VI.23 Pierre-Simon Laplace
b. Beaumont-en-Auge, France, 1749; d. Paris, 1827
Celestial mechanics; probability; mathematical physics
Laplace is known to later mathematicians for many concepts of fundamental importance to mathematics, including the LAPLACE TRANSFORM [III.91], the Laplace expansion, Laplace’s angles, Laplace’s theorem, Laplace functions, inverse probability, GENERATING FUNCTIONS [IV.18 §§2.4,3], a derivation of the Gauss/Legendre least-squares rule of error by means of a linear regression, and the LAPLACIAN [I.3 §5.4] or potential function. Laplace developed the fields of celestial mechanics (the phrase was his coinage) and probability, and along the way the mathematics to service and advance them. For Laplace, celestial mechanics and probability were complementary instruments that implemented a unified vision of a fully determined universe. Celestial mechanics vindicated the Newtonian system of the world. Probability was the measure, not of the operations of chance in nature, for there are none, but of human ignorance of causes, which was to be reduced to virtual certainty by calculation. The third reason for Laplace’s importance to the history of science was the mathematization of physics in the first two decades of the nineteenth century. Apart from a few formulations—speed of sound, capillary action, refractive indices of gases—his role there was that of instigator and patron rather than of major contributor.
Laplace came up with the majority of the above concepts in a probabilistic context. The earliest hints of the method of solving difference, differential, and integral equations, later known as the Laplace transform, appeared in “Mémoire sur les suites” (1782a), where Laplace introduced generating functions. Laplace considered generating functions to be the approach of choice in solving problems that involved the development of functions in series and evaluation of the sums. Years later, in composing Théorie Analytique des Probabilités (1812), he subordinated all the analytical part to the theory of generating functions and treated the entire subject as their field of application. In the early memoir, however, he emphasized what he expected to be their applicability to problems of nature.
In an even earlier paper, “Mémoire sur la probabilité des causes par les événements” (1774), Laplace stated the theorem permitting the analysis later termed BAYESIAN [III.3]. Unknown to Laplace, Thomas Bayes had arrived at the same theorem eleven years earlier but had not developed it. Laplace for his part proceeded, in further investigations over some thirty years, to develop inverse probability into the basis for statistical inference, philosophical causality, estimation of scientific error, quantification of the credibility of evidence, and optimal voting rules in the proceedings of legislative bodies and judicial panels. His initial attraction to the approach was its applicability to human concerns. It was in the course of these papers, most notably “Mémoire sur les probabilités” (1780), that the word probability came to connote not merely the basic quantity in the theory of games and chance but a subject in itself.
Laplace first addressed error theory in the above paper on causality. The problem was to estimate the most appropriate mean value to be taken in a series of astronomical observations of the same phenomenon. He also determined how the limits of the error were related to the number of observations (“Mémoire sur les probabilités” above). In “Essai pour connaître la population du Royaume” (1783-91) Laplace turned to demographic applications. In the absence of census data, one needed to determine the multiplier to be applied to the number of births at any one time in order to estimate the approximate size of a population. The specific problem Laplace solved was the size of the sample required for the probability of error to fall within given limits.
Laplace then put probabilistic investigation aside. Only twenty-five years later did he return to the subject, in the course of preparing the comprehensive Théorie Analytique des Probabilités. In 1810 he returned to the problem of determining the mean value from a large number of observations, which he interpreted as the problem of the probability that the mean value falls within certain limits. He proved a law of large numbers, stating that if positive and negative errors in an indefinitely large number of observations are assumed to be equally possible, then their mean result converges to a limit in a precise way. From this analysis followed the least-squares law of error. A priority dispute over the discovery of that law was even then simmering between GAUSS [VI.26] and LEGENDRE [VI.24].
In a long series of investigations brought together in Traité de Mécanique Céleste (1799-1825, five volumes), the two part “Mémoire sur la Théorie de Jupiter et de Saturne” (1788) demonstrates the most famous of his findings in planetary astronomy. He established that the current acceleration in orbital motion of Jupiter and the deceleration of Saturn are the reciprocal effects of their mutual gravitation, which are cyclical over many hundreds of years and not cumulative. From this and analysis of other phenomena that Laplace explored, it followed that the so-called secular inequalities of planetary motions are periodic over many centuries. Thus, they are not derogations from the law of gravity but evidence that its validity extended beyond the Sun-planet attractions that had been studied by NEWTON [VI.14]. He was never able to prove, however, that lunar acceleration is self-correcting over time.
The expansion known by Laplace’s name in the theory of determinants first appears in the background of the Jupiter-Saturn memoir in an analysis of the eccentricities and inclinations of orbits, “Recherches sur le calcul intégral et sur le système du monde” (1776). Except for that, Laplace’s mathematical originality is less notable in his analysis of planetary motion than in his development of the theory of probability. More in evidence in his astronomical work is his motivational drive and his power and virtuosity in calculation, which may indeed have been more important throughout his long career. Laplace was masterful in finding rapidly convergent series, in obtaining mathematical expressions incorporating terms to represent a multitude of physical phenomena, in justifying the neglect of inconvenient quantities in order to reach solutions, and in giving the widest possible generality to his conclusions.
The attraction exerted by a spheroid on an external or internal point proved to be mathematically the most fertile set of problems in Laplacian planetary astronomy. In “Théorie des attractions des sphéroïdes et de la figure des planètes” (1785) Laplace employed LEGENDRE POLYNOMIALS [III.85] in a form later called Laplace functions. He also proved a theorem that stated that all ellipsoids with the same foci for their principal sections attract a given point with a force proportional to their masses. Laplace’s angles appear in his development of the equation for the attraction of a spheroid on a given point. Laplace used polar coordinates in this analysis. He transformed the equation into one in Cartesian coordinates in “Mémoire sur la théorie de l’anneau de Saturne” (1789). In 1828 George Green dubbed Poisson’s application of the formula to electrostatic and magnetic forces the potential function, the term used thereafter in classical physics.
Further Reading
The memoirs cited in this article can be found in the bibliography of C. C. Gillispie’s Pierre-Simon Laplace: A Life in Exact Science (Princeton University Press, Princeton, NJ, 1997).
For the mathematical content of Laplacian physics, see pp. 440-55 (and elsewhere) of I. Grattan-Guinness’s Convolutions in French Mathematics (Birkhäuser, Basel, 1990, three volumes).