D'Alembert's Axioms of Motion

Date 1743-1744

Drawing upon elements of Cartesian and Newtonian thought, d’Alembert formulated a set of laws describing the behavior of bodies in motion. The laws, all derived completely through mathematical calculation, combined to produce a general principle for solving problems in rational mechanics.

Locale Paris, France

Key Figures

• Jean le Rond d’Alembert (1717-1783), French mathematician, physicist, and Encyclopedia contributor
• Alexis-Claude Clairaut (1713-1765), French Newtonian mathematician
• Sir Isaac Newton (1642-1727), English scientist and mathematician
• Daniel Bernoulli (1700-1782), Swiss physicist
• René Descartes (1596-1650), French philosopher and mathematician

Summary of Event

Jean le Rond d’Alembert is probably best known for his collaboration with Denis Diderot on the Encyclopédie: Ou, Dictionnaire raisonné des sciences, des arts, et des métiers (1751-1772; partial translation Selected Essays from the Encyclopedy, 1772; complete translation Encyclopedia, 1965). His “Discours préliminaire” (preliminary discourse), which prefaced the work, was known and admired throughout Europe, and he was responsible for many of the Encyclopedia’s technical articles. A favorite in the salons of Paris, d’Alembert was involved in all aspects of the intellectual life of his century. Beyond these pursuits, however, d’Alembert was a mathematician and scientist of considerable expertise who made significant contributions in the field of rational mechanics. In 1741, he was admitted as a member to the French Academy of Sciences. There, he met, discoursed with, and competed with men such as Alexis-Claude Clairaut and Daniel Bernoulli.

D’Alembert received his first instruction in mathematics at the Jansenist Collège des Quatres Nations. In his classes, he was introduced to the work of Cartesian thinkers such as Pierre Varignon, Charles Reyneau, and Nicolas Malebranche. Thus, his early education in mathematics was strongly influenced by the ideas of René Descartes. This background did not, however, prevent d’Alembert from recognizing the value of Sir Isaac Newton’s work. He read Newton’s Philosophiae Naturalis Principia Mathematica (1687; The Mathematical Principles of Natural Philosophy, 1729; best known as the Principia) shortly after 1739 and Colin Maclaurin’s A Treatise of Fluxions (1742), which gave detailed explanations of Newton’s methods, before publishing his own Traité du dynamique (1743; treatise on dynamics) and Traité de l’équilibre et du movement des fluides (1744; treatise on equilibrium and on movement of fluids). D’Alembert believed that mathematics was the key to solving all problems. He rejected the use of experiments and observation. He maintained that rational mechanics was a component of mathematics along with geometry and analysis.

When d’Alembert set about writing his Traité du dynamique, an enormous amount of work had already been done on the laws of motion. Much of existing theory was contradictory, however, because of the problems involved in defining terms such as force, motion, and mass. D’Alembert was convinced that a logical foundation applicable to all mechanics could be found through the use of mathematics. Although d’Alembert insisted that he had rejected the theories of Descartes that he had studied in his youth, his approach to mechanics still relied heavily on Descartes’s method of deduction. D’Alembert wished to discover laws of mechanics that would be as logical and self-evident as the laws of geometry. Above all, he was determined to “save” mechanics from being an experimental science.

D’Alembert, like his fellow scientists, was a great admirer of Newton, and Newton’s Principia was for him the starting point in a study of mechanics. Thus, he developed his laws of mechanics using Newton’s work as a model. In his first law, d’Alembert expressed his agreement with Newton’s law of inertia, that is, that bodies do not change their state of rest or motion by themselves. They tend to remain in the same state; Newton would say, they remain in the same state until acted upon by a force. D’Alembert also was in accord with Newton’s concept of hard bodies moving in a void.

