b. Paris, 1717; d. Paris, 1783
Algebra; infinitesimal calculus; rational mechanics; fluid mechanics; celestial mechanics; epistemology
D’Alembert spent his whole life in Paris, where he became one of the most influential members of the Académie Royale des Sciences and of the Académie Française. He became well-known as the scientific editor of the celebrated French Encyclopédie, the twentyeight-volume work on which he collaborated with Denis Diderot, and for which he wrote most of the mathematical and many of the scientific articles.
As a student at the Jansenist Collège des Quatre-Nations, he followed the usual curriculum of grammar, rhetoric, and philosophy, the latter including some Cartesian science, a little mathematics, and much theology framed by the then burning debate about predestination, freedom, and grace. Disgusted with the permanent climate of controversy and the endless metaphysical discussions among his Jansenist teachers, d’Alembert decided, after attaining his diploma in law, to devote himself to his personal passion, “géométrie” (that is, mathematics).
D’Alembert’s first communications to the French Académie were concerned with the analytic geometry of curves, the integral calculus, and fluid resistance, notably the problem of the deceleration and deflection of a disk entering a fluid, which was linked to the Cartesian explanation of refraction of light. He made a close reading of NEWTON’S [VI.14] Principia, his commentary on passages of the first book showing his clear preference for analytical methods over the synthetic geometry of Newton.
D’Alembert’s Traité de Dynamique (1743) made him famous in learned circles. He built up a systematic and rigorous theory of mechanics founded upon a short list of well-chosen principles—inertia, composition of motions (i.e., the addition of the effects of two forces or powers), and equilibrium—while at the same time trying to avoid metaphysical arguments. Most notably he proposed an important general principle, known today as “d’Alembert’s principle,” to simplify the investigation of constrained systems, such as the compound pendulum, vibrating rods, strings, rotating bodies, and even fluids, which he considered to be aggregates of parallel slices. The essential idea behind the principle was to reduce a problem in dynamics to one in statics, roughly speaking by introducing a fictitious force, the “kinetic reaction,” which was minus the mass times the acceleration. This allowed techniques from statics to be brought to bear on problems in dynamics.
His other books and memoirs were developments, some very innovative, in fluid theory, partial differential equations, celestial mechanics, algebra, and integral calculus. He devoted much thought to the use and status of imaginary numbers.
In his Réflexions sur la Cause Générale des Vents (1747) and his Recherches sur le Calcul Intégral (1748) he observed that numbers of the form a + bi (where 
retain the same form when subjected to the usual operations (addition, subtraction, multiplication, division, and exponentiation). He proved that, for a real polynomial, imaginary roots always occur in conjugate pairs, and that even if a real polynomial has no real root, there is still always a complex root. However, his work was not rigorous—for example, he presupposed the existence of roots—and consequently he did not provide a proof of THE FUNDAMENTAL THEOREM OF ALGEBRA [V.13].
At the end of the 1740s there was a crisis in Newtonian science, with d’Alembert’, Clairaut, and EULER [VI.19] each independently coming to the conclusion that Newton’s theory of gravitation could not account for the motion of the Moon. In 1747 d’Alembert discussed various possibilities for solving the problem—an additional force, or a very irregular shape for the Moon, or some vortices between Earth and Moon—and produced a long study on celestial mechanics and planetary perturbations that has only recently been rediscovered and published (see d’Alembert 2002). By 1749 an improved mathematical analysis of the problem had shown that Newton’s theory was correct. The rest of d’Alembert’s extensive work on celestial mechanics was published in his Recherches sur la Précession des Équinoxes (1749), Recherches sur Différents Points du Système du Monde (1754-56), and in some of the eight volumes of his Opuscules (1761-83).
In 1747 d’Alembert presented a paper on the famous problem of vibrating strings, Recherches sur la Courbe que Forme une Corde Tendue Mise en Vibration (1749). This paper contained a solution of THE WAVE EQUATION [I.3 §5.4]. This was the first solution of a partial differential equation—partial differential equations were a new tool that he had already used in his 1747 Réflexions sur la Cause Générale des Vents. It led to a lengthy debate with Euler and DANIEL BERNOULLI [VI.18] about the possible form of the solutions and the general notion of function.
D’Alembert’s work for the Encyclopédie (1751-65) and his efforts to find rigorous foundations for the sciences led him into the field of philosophy, where his main contributions concerned the classification of various sciences. He also worked on the study of cognition following the lines proposed by DESCARTES [VI.11], Locke, and Condillac.
D’Alembert, J. le R. 2002. Premiers Textes de Mécanique Céleste, edited by M. Chapront. Paris: CNRS.
Hankins, T. 1970. Jean d’Alembert, Science and the Enlightenment. Oxford: Oxford University Press.
Michel, A., and M. Paty. 2002. Analyse et Dynamique. Études sur l’Oeuvre de d Alembert Laval, Québec: Les Presses de l’Unwersité Laval.
Francois de Gandt