Jean le Rond d’Alembert

How to use this course

Jean le Rond d’Alembert is remembered by two audiences that rarely meet. Historians of ideas know him as co-editor of the Encyclopédie and author of its Discours préliminaire. Physicists and mathematicians know his principle of dynamics, his solution of the wave equation, his paradox of fluid resistance and his ratio test. This course treats both, and shows how the same taste for clear first principles runs through his science and his philosophy.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements in modern notation, derivations and exercises with folded solutions. D’Alembert wrote in the language of the 1740s: geometric ratios, infinitesimals and long verbal arguments. The lessons translate his results into today’s notation and say plainly where his own arguments had gaps.

Timeline

A foundling, 1717–1729

D’Alembert was born in Paris on 16 November 1717. Some sources give the 17th, the day he was found. His mother was Claudine-Alexandrine Guérin de Tencin, a former nun released from her vows who later ran one of the most influential salons in Paris. She did not want the child, and the day after his birth he was left on the steps of the small church of Saint-Jean-le-Rond, beside Notre-Dame. The church gave him his first name. He never received any acknowledgement from her.

His father was Louis-Camus Destouches, an artillery officer who was abroad at the time of the birth. On his return Destouches traced the child and placed him with a foster mother, Mme Rousseau, the wife of a glazier. D’Alembert lived in her modest house for nearly fifty years and remained attached to her all his life. Destouches paid for his upbringing and, when he died in 1726, left him an income large enough to live on simply. The Destouches family continued to look after his education.

The circumstances of his birth shaped his public character. He had no family name, no rank and no inheritance beyond a small annuity. Everything he gained came through talent and through the institutions of learning. His later insistence that merit, and not birth, should decide places in the Republic of Letters had a personal source.

College, law and medicine, 1729–1739

At about twelve he entered the Collège des Quatre-Nations, founded by Cardinal Mazarin and strongly Jansenist in outlook. He was registered there as Jean-Baptiste Daremberg; the name d’Alembert appears a little later, and nobody is sure where it came from. The college gave a classical and theological education. Its mathematics course, taught by a professor named Carron, followed the lectures of Pierre Varignon. His teachers hoped he would become a theologian, and he did write a commentary on St Paul’s Epistle to the Romans. He came away instead with a lasting distrust of theology and of Cartesian metaphysics.

He graduated as bachelor of arts in 1735. Like many young men without a career, he studied law and qualified as an advocate in 1738. He then spent a year on medicine. Neither held him. He taught himself mathematics from borrowed books, mostly the works of Isaac Newton and of the French Leibnizian school, and he later said he discovered for himself results he afterwards found in print.

First papers

In July 1739, aged 21, he sent the Académie des Sciences a memoir pointing out errors in Charles Reyneau’s Analyse démontrée, then a standard textbook of the calculus. A second memoir in 1740 drew the attention of Alexis Clairaut, only four years older and already an academician. The work was competent rather than brilliant, but it showed exactly the skill the Academy needed: fluency in the new calculus and the willingness to apply it to physical problems.

The Academy, 1739–1746

After several unsuccessful applications in quick succession, d’Alembert was admitted to the Académie des Sciences in May 1741 as an adjoint astronome, the lowest of its ranks. The next five years were the most creative of his life. In 1743 he published the Traité de dynamique, which reduced the dynamics of constrained systems to statics (Lesson 04). In 1744 he applied the same principle to fluids in the Traité de l’équilibre et du mouvement des fluides, a rival to Daniel Bernoulli’s Hydrodynamica of 1738.

His rise brought conflict. Clairaut, who had welcomed him, became a rival once their interests overlapped in mechanics and celestial mechanics. D’Alembert was quick, sharp-tongued and ready to claim priority, and his writing was often hard to follow. He was nevertheless recognised: in 1746 he became associé géomètre in Paris and was elected to the Berlin Academy, where Pierre-Louis Moreau de Maupertuis was president and Leonhard Euler director of the mathematics class.

The same years opened Paris society to him. Introduced into the salon of Marie-Thérèse Geoffrin in 1746, and later a fixture of the salon of Mme du Deffand, the foundling became a celebrated talker and mimic. It was through these circles that the publisher André Le Breton and the writer Denis Diderot brought him into the project that would make him famous beyond science: the Encyclopédie.