D’Alembert, however, found Newton’s second and third laws unacceptable, because they acknowledged force as real and relied upon experiments and observation. The logical geometric basis that he sought for the foundation of mechanics allowed no room for experiments and observations. Force was for d’Alembert a concept to be avoided because it did not lend itself to definition. He rejected not only innate force but all force. In contrast, Newton recognized force as having real existence. D’Alembert acknowledged that bodies would not move unless some external cause acted upon them but defined causes only in terms of their effects. His third law was similar to Newton’s third law. Newton had stated that two bodies must act on each other equally. D’Alembert proposed the concept of equilibrium, resulting from two bodies of equal mass moving in opposite directions at equal velocities.

Because of his rejection of force as a scientific concept, d’Alembert was closer in his theories to Malebranche, who viewed the laws of motion as entirely geometrical, than he was to Newton. D’Alembert’s laws of motion dealt with idealized geometrical figures rather than real objects. These figures moved through space until they impacted, causing them either to stop or to slip past one another. Change of motion was necessitated by geometry; force was an unnecessary element and only brought into play disturbing metaphysical concepts.

From the last two laws of his axioms of motion, d’Alembert derived what is now known as d’Alembert’s principle : The impact of two hard bodies either is direct or is transmitted by an intermediate inflexible object or constraint. He applied his principle the next year in his Traité de l’equilibre et du mouvements des fluides, which was for the most part a criticism of Daniel Bernoulli’s work on hydrodynamics. Although d’Alembert had used his principle successfully in his 1743 treatise, it failed to be very useful in fluid mechanics.

Significance

During the eighteenth century, opinions about d’Alembert’s contributions to science were many and varied. Some of his contemporaries credited him with having found a set of principles for rational mechanics; for some, his work verified Descartes’s beliefs that the laws of mechanics could be deduced from matter and motion and that there was no force involved in movement. However, others criticized and rejected d’Alembert, because he refused to accept experimentation and simply eliminated concepts that he found resistant to mathematical expression. His most important contribution was d’Alembert’s principle, which provided a general approach to solving mechanical problems. It was one of the first attempts to find simple and general rules for the movements of mechanical systems.

D’Alembert’s laws of motion were accepted as the logical foundation of mechanics well into the nineteenth century. Ultimately, however, his refusal to discuss force proved to be a fatal flaw. Today, Newton’s Principia is viewed as containing the basic laws of classical mechanics.

Bibliography

Greenberg, John L. The Problem of the Earth’s Shape from Newton to Clairaut: The Rise of Mathematical Science in Eighteenth Century Paris and the Fall of “Normal” Science. New York: Cambridge University Press, 1995. Chronicles the spread of Newtonian physics in France and discusses d’Alembert’s treatises. Readers need some scientific background.

Grimsley, Ronald. Jean d’Alembert, 1717-1783. Oxford, England: Clarendon Press, 1963. Considered the best biography of d’Alembert. Treats his literature and philosophy.

Hankins, Thomas L. Jean d’Alembert: Science and the Enlightenment. Oxford, England: Clarendon Press, 1970. Treats d’Alembert’s many-faceted role in the science, culture, politics, and philosophical activities of the century. Useful for all readers.

‗‗‗‗‗‗‗. Science and the Enlightenment. Reprint. New York: Cambridge University Press, 1991. General history of science in the eighteenth century, placing it in the cultural context of the time and stressing the impact of science on thought of the period.

Porter, Roy, ed. Eighteenth Century Science. Vol. 4 in The Cambridge History of Science. New York: Cambridge University Press, 2003. Survey of the development of science; explores the implications of the Scientific Revolution of the seventeenth century and treats the social, political, and economic significance of science in the eighteenth century.

Shectman, Jonathan. Groundbreaking Scientific Experiments, Investigations, and Discoveries of the Eighteenth Century. Westport, Conn.: Greenwood Press, 2003. Written for middle and high school students, college non-science majors, and general readers. Discusses science in the social and political climate of the period.

Yolton, John W., ed. Philosophy, Religion, and Science in the Seventeenth and Eighteenth Centuries. Rochester, N.Y.: University of Rochester Press, 1990. Examines the relationships of philosophy, science, and religion and their interactions in the works of René Descartes, Gottfried Wilhelm Leibniz, David Hume, and Sir Isaac Newton.