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of the first two volumes of his œuvres complètes, concerning celestial mechanics, has put into evidence some of his unpublished early texts about lunar theory and it has given rise to critical studies.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 In addition, he left several unpublished works: early memoirs of mathematics and celestial mechanics, plis cachetés deposited at the Paris Academy, and a ninth volume of his Opuscules.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