D’Alembert’s principle

For the general reader. Newton’s laws tell you how a free particle moves. Real machines are full of rods, strings, hinges and surfaces that constrain motion, and the forces they exert are unknown in advance. D’Alembert found a way to remove those forces from the problem. He turned every question in dynamics into a question of equilibrium, which mechanics already knew how to answer.

The idea of 1743

In the Traité de dynamique d’Alembert considered the motion each body would take if it were free, and split it into two parts: the motion it actually takes, and a motion that is “destroyed” by its connection with the other bodies. His principle says that the destroyed motions alone would leave the system in equilibrium. He stated it in words and with geometric constructions, and he deliberately avoided talking about force as a cause, which he regarded as obscure. Joseph-Louis Lagrange later combined it with the principle of virtual work and gave it the form used today.

Modern statement

Consider \(N\) particles with masses \(m_i\) and positions \(\mathbf r_i\), acted on by applied forces \(\mathbf F_i\) and constraint forces \(\mathbf R_i\). A virtual displacement \(\{\delta\mathbf r_i\}\) is an infinitesimal change of configuration consistent with the constraints at a fixed instant. The constraints are ideal if \(\sum_i\mathbf R_i\cdot\delta\mathbf r_i=0\) for every virtual displacement: rigid rods, inextensible strings, frictionless surfaces and rolling without slipping all qualify.

Derivation. Newton’s second law gives \(m_i\ddot{\mathbf r}_i=\mathbf F_i+\mathbf R_i\). Dot with \(\delta\mathbf r_i\) and sum. The constraint terms vanish by ideality, leaving the principle. Its power is that the unknown \(\mathbf R_i\) never appear.

From the principle to Lagrange’s equations

For holonomic constraints, write \(\mathbf r_i=\mathbf r_i(q_1,\dots,q_n,t)\) with independent generalized coordinates \(q_k\), so \(\delta\mathbf r_i=\sum_k\frac{\partial\mathbf r_i}{\partial q_k}\delta q_k\). With kinetic energy \(T=\tfrac12\sum_i m_i|\dot{\mathbf r}_i|^2\) and generalized forces \(Q_k=\sum_i\mathbf F_i\cdot\partial\mathbf r_i/\partial q_k\), the identities \(\partial\dot{\mathbf r}_i/\partial\dot q_k=\partial\mathbf r_i/\partial q_k\) and \(\frac{d}{dt}\partial\mathbf r_i/\partial q_k=\partial\dot{\mathbf r}_i/\partial q_k\) turn the principle into

Since the \(\delta q_k\) are independent, each bracket vanishes. If \(Q_k=-\partial V/\partial q_k\), then with \(L=T-V\) these are the Euler–Lagrange equations \(\frac{d}{dt}\frac{\partial L}{\partial\dot q_k}=\frac{\partial L}{\partial q_k}\). This is the route Lagrange took in the Mécanique analytique of 1788.

The vibrating string and the wave equation

For the general reader. Pluck a guitar string and its shape changes in a complicated way. In 1747 d’Alembert wrote down the equation the string obeys and found every solution: the motion is always the sum of two shapes, one sliding to the right and one to the left, each keeping its form. This was the first partial differential equation of physics to be solved in general, and the argument it started shaped analysis for a century.

The equation

A string of linear density \(\rho\) under tension \(T\) lies along the \(x\)-axis; let \(y(x,t)\) be its small transverse displacement. The net vertical force on a short piece \([x,x+\Delta x]\) is \(T\,[y_x(x+\Delta x,t)-y_x(x,t)]\approx T\,y_{xx}\,\Delta x\) for small slopes, and Newton’s law gives

Brook Taylor had found the fundamental frequency of a string in 1713 by assuming a particular shape. D’Alembert’s memoir, Recherches sur la courbe que forme une corde tendue mise en vibration, written for the Berlin Academy in 1747, was the first to state the equation in this partial-derivative form and to solve it without assumptions on the shape. The operator \(\square=\partial_t^2-c^2\nabla^2\) is still called the d’Alembertian.

Proof. Put \(\xi=x-ct\), \(\eta=x+ct\). By the chain rule \(\partial_x=\partial_\xi+\partial_\eta\) and \(\partial_t=c(\partial_\eta-\partial_\xi)\), so \(y_{tt}-c^2y_{xx}=-4c^2\,y_{\xi\eta}\). The equation becomes \(y_{\xi\eta}=0\): \(y_\xi\) does not depend on \(\eta\), so \(y_\xi=f(\xi)\) and \(y=F(\xi)+G(\eta)\). For the initial value problem, \(F+G=\varphi\) and \(c(G'-F')=\psi\). Integrating the second relation, \(G-F=\frac1c\int_0^x\psi+K\); solving the two linear equations for \(F\) and \(G\) gives the formula.

