## Traité de dynamique, dans lequel les loix de l'equilibre & du mouvement des corps sont réduites au plus petit nombre possible, & démonstrées d’une manière nouvelle, & où l’on donne un principe général pour trouver le mouvement de plusieurs corps qui agissent les uns sur les autres, d’une manière quelconque. [Bound after:] Réflexions sur la cause générale des vents.

Paris: David l’ainé, 1743; 1747.

First edition of one of the classics in the history of science, d’Alembert’s formalization of the new science of mechanics. “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. It has become useful in the practical solution of many technical and mechanical problems, and is as important for the motion of bodies as is the principle of virtual velocities for their equilibrium – the latter formulated by Johann Bernoulli in 1717. It was left to Lagrange to combine both these principles and to construct mechanical equations applicable to the motions of any system of bodies” (*Printing and the Mind of Man*, 195). “In the first part of the treatise Alembert developed his own three laws of motion: inertia, the parallelogram of motion, and equilibrium; his third law assumed conservation of momentum and defined mass accordingly. The second part contains the first statement of what is now known as ‘Alembert’s principle’ (Norman). “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 … The *Traité* also contained the first statement of what is 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 (whether, in modern terms, it is momentum or kinetic energy that measures the ‘quantity of motion’), a statement with just enough truth in it to be plausible” (DSB). The second work, in which d’Alembert *laid the foundations of scientific meteorology,* is an application of the methods set out in the *Traité de dynamique**. He rejected the conception of Edmund Halley that the general circulation of the atmosphere is significantly controlled by the distribution of solar heating, and instead explained the origin of winds by means of the gravitational effects of the sun and moon**.* “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 appears the first general use of partial differential equations in mathematical physics. Euler later perfected the techniques of using these equations … D’Alembert’s treatise on winds was the only 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" (DSB). “This was the first attempt to express the motions of the atmosphere in mathematical terms; a new fruitful trend in meteorology” (Persson).

“The [*Traité de dynamique*] was published in 1743 by David, the great bookselling and printing house, … presented to the *Académie des Sciences *on 22 June and given a favourable review by the commissioners P. L. Maupertuis and F. Nicole. It includes a letter to Count de Maurepas, a 26-page preface summarizing and commenting on the main general ideas, and finally the body of the text with all its trimmings (table of contents, plates, corrections, extract from the records of the *Académie des Sciences*, royal favour).

“Contrary to the author’s current custom, the book is clearly structured. Following the definitions and preliminary notions (pp. 1–2), the first Part is entitled ‘General laws of motion and equilibrium of bodies’ (pp. 3–48). It consists of three chapters, each of which is subdivided into articles numbered continuously. These chapters represent the three great principles on which dynamics is based: I. ‘On the force of inertia’ (arts. 2– 20); II. ‘On composite motion’ (arts. 21–26); III. ‘On motion destroyed or changed by obstacles’ (arts. 27–49). This last chapter contains, in particular, the theory of equilibrium.

“The second Part, which is much larger (pp. 49–186), is entitled ‘A general principle for finding the motion of many bodies that act on each other in an arbitrary way, with many applications of this principle’. It consists of four chapters of disparate length and status: I. ‘Exposition of the principle’ (art. 50); II. ‘Properties of the centre of gravity of many bodies combined, deduced from the preceding principle’ (arts. 51–72); III. ‘Problems illustrating the application of the preceding principle’ (arts. 73–153); IV. ‘On the principle of conservation of live forces’ (arts. 154–175). What is today called ‘D’Alembert’s principle’ constitutes the single article of Chapter I. The rest of the second Part consists of what the author calls ‘applications’ …

“‘D’Alembert’s principle’ (Chapter I of Part II) plays a pivotal role in the book: Part I paves the way for it and Chapters II–IV of Part II consist of applications. Moreover, it is this principle that posterity has universally accepted as one of D’Alembert’s main contributions to science” (Crépel, pp. 161-3).

“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. 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).

