Marquis de Laplace

The French mathematician Pierre Simon, Marquis de Laplace (1749-1827), made outstanding contributions to pure mathematics, probability theory, and dynamical astronomy.

Pierre Simon Laplace was born on March 23, 1749, at Beaumont-en-Auge in Normandy. Little is known of his early life, for when he became famous, he was disinclined to talk about his humble origin as the son of a farmer. Rich neighbors recognized the boy's talent and helped to secure his education, first in Caen and then at the military school in Beaumont. At the conclusion of his course, Laplace remained for a short time as a teacher. Desiring to further his studies in mathematics, however, he soon set out for Paris. He sent a letter concerning the general principles of mechanics to Jean d'Alembert, who responded immediately and invited Laplace to an interview. Through the efforts of D'Alembert, Laplace obtained a professorship of mathematics at the military school in Paris.

In 1773 Laplace communicated the first of many memoirs to the Academy of Sciences in Paris. This contribution, despite his youth, placed him among the notables of science and secured his election as a member of the academy. In this memoir he treated two subjects: particular solutions of differential equations and the mean motions of the planets.

Physical problems on which Laplace worked included heat, the theory of vapors, and electrostatics. He made the first advance toward the understanding of molecular forces in liquids; in particular, he explained the inverse proportionality between the capillary rise in tubes and their diameters. In acoustics, he suggested that the excess of the speed of sound over that expected could be explained in terms of the heat changes induced by the vibrations. Dynamical astronomy and the theory of probability, however, were Laplace's major interests. He advocated the application of probability theory not only in science but also to social and economic problems.

Laplace was an opportunist and something of a snob. His political allegiance shifted easily as expediency required; the successive revolutionary governments, the republic, the empire, and the Bourbon restoration rewarded his servility with numerous honors. Napoleon made him a count, and Louis XVII made him a marquis.

Throughout the troubled times, Laplace's devotion to science remained constant, and he continued to contribute memoirs to the academy. The composition and publication in five volumes of his great work, the Traitédelamécanique céleste, spanned the three regimes between 1799 and 1825. It was no doubt the Traité that inspired Siméon Poisson to describe Laplace as the Isaac Newton of France, for the brilliant results achieved by Laplace were an evident vindication of the principle of universal gravitation. Laplace's other major work, the Théorie analytique des probabilités, was published in 1812.

Laplace was elected to the Académie Française in 1816. In the same year he presided over the commission charged to reorganize the école Polytechnique. Although he encouraged able young mathematicians and scientists among his pupils, Laplace was not a successful teacher. He died in Paris on March 5, 1827, leaving a wife and a son.

Dynamical Astronomy

In 1787 Laplace completed his demonstration, started in the memoir of 1773, of the stability of the solar system. Observations had indicated a disturbing acceleration of the mean motion of Jupiter and a similar diminution of the mean motion of Saturn; it had appeared probable that Jupiter would eventually fall into the sun and that Saturn would leave the solar system altogether. Laplace succeeded in showing that these fears were unfounded by demonstrating that these variations from the mean motion, known as the great inequality, were in fact ordinary periodic perturbations explicable in terms of the law of gravitational attraction and the near commensurability of the mean motions of the two planets. Finally, he explained the relation between the lunar acceleration and secular changes in the eccentricity of the earth's orbit, thus disposing of the fear that the moon would eventually fall into the earth.

The motion of the moon was one of Laplace's favorite subjects. His discovery that two lunar perturbations were caused by the spheroidal figure of the earth was quite remarkable. His deduction of the solar distance from observations of the moon drew the admiration of Dominique Arago. Among other results obtained by Laplace for satellite systems were the masses of the satellites of Jupiter and the calculation of the period of revolution of the rings of Saturn, which corresponded to William Herschel's measurements.

Laplace's famous nebular hypothesis first appeared in his Exposition du système du monde. The solar system, he suggested, had evolved from a quantity of incandescent gas. As the gas cooled, it contracted, and successive rings detached from the outer edge. These in turn cooled and condensed into planets, the sun remaining as the central core.

An important terrestrial problem to which Laplace contributed significantly was that of the tidal oscillations. He first derived the dynamical equations for the motion of the oceans caused by the attraction of the sun and the moon in a memoir of 1775; the theory was elaborated in volumes 2 and 5 of the Traitédelamécanique céleste. Laplace's theory is the first that can truly be called dynamical and is notable for the analysis of the tidal oscillation into its main harmonic constituents, namely, the long-term inequalities, the daily inequality, and the principal twice-daily oscillation. Moreover, Laplace was the first to take into account the attraction of the ocean in the disturbed state, the effect of the earth's rotation, and the depth of the ocean. Finally, he demonstrated the stability of the tidal oscillations; the necessary condition was that the density of the ocean be less than the mean density of the earth.

Probability Theory

In the 17th century combinatorial analysis was applied to problems concerning games of chance; this marked the beginning of probability theory. Jacob Bernoulli contributed the first significant theorem, the law of large numbers. Laplace's contribution may be characterized as the extensive application of analysis to probability theory. Although his original results were numerous and of great importance, regrettably he did not give due credit to others. The basic concept employed by Laplace in his analysis was the generating function. Generating functions were used by Laplace not only in the solution of probability problems but also in applications to problems in pure mathematics, such as interpolation, the solution of finite difference equations, and the expression of functions in terms of definite integrals. Laplace developed a method of approximation for definite integrals with integrands containing factors of high powers, a type frequently occurring in probability theory.

Laplace was also the first to apply infinitesimal calculus to probability problems. This he did by showing that a certain partial differential equation was approximately equivalent to the finite difference equation giving the solution of a particular problem. Another important contribution of Laplace was the extension of the work of Thomas Bayes, who had been the first to use mathematical probability inductively, that is, to reason from observed events to their causes.

Further Reading

Extracts from Laplace's work on the nebular hypothesis, astronomy, and probability are available in Harlow Shapley and Helen E. Howarth, A Source Book in Astronomy (1929). A short account of his work is given in Roger Hahn, Laplace as a Newtonian Scientist (1967). See also Jerzy Neyman and Lucien M. LeCam, eds., Bernoulli, 1713; Bayes, 1763; Laplace, 1813 (1965). □