must be equal and opposite to the forces that produce the acceleration’ is still known as ‘d’Alembert’s principle’. This principle is applied to many phenomena and, in particular, to the theory of the motion of fluids” (*PMM* 195). D’Alembert had applied his principle to fluid mechanics in a brief section at the end of the *Traité** de dynamique*, but in the present work he gives a far more detailed treatment.“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 [*Hydrodynamica*, 1738], 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” (DSB).

D’Alembert himself gives an account of the *Trait**é** des fluides* in the article ‘Hydrodynamique’ in the *Encyclopédie*. “My object in this book has been to reduce the laws of fluid equilibrium and motion to the least possible number and to determine by an extremely simple general principle, everything that is concerned with the motion of fluid bodies. I have examined the theories given by M. Bernoulli and M. Maclaurin and I believe that I have revealed the difficulties as well as confusion. I also believe that on certain occasions M. Daniel Bernoulli has used the principle of live forces in cases where he should not have done so. I must add that this great geometer has used this principle without having proved it, or rather that the proof that he has provided is unsatisfactory. However this should not stop from providing this work with the merit that other scientists as well as I should give to this work. I deal as well in this work on the resistance of fluids to bodily motion, of refraction or the motion of a body as it enters into a fluid and finally concerning laws of motion governing fluids which move in vortices” (*The Encyclopedia of Diderot and d’Alembert Collaborative Translation Project*).

The *Traité** des fluides* is based on the principle set forth in the *Trait**é** de dynamique*. In the first part of the earlier work, “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 can 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)” (DSB).

“The *Traité de dynamique* permeates a large part of his work, quite explicitly so in the case of the *Traité des fluides*” (Crépel, p. 165). “At the end of his treatise on dynamics, d’Alembert considered the hydraulic problem of efflux through the vessel … D 'Alembert intended his new solution of the efflux problem to illustrate the power of his principle of dynamics. He clearly relied on the long-known analogy with a connected system of solids. Yet he believed this analogy to be imperfect. Whereas in the case of solids the condition of equilibrium was derived from the principle of virtual velocities, in the case of fluids d’Alembert believed that only experiments could determine the condition of equilibrium. As he explained in his treatise of 1744 on the equilibrium and motion of fluids, the interplay between the various molecules of a fluid was too complex to allow for a derivation based on the only a priori known dynamics, that of individual molecules.

“In this second treatise, d’Alembert provided a similar treatment of efflux, including his earlier derivations of the equation of motion and the conservation of live forces, with a slight variant: he now derived the equilibrium condition by setting the pressure acting on the bottom slice of the fluid to zero. Presumably, he did not want to base the equations of equilibrium and motion on the concept of internal pressure, in conformance with his general avoidance of internal contact forces in his dynamics. His statement of the general conditions of equilibrium of a fluid, as found at the beginning of his treatise, only required the concept of wall pressure …

“In the rest of his treatise, d’Alembert solved problems similar to those of Daniel Bernoulli’s *Hydrodynamica*, with nearly identical results. The only important difference concerned cases involving the sudden impact of two layers of fluids. Whereas Daniel Bernoulli still applied the conservation of live forces in such cases (save for possible dissipation into turbulent motion), d’Alembert’s principle of dynamics there implied a destruction of live force. Daniel Bernoulli disagreed with these and a few other changes. In a contemporary letter to Euler he expressed his exasperation over d’Alembert’s treatise:

‘I have seen with astonishment that apart from a few little things there is nothing to be seen in his hydrodynamics but an impertinent conceit. His criticisms are puerile indeed, and show not only that he is no remarkable man, but also that he never will be.’

“In this judgment, Daniel Bernoulli overlooked that d’Alembert’s hydrodynamics, being based on a general dynamics of connected systems, lent itself to generalizations beyond parallel-slice flow. D’Alembert offered striking illustrations of the power of his approach in a prize-winning memoir published in 1747 on the cause of winds [*Réflexions sur la cause générale des vents*, 1747]” (Darrigol, pp. 14-16).