Fixed ends

For a string fixed at \(x=0\) and \(x=L\) the conditions \(y(0,t)=y(L,t)=0\) force \(F(-ct)=-G(ct)\) and a similar relation at \(L\). Equivalently, extend \(\varphi\) and \(\psi\) from \([0,L]\) to odd, \(2L\)-periodic functions on \(\mathbb R\) and use the formula unchanged. The motion is then periodic in time with period \(2L/c\), so the fundamental frequency is

the law of Marin Mersenne, now derived from mechanics.

The controversy

D’Alembert insisted that the initial shape be given by a single analytic expression, so that the equation made sense everywhere. In 1748 Euler replied that any shape one can draw, such as a plucked string with a corner, must be admissible, since the travelling-wave formula still describes it. In 1753 Daniel Bernoulli argued from physics that every motion is a superposition of the modes \(\sin(n\pi x/L)\cos(n\pi ct/L)\). Euler objected that a sum of sines could not represent an arbitrary curve; d’Alembert rejected both views. Lagrange joined the debate in 1759. The questions it raised, what a function is and which functions are sums of trigonometric series, were settled only with the work of Joseph Fourier, Peter Gustav Lejeune Dirichlet and Bernhard Riemann. In modern terms, Euler’s corner shapes are weak solutions, and Bernoulli’s series converge to them.

The fundamental theorem of algebra

For the general reader. Every polynomial equation, however high its degree, has a solution among the complex numbers. D’Alembert gave the first serious attempt at a proof in 1746. His key idea, that a polynomial’s size can always be pushed lower unless it is already zero, is the heart of the shortest proof taught today. In France the result is still called the d’Alembert–Gauss theorem.

Proof. Expand about \(z_0\): \(p(z_0+h)=p(z_0)+b_kh^k+b_{k+1}h^{k+1}+\cdots+b_nh^n\), where \(b_k\ne0\) is the first non-zero coefficient after the constant term (one exists because \(p\) is not constant). Choose \(\omega\) with \(\omega^k=-p(z_0)/b_k\); complex \(k\)-th roots always exist. For real \(t\in(0,1)\),

which is less than \(|p(z_0)|\) once \(t<|p(z_0)|/C\).

Completing the proof

Because the leading term dominates, \(|p(z)|\to\infty\) as \(|z|\to\infty\) (Exercise 6.2). Choose \(R\) so that \(|p(z)|>|p(0)|\) for \(|z|>R\). The continuous function \(|p|\) attains a minimum on the compact disc \(|z|\le R\) at some \(z_0\), and this is a global minimum. By the lemma, \(p(z_0)\) cannot be non-zero. So \(p(z_0)=0\).

What d’Alembert did and did not prove

D’Alembert’s argument appeared in a memoir on integral calculus for the Berlin Academy for 1746. He needed the theorem to integrate rational functions by partial fractions, which requires that every real polynomial factor into real linear and quadratic factors. His version of the lemma relied on expanding a branch of the inverse function in fractional power series, a step he did not justify. More seriously, nothing in the mathematics of his time guaranteed that \(|p|\) attains a minimum. Carl Friedrich Gauss criticised his proof, along with those of Euler and Lagrange, in his 1799 doctoral thesis. The proof above, with the minimum taken on a compact set, is essentially the one Jean-Robert Argand published in 1814; full rigour needed the extreme value theorem of the nineteenth century.

Linear differential equations

For the general reader. Many of d’Alembert’s physical problems, from strings loaded with beads to the oscillations of the atmosphere, led to systems of linear differential equations. He developed two techniques that every student still learns: how to find a second solution from a known one, and how to split a coupled system into independent pieces by taking the right combinations.

Reduction of order

Proof. Substitute \(y=vy_1\): \(y''+Py'+Qy=v(y_1''+Py_1'+Qy_1)+v''y_1+v'(2y_1'+Py_1)\). The first bracket vanishes. The remaining first-order equation for \(w=v'\) is separable: \(w=C\,e^{-\int P}/y_1^2\). The Wronskian of \(y_1,y_2\) is \(y_1^2v'=e^{-\int P}\ne0\), so they are independent.

