VI.27 Siméon-Denis Poisson
b. Pithiviers, France, 1781; d. Paris, 1840
Analysis; mechanics; mathematical physics; probability
A brilliant graduate of the École Polytechnique in 1800, Poisson was quickly appointed to the staff, and became professor and graduation examiner there until his death. He was also founder professor of mechanics at the new Paris Faculté des Sciences of the Université de France; from 1830 he was also a member of the governing Council of the Université.
Poisson’s research output was dominated by his adherence to the traditions established by LAGRANGE [VI.22] and LAPLACE [VI.23]. Like Lagrange he preferred to algebraize theories, and to rely if possible upon power series and variational methods. From the mid 1810s he challenged the new theories of FOURIER [VI.25] (especially the solving of differential equations by trigonometric series and the Fourier integral) and of CAUCHY [VI.29] (the new approach to real-variable analysis using limits, and his innovation of complexvariable analysis). His overall achievements were much less significant than theirs: the main novelties were the “Poisson integral,” which embedded Fourier series within a power series; and a summation formula. He also studied the general and singular solutions of differential, difference, and mixed equations.
In physics Poisson tried to justify Laplace’s claim that all physical phenomena were molecular, and that the cumulative action upon a molecule of all its companions should be expressed mathematically in terms of an integral. He applied this approach to heat diffusion and to elasticity theory by the mid 1820s, but then decided that integrals should be replaced by sums; he elaborated this alternative especially in capillary theory (1831). Curiously, molecularism did not dominate his most important contributions to physics: to electrostatics (1812-14) and to magnetic bodies and the process of magnetization (1824-27). His mathematical contributions to these topics included modifying Laplace’s equation to what we now call Poisson’s equation, which deals with the potential at points inside a charged body or region of charge (1814); and also a divergence theorem (1826).
In mechanics, between 1808 and 1810 Poisson and Lagrange developed the brackets theory (named after them) of canonical solutions to the equations of motion. Poisson’s motivation was to extend, to secondorder terms in masses of the planets, Lagrange’s superb attempt to prove that the planetary system was stable; in later work he examined this (first-order) problem specifically, as well as other aspects of perturbation theory. He also analyzed rotating bodies by using moving frames of reference (1839), in an analysis that was to inspire Léon Foucault to propose his famous long pendulum in 1851. His best-known publications include a substantial and wide-ranging two-volume Traité de Mécanique (editions in 1811 and 1833), which did not, however, have room for Louis Poinsot’s beautiful recent theory (1803) of the couple in statics. In the mid 1810s, he studied deep-body fluid dynamics, in rivalry with Cauchy.
Poisson was one of the few contemporaries to take up Laplace’s work in probability theory and mathematical statistics. He studied various PROBABILITY DISTRIBUTIONS [III.71]: not only the one named after him (1837, rather in passing) but also the so-called Cauchy (1824) and Rayleigh (1830) distributions. He also examined proofs of the CENTRAL LIMIT THEOREM [III.71 §5], and formulated THE LAW OF LARGE NUMBERS [III.71 §4] (his term). One of his main applications was to the old problem of determining the probability that a triad of judges would come to the correct decision in court cases (1837).
Further Reading
Grattan-Guinness, I. 1990. Convolutions in French Mathematics 1800-1840. Basel: Birkhäuser.
Métivier, M., P. Costabel, and P. Dugac, eds. 1981. Siméon-Denis Poisson et la Science de son Temps. Paris: École Polytechnique.