Poisson, Simeon-Denis

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Siméon-Denis POISSON

b. 21 June 1781, d. 25 April 1840

Summary Poisson, the apostle of Laplacian science, was the master of French mathematics from 1815 to 1840. His contribution to probability theory is not confined to the distribution which bears his name or to the expression "Law of Large Numbers", but bears on various areas ranging from pure mathematics to the mathematics of artillery.

Siméon-Denis Poisson who was born in Pithiviers and died in Sceaux, came from a modest background. His father, who had been a soldier, held a minor administrative post and became president of the Pithiviers district (Loiret) during the French Revolution. As a child, Siméon-Denis was first educated within his family, and later in the "école centrale at Fontainebleau, one  of the secondary schools created by the Directoire between 1795 and 1799. He owed his social and scientific rise to the new scientific institutions such as the Institut de France, the École Polytechnique, and the Bureau des Longitudes, created by the French Revolution, as well as the unconditional and very effective support of Laplace, his mentor in all things.

Poisson was admitted as the top candidate to the École Polytechnique in 1798\. Singled out by his teachers, he was appointed as Associate Professor in 1802, and Professor in 1806 to replace Fourier who had been named Prefect. In 1809, he was the first Professor of Mechanics in the new Faculty of Sciences of the University of Paris, which Napoleon had just created. He became an astronomer at the Bureau des Longitudes in 1808, and a member of the Institut de France in 1812. In short, Poisson gathered all the possible laurels available under the new French academic nomenclature. At the fall of the Emperor Napoleon in 1815, Poisson, whose political flexibility was as remarkable as the rigidity of his scientific convictions, was given supreme powers over the French universities. For 25 years, he was to reign without sharing of power over scientific education at all levels. It was he alone who decided on the appointment of Professors at every scientific institution, large or small, and he determined their teaching programs in the minutest detail. One can well imagine that this concentration of power earned him many enemies. Nevertheless, his administrative achievement was considerable, and devoted entirely to the service of a cause: the development and use of Laplacian science, of which he was the main heir after his mentor's death in 1827.

Poisson's scientific achievement is impressive both in its quantity and quality: it covers all of the physics and mathematical analysis of the period. Poisson was a very thorough analyst who solved problems by direct attack on the most complex and delicate of points. We may mention his work on the differential equations of mechanics, which are at the very basis of the fundamental researches of Hamilton and Jacobi 30 years later and also his work on domains of attraction which inspired the most beautiful discoveries of Green. One could equally well single out his work on the various "definite integrals" with which he toyed. Poisson is remembered in history more for his mathematical virtuosity than for his corpuscular physical models. These were the cause of many of his controversies, for example with Laplace on capillarity, with Fourier on heat, with Fresnel on light and with Navier on elasticity. According to P. Costabel, his strength lay rather in his ``sense of formalism, which uncovers analogies, unifies problems or areas which have so far been distinct, and extends decisively the use of the calculus."

The clarity of his exposition, whether oral or written, was such that his teachings encapsulated in his "Traité de Mécanique (1st edition  1811) served as a basic text during the 19th century.

Together with Fourier, Poisson was among the first to understand the fundamental importance of Laplace's probabilistic opus. It was Poisson who gave the first convincing if not definitive proofs of Chapter 4 of the second volume of the Théorie  analytique of 1812 containing "Laplace's Theorem". This states that the  sum of a large number of errors having any distributions is asymptotically Gaussian, whence ``Laplace's formula" allows us to calculate the parameters of the limiting distribution to good precision from the observations alone. Poisson, who helped to create the first chair in the Calculus of Probabilities for his friend Libri in the Faculty of Sciences of the University of Paris, was also the first to teach Laplace's formulae in his lectures at the Sorbonne in 1836. These lectures, later published as a large 415 page book in 1837, served as a model for the teaching of the asymptotic theory of probabilities throughout Europe, particularly in Berlin, Cambridge and St Petersburg. The title "Recherches sur la probabilité des jugements of this treatise is misleading. Only the last quarter of  the volume is concerned with judgements; the rest of the work is a complete treatise on probability, in which several important theoretical concepts are to be found, at least between the lines and without specific announcements. Among them are those of random variable and distribution function (p.140).