The method of multipliers

For a system \(\dot x_i=\sum_j a_{ij}x_j\) with constant coefficients, d’Alembert multiplied the equations by constants \(\lambda_i\) and added them, seeking a combination \(u=\sum_i\lambda_ix_i\) that satisfies a single equation \(\dot u=\mu u\).

The condition is an eigenvalue problem, \(\det(A^{\mathsf T}-\mu I)=0\), more than a century before matrices existed. For second-order systems \(\ddot{\mathbf x}=-K\mathbf x\) the same idea gives combinations with \(\ddot u=-\omega^2u\): the normal modes of oscillation that d’Alembert, Daniel Bernoulli and Euler used for beaded strings and hanging chains, and that Lagrange made general.

Fluids, complex functions and the paradox

For the general reader. Why does a moving body feel the resistance of the air or water? In 1752 d’Alembert set out to compute this from first principles for an ideal fluid with no internal friction. He found that the answer was exactly zero, in flat contradiction with experience. On the way he wrote down, for the first time, the equations that link fluid flow in a plane to functions of a complex variable.

Plane flow and analytic functions

D’Alembert’s Essai d’une nouvelle théorie de la résistance des fluides (1752) grew out of an entry for a Berlin prize competition on fluid resistance for which no prize was awarded; he blamed Euler’s influence. For a steady plane flow with velocity \((u,v)\) that is incompressible and has no rotation, he obtained

These say that \(u-iv\) satisfies the Cauchy–Riemann equations: it is an analytic function \(w'(z)\) of \(z=x+iy\). The complex potential \(w=\phi+i\psi\) combines the velocity potential \(\phi\) (with \(\nabla\phi=(u,v)\)) and the stream function \(\psi\), whose level curves are streamlines. Euler took up the same relations soon after, and Augustin-Louis Cauchy and Bernhard Riemann made them the foundation of complex analysis.

The paradox

D’Alembert found the vanishing resistance in 1752 for symmetric bodies and stated it as a general paradox for the geometers to explain in the fifth volume of his Opuscules (1768). The general proof uses conservation of momentum in a large sphere around the body: the potential decays fast enough at infinity that no momentum flux escapes to infinity. The two-dimensional lift formula is the Kutta–Joukowski theorem of the early twentieth century.

The cylinder

For a circular cylinder of radius \(a\) in a stream of speed \(U\), take \(w(z)=U\big(z+\frac{a^2}{z}\big)\). On \(|z|=a\) the flow is tangential with speed \(2U|\sin\theta|\). Bernoulli’s equation \(p+\tfrac12\rho|\mathbf u|^2=\text{const}\) gives the surface pressure

This is symmetric under \(\theta\mapsto\pi-\theta\): the pressure on the rear face equals that on the front, so it pushes back exactly as hard as it pushes forward. Real flows separate behind the body and leave a low-pressure wake. The resolution came in 1904, when Ludwig Prandtl showed that a thin boundary layer, where viscosity cannot be neglected however small it is, controls separation and drag.

Precession, nutation and the system of the world

For the general reader. The Earth’s axis does not point in a fixed direction. Like a tilted spinning top, it slowly sweeps out a cone, completing a circle in about 26,000 years, so the pole star changes over the ages. Newton explained the cause but got the details wrong. In 1749 d’Alembert gave the first correct dynamical theory, treating the Earth as a rigid, slightly flattened spinning body pulled by the Sun and Moon.

The torque on an oblate Earth

Model the Earth as a rigid body with axial symmetry, principal moments \(A=B<C\), spinning with angular velocity \(\omega\) about the symmetry axis, which is tilted at angle \(\varepsilon\) (the obliquity) to the normal of the ecliptic. A body of mass \(M\) on a circular orbit of radius \(a\) exerts a torque whose average over one orbit has magnitude

directed along the line of the equinoxes. It tries to pull the equatorial bulge into the plane of the orbit.

Derivation. For fast spin the angular momentum is essentially \(\mathbf L=C\omega\,\hat{\mathbf k}\). The torque is perpendicular to \(\mathbf L\), so it rotates \(\mathbf L\) without changing its length; the tip moves on a circle of radius \(L\sin\varepsilon\). Hence \(\Omega=\langle N\rangle/(C\omega\sin\varepsilon)\), which gives the formula. The fraction \(H=(C-A)/C\approx1/305\) is the Earth’s dynamical ellipticity.