“D’Alembert’s *Traité** des fluides* was largely a criticism of Bernoulli’s *Hydrodynamica*. Where Bernoulli had used the conservation of *vis viva* to arrive at his results, d’Alembert employed his ‘Principle’, first used successfully in the *Trait**é** de dynamique*. On a few points of detail d’Alembert’s criticisms were valid (as, for instance, his clarification of negative pressures), but he retained Bernoulli’s hypothesis of parallel sections (that is, the assumption that a fluid moving through any conduit may be divided into layers perpendicular to its axis of flow and that these layers always retain a plane surface perpendicular to the axis), and therefore his method failed just where Bernoulli’s had failed. What particularly annoyed Bernoulli was d’Alembert’s complete disregard for all the experimental evidence that he had brought forward in his own work – a criticism of d’Alembert that later became quite characteristic. Bernoulli was more favourably impressed by d’Alembert’s *Trait**é** de dynamique* which he had received after the *Trait**é** des fluids*. He told Euler that this work had given him a ‘rather good opinion’ of its author, although he had not changed his mind about d’Alembert’s hydrodynamics” (Hankins, p. 45).

In Book I of the *Traité des fluides*, d’Alembert discusses various cases of fluid equilibrium such as fluid equilibrium with immersed solids, pressure distribution in the fluid layers with gravity in a constant direction, the equilibrium of fluids of different densities, the law relating gravity and density in different layers of a fluid, the equilibrium of a fluid in which the layers vary in density, the “adherence” of fluids, the equilibrium of a fluid with a curved upper surface (with application to the figure of the earth), and the equilibrium of elastic fluids. Book II is devoted to the movement of fluids contained in vessels. D’Alembert contrasts his own approach with that in Bernoulli’s *Hydrodynamica*, and also with that in Colin Maclaurin’s *Treatise of Fluxions* (1742). Book III deals with the resistance experienced by bodies moving through fluids. It ends with a chapter on vortex flow, including the motion of bodies in rotating fluid masses.

“A natural son of the chevalier Destouches and Mme. De Tencin, D’Alembert was born on 16 or 17 November 1717 and was placed (rather than abandoned) on the steps of the church of Saint-Jean-le-Rond in Paris—whence his given name, although much later he preferred ‘Daremberg’, then ‘Dalembert’ or ‘D’Alembert’. He followed his secondary studies at the *Quatre-Nations *College in Paris, and later studied law and probably a little medicine. His first memoir was submitted to the *Académie des Sciences *in Paris in 1739, and he became a member of that institution in 1741. The *Traité de dynamique *was D’Alembert’s first major work, to be followed by many others in the 1740s and early 1750s. He was co-editor with Dénis Diderot of the *Encyclopédie*, for which he wrote the introduction (1751) and around 1700 articles, mainly scientific, the majority of them before the work was banned following his article ‘Genève’ in 1758–1759. He was appointed to membership of the *Académie Française *in 1754 and quickly became second to Voltaire in the group for *philosophes*. In 1772 he became permanent secretary of this academy (but not that for sciences). He died of gall-stones on 29 October 1783” (Crépel, p. 160).

*Bibliotheca Mechanica*, pp. 7-8; *En français dans le texte*, p. 167; Honeyman 7; Norman 33; Sotheran I, 74. Darrigol, *Worlds of Flow*, 2008. Hankins, *Jean d’Alembert*, 1970. Crépel, ‘Jean le Rond d’Alembert, *Trait**é** de dynamique*,’ Chapter 11 in *Landmark Writings in Western Mathematics 1640-1940 *(Grattan-Guinness, ed.), 2005.

4to (220 x 169 mm), pp. xxxii, [8], 458, [2], with engraved printer’s device on title and 10 folding engraved plates. Contemporary vellum.

Item #5320

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Price:
$2,800.00
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