“D’Alembert deduces from [his principle] that the determination of all motions reduces to applying the principle of equilibrium and that of composite motion. That is why it is often said that D’Alembert’s principle reduces dynamics to statics. The simplest example is that of a body without elasticity obliquely striking a fixed impenetrable wall: the only component of motion preserved after the impact is that parallel to the wall, the component perpendicular to the wall being destroyed (Part I, Chapter III). A typical theorem from Chapter II is as follows: ‘The state of motion or rest of the centre of gravity of many bodies does not change under the mutual action of these bodies provided that the system is entirely free, that is, it is not subject to motion around a fixed point’.

“Chapter III, which is by far the longest and takes up more than half of the book, contains a detailed treatment of 14 problems, divided into Sections as follows: I. ‘Bodies pulled by wires or rods’ (Problems I–VI); II. ‘Bodies moving in the plane’ (Problem VII); III. ‘Bodies acting on one another via wires along which they can run freely’ (Problem VIII); IV. ‘Bodies which move or collide’ (Problems IX–XIV). These problems are treated at unequal length, some being more famous than others. For example, Problem V (arts. 98– 112), on the period of oscillation of a composite pendulum, formed a part of the immediate prehistory of the problem of the vibrating string.

“In Chapter IV, the author emphasizes the fact that, contrary to the Bernoullis, he does not assume the conservation of live forces, but that it can be deduced from his principle and methods. He states that he gives, ‘if not a general proof for all cases, at least a sufficient number of principles for finding the proof in each particular case’. He sketches these proofs for bodies on wires or rods in the case of elastic impacts, and for fluids” (Crépel, pp. 163-4).

“A major question that beset all philosophers of the Enlightenment was that of the nature of matter … [D’Alembert] 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” (DSB).

“The *Traité de dynamique* permeates a large part of his work, quite explicitly so in the case of the *Traité des fluides*, which was published in the following year (1744) as a continuation of the earlier work. It is also true that the majority of his work on the physical sciences make use of D’Alembert’s principle, as do the *Recherches sur la cause des vents *of 1747 of those on the *Précession des equinoxes *two years later, or the *Essai sur la résistance des fluides *of 1752. It may be said that, even if these treatises are bursting with other interesting discoveries, they are also ‘applications’ of the treatise” (Crépel, p. 165).

Daniel Bernoulli criticized d’Alembert’s *Traité des fluids* for containing little that was not in his own *Hydrodynamica *(1738). “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” (Darrigol, pp. 14-16).

*Réflexions sur la cause générale des Vents* is d’Alembert’s prize-winning response to the question posed by the Berlin Academy as to the chief cause of winds. “This memoir of d’Alembert constitutes one of the great moments in the history of mathematics and mathematical physics. It is the true foundation of the theory of partial differential equations and the analytic treatment of the mechanics of deformable bodies. From 1750 onwards, these theories interested the most eminent mathematicians. In the 18^{th} century, d’Alembert was followed by Euler, Lagrange, Laplace, Monge and many others; they made the theory of partial differential equations one of the most important branches of pure mathematics and one of the most powerful tools on mathematical physics. It is impossible to overestimate the importance of this theory. It is because of it that mathematical analysis was applied to a vast number of scientific questions and practical and technical problems, such as the aerodynamics of aeroplanes, and the hydrodynamics of ocean currents. D’Alembert was at the birth of this development” (Engelsman, p. 27 – our translation). “Dumas notes that in this celebrated work, d’Alembert founded the new analysis of partial derivatives, and attacked the equation for vibrating cords, indicating the general solution” (Roberts & Trent, p. 8).

D’Alembert’s essay on winds may have been prompted indirectly by one of the Scottish mathematician Colin Maclaurin (1698-1746), whom d’Alembert admired. “Maclaurin argued, without mathematics, in his 1740 work ‘De Causa physica fluxus et refluxus maris’ (On the cause of tides) that the sea currents were affected by ‘the uneven velocity of a body carried by the earth in its daily motion around its axis’ … MacLaurin was aware that this explanation could be extended to other motions in the atmosphere and the sea. ‘But it is not possible to go into this in any detail’ … MacLaurin’s paper was one of the prize winning contributions to a competition launched by the French Royal Academy of Sciences. This might have inspired the Berlin Academy of Sciences in 1746 to announce a prize for anybody who could determine ‘the nature and the law’ which the wind ought to obey in case the earth was covered by an ocean. The solution had to be presented in a form that allowed predictions” (Persson).