Nutation

The Moon’s orbit is tilted by about 5° to the ecliptic, and its nodes regress with a period of about 18.6 years. The lunar torque therefore varies, and the axis nods about its mean cone. James Bradley announced this nutation from two decades of observations in 1748. D’Alembert’s treatise of 1749 derived both the precession and the nutation from the laws of mechanics. Euler published on the same problem in 1751 without citing him; he later conceded d’Alembert’s priority on this question, and the episode added to their growing hostility.

The Moon and the system of the world

In 1747 Clairaut, Euler and d’Alembert independently found that first-order perturbation theory gave only half the observed motion of the Moon’s apsides, and Clairaut briefly proposed modifying the inverse-square law. Clairaut showed in 1749 that higher-order terms account for the rest. D’Alembert published his own lunar theory and tables in the Recherches sur différens points importans du système du monde (1754–56). The quarrels over priority and method between the three men were bitter, but between them they turned Newton’s geometric arguments into the analytical celestial mechanics of Lagrange and Laplace.

Limits, series and chance

For the general reader. The calculus of the eighteenth century rested on “infinitely small” quantities that nobody could define. D’Alembert argued that it should rest on limits instead, and he gave a simple test for when an infinite sum adds up to a finite number. He also wrote on probability, where his scepticism led him to famous mistakes as well as to sharp questions.

Limits

In the article “Différentiel” of the Encyclopédie (vol. 4, 1754) d’Alembert explained the derivative as the limit of the ratio of finite differences: the ratio \(\frac{f(x+h)-f(x)}{h}\) approaches a value as closely as one pleases as \(h\) shrinks, and that value is the differential coefficient. He wrote that the metaphysics of the calculus lies entirely in the notion of limit. His definition of limit in the article “Limite” was still informal, and Cauchy and Karl Weierstrass gave it its modern form, but the programme was his.

The ratio test

In the fifth volume of the Opuscules (1768), studying the binomial series, d’Alembert used the ratio of consecutive terms to decide convergence.

Proof. Choose \(r\) with \(\limsup a_{n+1}/a_n<r<1\). For \(n\ge N\), \(a_{n+1}\le ra_n\), so \(a_{N+k}\le r^ka_N\) and the tail is dominated by the geometric series \(a_N\sum r^k\). If \(a_{n+1}\ge a_n\) eventually, the terms do not tend to 0.

Croix ou pile

In the Encyclopédie article “Croix ou pile” (1754) d’Alembert asked for the probability of getting heads at least once in two tosses of a coin. He argued that the game stops as soon as heads appears, so there are three outcomes: heads at once, tails then heads, tails twice. Two are favourable, so he gave \(2/3\). The standard answer is \(3/4\). He later raised more serious doubts about expectation in the St Petersburg problem and criticised Daniel Bernoulli’s mathematical case for smallpox inoculation, arguing that a long-run average gain need not persuade an individual facing an immediate risk.

The Encyclopédie and the Academies, 1746–1765

From the late 1740s d’Alembert was co-editor, with Denis Diderot, of the Encyclopédie, ou Dictionnaire raisonné des sciences, des arts et des métiers, responsible for mathematics and the physical sciences. He wrote or revised more than a thousand articles, among them most of the mathematical entries. The first volume appeared in 1751 with his Discours préliminaire. It sets out a genealogy of human knowledge from sensation, and a classification of the sciences under memory, reason and imagination adapted from Francis Bacon. It made him a public figure across Europe.

He was elected to the Académie française in 1754, a rare honour for a man of science. His literary and philosophical essays were collected in the Mélanges de littérature, d’histoire et de philosophie (from 1753), and his Éléments de musique (1752) presented the harmonic theory of Jean-Philippe Rameau.

Crisis and withdrawal

In 1757 the seventh volume carried his article “Genève”, written after a visit to Voltaire. It praised the Genevan pastors for a rational faith close to Socinianism and urged the city to allow a theatre. The pastors protested, Jean-Jacques Rousseau answered with his Lettre à d’Alembert sur les spectacles (1758), and in Paris the attacks on the Encyclopédie intensified until its privilege was revoked in 1759. D’Alembert gave up the co-editorship early in 1758, leaving Diderot to finish the work largely alone, though he continued to supply mathematical articles.

Kings and empresses

Frederick II of Prussia offered him the presidency of the Berlin Academy in 1752 and again later; he refused, and in 1763 spent three months with the king at Potsdam, again declining. Catherine II of Russia invited him to St Petersburg as tutor to her son; he declined that too. He preferred his independence in Paris, his small lodgings and the salons. From the 1750s he was close to Julie de Lespinasse, who ran her own salon from 1764.

