Poisson had made considerable progress with the problem before Malus died on 24 February 1812. Poisson submitted the first part of his solution to the Academy on 9 March entitled Sur la distribution de l'йlectricitй а la surface des corps conducteurs. As the mathematicians had intended, this was the deciding factor in Poisson being elected to the physics section of the Institute to replace Malus. It also marked a move away from experimental research towards theoretical research in what was considered to constitute physics, and in this the Institute was following the lead given by Laplace.
Poisson continued to add various responsibilities to his already busy life. In 1815 he became examiner for the Йcole Militaire and in the following year he became an examiner for the final examinations at the Йcole Polytechnique.
It is remarkable how much work Poisson put in; to his research, to his teaching and to playing an ever increasingly important role in the organisation of mathematics in France. When he married Nancy de Bardi in 1817 he found that family life put yet another pressure on him yet somehow he survived the pressures continuing to take on further duties. His research contributions covered a wide range of applied mathematics topics. Although he devised no innovative new theories, he made major contributions to further developing the theories of others often being the first to exhibit their real significance. We mention now just a few of the topics he studied after his election to the Academy.
In 1813 Poisson studied the potential in the interior of attracting masses, producing results which would find application in electrostatics. He produced major work on electricity and magnetism, followed by work on elastic surfaces. Papers followed on the velocity of sound in gasses, on the propagation of heat, and on elastic vibrations. In 1815 he published a work on heat which annoyed Fourier who wrote:-
Poisson has too much talent to apply it to the work of others. to use it to discover what is already know is to waste it ...
Fourier went on to make valid objections to Poisson's arguments which he corrected in later memoirs of 1820 and 1821.
In 1823 Poisson published on heat, producing results which influenced Sadi Carnot. Much of Poisson's work was motivated by results of Laplace, in particular his work on the relative velocity of sound and his work on attractive forces. This latter work was not only influenced by Laplace's work but also by the earlier contributions of Ivory. Poisson's work on attractive forces was itself a major influence on Green's major paper of 1828 although Poisson never seems to have discovered that Green was inspired by his formulations.
In Recherchйs sur la probabilitй des jugements en matiиre criminelle et matiиre civile, an important work on probability published in 1837, the Poisson distribution first appears. The Poisson distribution describes the probability that a random event will occur in a time or space interval under the conditions that the probability of the event occurring is very small, but the number of trials is very large so that the event actually occurs a few times. He also introduced the expression "law of large numbers". Although we now rate this work as of great importance, it found little favour at the time, the exception being in Russia where Chebyshev developed his ideas.
It is interesting that Poisson did not exhibit the chauvinistic attitude of many scientists of his day. Lagrange and Laplace recognised Fermat as the inventor of the differential and integral calculus; he was French after all while neither Leibniz nor Newton were! Poisson, however, wrote in 1831:-
This [differential and integral] calculus consists in a collection of rules ... rather than in the use of infinitely small quantities ... and in this regard its creation does not predate Leibniz, the author of the algorithm and of the notation that has generally prevailed.
He published between 300 and 400 mathematical works in all. Despite this exceptionally large output, he worked on one topic at a time. Libri writes :-
Poisson never wished to occupy himself with two things at the same time; when, in the course of his labours, a research project crossed his mind that did not form any immediate connection with what he was doing at the time, he contented himself with writing a few words in his little wallet. The persons to whom he used to communicate his scientific ideas know that as soon as he had finished one memoir, he passed without interruption to another subject, and that he customarily selected from his wallet the questions with which he should occupy himself. To foresee beforehand in this manner the problems that offer some chance of success, and to be able to wait before applying oneself to them, is to show proof of a mind both penetrating and methodical.
Poisson's name is attached to a wide variety of ideas, for example:- Poisson's integral, Poisson's equation in potential theory, Poisson brackets in differential equations, Poisson's ratio in elasticity, and Poisson's constant in electricity. However, he was not highly regarded by other French mathematicians either during his lifetime or after his death. His reputation was guaranteed by the esteem that he was held in by foreign mathematicians who seemed more able than his own colleagues to recognise the importance of his ideas. Poisson himself was completely dedicated to mathematics. Arago reported that Poisson frequently said:-
Life is good for only two things: to study mathematics and to teach it.
J J O'Connor and E F Robertson
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