Poisson's interest in the calculus of probabilities was probably evoked by the abstracts (comptes rendus) of Laplace's and Fourier's papers, which he prepared for the Bulletin de la Société Philomatique, dating from 1808-1810. His first publication on probabilities discusses the banker's advantage in the game of "trente et quarante" which was very popular in 1820; in it he adapts some Laplacian ideas on characteristic functions. A short time after, he studies certain noteworthy Fourier transforms, such as that of the Cauchy distribution. Poisson was thus one of the first to consider a theory of errors where the error distribution was not necessarily Gaussian (Normal).

Poisson lived through the extraordinary development of statistics in the first half of the 19th century. In every one of the positions which he held, he tried to demonstrate how Laplacian theory could be used to validate statistical data. It was he who introduced Laplacian theory into the probabilistic problems of target practice. In his articles in the "Mémorial de l'Artillerie of 1829, he  defined the problem of dispersion of shots in a probabilistic setting, where these shots follow what we would now refer to as a bivariate Gaussian distribution. This was the starting point of the remarkable and unacknowledged works of French artillery men of the 19th century, trained at the École Polytechnique and the École d'Application d'Artillerie et du Génie in Metz, such as I. Didion, G. Piobert, or E. Jouffret. These works foreshadow many aspects of modern mathematical statistics.

Poisson's name remains best known for the most famous scientific metaphor in the history of statistics, namely the ``Law of Large Numbers", which he published in 1835. This was to become the main inspirational source for Quetelet, and later of the so-called Continental School of statistics. By the Law of Large Numbers, Poisson meant that the proportion of successes in independent trials shows statistical regularity even if the probability of success in each trial is not the same. This version of the Law has come to be known more specifically as Poisson's Law of Large Numbers, being a generalization of the famous Law of J. Bernoulli. It explained theoretically the fact that descriptive statistics in the moral sciences had been observed in the 1830's to display numerical regularity, but gave rise to many memorable controversies, particularly in the decade following 1830, not only at the Academy of Sciences but also in the Chamber of Deputies. The more general question in dispute was the relevance of the calculus of probabilities, or even more simply ``quantitative methods" in medicine (with Louis, Gavarret, etc.) as well as in criminology and demography. Poisson's Law was, in particular, unjustly attacked by Bienaymé after Poisson had devoted the last hundred pages of his famous work of 1837 to it, extending the work of Condorcet.

As for the Poisson distribution, it may be found highlighted in his treatise of 1837, but he himself would have been very surprised to learn that in probability theory he was to be remembered in our time almost entirely for this single minor contribution, especially since the distribution itself had already been found by de Moivre. It was mainly around 1900, following the researches of Bortkiewicz, Erlang, Bateman and others, that the Poisson distribution acquired a renewed practical significance. It is particularly prominent in connection with the ``Poisson process". Of course, Poisson's name is attached to a number of concepts outside of probability. To mention two: Dirac has acknowledged ``Poisson brackets", and students of analysis continue to learn about "Poisson's integral". Thus Poisson may well rest in peace.

#### References[edit]

[1] | Bru, B. (1996). Le problème de l'efficacité du tir à l'Ecole de Metz: aspects théoriques et expérimentaux. In B. Belhoste and A. Picon, L'École d'application de l'artillerie et du génie de Metz (1802-1870), Musée des plans-reliefs, Paris, pp. 61-70.

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[2] | Haight, F. (1967). Handbook of the Poisson Distribution. Wiley, New York.
[3] | Métivier, M., Costabel, P., Dugac, P. (dir.) (1981). Siméon-Denis Poisson et la science de son temps, École polytechnique, Palaiseau.
[4] | Sheynin, O.B. (1978). S.D. Poisson's work in probability. Archive for the History of Exact Sciences, 18, 245-300.
[5] | Stigler, S. (1986). The History of Statistics. Belknap, Harvard, pp. 182-194.

Reprinted with permission from Christopher Charles Heyde and Eugene William Seneta (Editors), Statisticians of the Centuries, Springer-Verlag Inc., New York, USA.