Last years, death and legacy, 1765–1783

In 1765 a serious illness led d’Alembert to leave Mme Rousseau’s house, where he had lived since infancy, and move to Julie de Lespinasse’s apartment; the same year he became a full pensionnaire of the Académie des Sciences. From about then he complained that he could no longer concentrate on long calculations, though the Opuscules mathématiques continued until 1780.

Patron of the next generation

His influence shifted to the people he helped. He corresponded for years with Joseph-Louis Lagrange, whom he championed with Frederick II; in 1766 Lagrange succeeded Euler in Berlin. In 1768 the nineteen-year-old Pierre-Simon Laplace arrived in Paris with a letter of introduction; d’Alembert found him a post teaching mathematics at the École Militaire and pressed his candidacy at the Academy. Nicolas de Condorcet was another protégé, and became his heir and the author of his éloge. In a letter of 1777 he told Lagrange that he could no longer do geometry.

In 1772 he was elected perpetual secretary of the Académie française, and he wrote many of its éloges of deceased members. The death of Julie de Lespinasse in 1776, and the discovery from her papers that she had loved other men, left him grief-stricken. He died of a bladder disease at his lodgings in the Louvre on 29 October 1783, after refusing an operation. As a known unbeliever he was refused burial in the church and was buried in an unmarked grave in the Porcherons cemetery.

Legacy

• Mechanics: d’Alembert’s principle, through Lagrange’s Mécanique analytique, underlies analytical mechanics.
• Partial differential equations: the wave equation, its general solution, the d’Alembertian \(\square\), and the debate over functions that led to Fourier analysis.
• Fluids and complex analysis: the Cauchy–Riemann relations in plane flow, and a paradox that shaped fluid mechanics until boundary-layer theory.
• Analysis: the limit concept as the foundation of the calculus, the ratio test, and the d’Alembert–Gauss theorem.
• Astronomy: the first dynamical theory of precession and nutation.
• Honours: Fellow of the Royal Society (1748), member of the Berlin Academy (1746), Foreign Honorary Member of the American Academy of Arts and Sciences (1781); a lunar crater bears his name.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Jean le Rond d’Alembert, MacTutor History of Mathematics.
2. Jean le Rond d’Alembert, Wikipedia; and the French article Jean Le Rond d’Alembert.
3. J. Pappas and others, Alembert, Jean le Rond d’, Complete Dictionary of Scientific Biography.
4. T. L. Hankins, Jean d’Alembert: Science and the Enlightenment, Oxford University Press, 1970.
5. V. Le Ru, Jean Le Rond d’Alembert philosophe, Vrin, 1994.
6. M. J. A. N. Caritat de Condorcet, Éloge de M. d’Alembert, Académie des Sciences, 1783.

Mathematics and physics

1. J. d’Alembert, Traité de dynamique, Paris, 1743.
2. J. d’Alembert, Recherches sur la courbe que forme une corde tendue mise en vibration, Hist. Acad. Roy. Sci. Berlin 3 (1747, publ. 1749).
3. J. d’Alembert, Recherches sur le calcul intégral, Hist. Acad. Roy. Sci. Berlin 2 (1746, publ. 1748).
4. J. d’Alembert, Recherches sur la précession des équinoxes et sur la nutation de l’axe de la terre, Paris, 1749.
5. J. d’Alembert, Essai d’une nouvelle théorie de la résistance des fluides, Paris, 1752.
6. J. d’Alembert, Opuscules mathématiques, 8 vols., Paris, 1761–1780.
7. Œuvres complètes de d’Alembert, critical edition, CNRS Éditions, from 2002.
8. C. Truesdell, The rational mechanics of flexible or elastic bodies 1638–1788, in L. Euler, Opera Omnia II.11.2, 1960.
9. J. Ravetz, Vibrating strings and arbitrary functions, in The Logic of Personal Knowledge, 1961.
10. G. Grimberg, W. Pauls and U. Frisch, Genesis of d’Alembert’s paradox and analytical elaboration of the drag problem, arXiv:0801.3014.
11. H. Goldstein, C. Poole and J. Safko, Classical Mechanics, 3rd ed., Addison-Wesley, 2002 (d’Alembert’s principle; precession of the equinoxes).