Both d’Alembert and Daniel Bernoulli submitted essays for the prize. “Euler wrote to his friend Goldbach that the Academy had received ten papers of which one was ‘sufficiently above all the others in the importance of its considerations to be worthy of the prize,’ and on 2 June 1746, at the very first session of the new Academy, he announced that the commission had unanimously chosen [d’Alembert’s] paper … After a short deliberation, the Academy unanimously voted to welcome him as a member. Bernoulli was chagrined to lose the prize to the ‘impertinent and puerile’ young mathematician from Paris, particularly since Euler had practically assured him of victory. He immediately protested that the decision was absurd and was still protesting as late as January 1750 … Euler kept his piece, however, since he was fully aware of d’Alembert’s merits” (Hankins, pp. 45-6).

“For simplicity, [d’Alembert] confined his analysis to the case of a constant-density layer of air covering a spherical globe with uniform thickness. He further assumed that fluid particles originally on the same vertical line remained so in the course of time (owing to the thinness of the air layer) and that the vertical acceleration of these particles was a negligible fraction of gravity, and he neglected second-order quantities with respect to the fluid velocity and to the elevation of the free surface. His strategy was to apply his principle of dynamics to the motion induced by the tidal force and the force of gravity, both of which depend on the location on the surface of the Earth … For the first time, [d’Alembert] gave the complete equations of motion of an incompressible fluid in a genuinely bidimensional case. Thus emerged the velocity field and the corresponding partial derivatives with respect to two independent spatial coordinates. D'Alembert pioneered the application to the dynamics of continuous media of the earlier calculus of differential forms by Alexis Fontaine and Leonhard Euler” (Darrigol, pp. 16-19).

D’Alembert’s dissertation on winds was originally submitted in Latin to the Berlin Academy in 1746 (*Meditationes in generalem ventorum causam*). The present work contains this Latin dissertation, together with d’Alembert’s own French translation. In the introduction, d’Alembert states that he has added to the translation “various more or less considerable additions relating to several curious consequences one can draw from my theory” – these are enclosed within brackets to distinguish them from the original dissertation. He also includes a general introduction which gives an extensive discussion of his principle. Another version of d’Alembert’s treatise on winds was published at Berlin in the same year (1747); this contains the original Latin dissertation, but without the French translation. It also includes two other submissions for the same prize (one in French and one in German); these are not included in the Paris version. Contrary to d’Alembert’s assertion in the introduction to the present work, there are in fact minor differences between the Latin dissertations printed in the Paris and Berlin versions.

“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).

I. *En Français dans le texte* 147; Norman 31; Parkinson *Breakthroughs*, p. 159; PMM 195; Roberts and Trent, p. 7; Roller and Goodman I, p. 26. Crépel, ‘Jean le Rond d’Alembert, *Trait**é** de dynamique*,’ Chapter 11 in *Landmark Writings in Western Mathematics 1640-1940 *(Grattan-Guinness, ed.), 2005. II. Norman 34. Darrigol, *Worlds of Flow*, 2008. Engelsman, ‘D’Alembert et les équations aux dérivées partielles,’ *Dix-Huitieme Siècle* 16 (1984), pp. 27-37. Hankins, *Jean d’Alembert*, 1970. Persson, ‘Hadley’s principle: understanding and misunderstanding the trade winds,’ *History of Meteorology* 3 (2006), pp. 17-42.

Two works bound in one vol., 4to (218 x 165 mm). I. pp. [iv], xxvi, [ii], 186, [2], with engraved vignette on title and four folding engraved plates. II. pp. [viii], xxviii, 194, 138, with two folding engraved plates. Text in both French and Latin. Contemporary vellum.

Item #5322

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