# Fourier, Jean Baptiste Joseph (1768–1830)

# FOURIER, JEAN BAPTISTE JOSEPH (1768–1830)

Every physical scientist knows the name Fourier; the Series, Integral, and Transform that bear his name are essential mathematical tools. Joseph Fourier's great achievement was to state the equation for the diffusion of heat. The Fourier Series is a series of origonometric terms which converges to a periodic function over one period. The Fourier Integral is the limiting form of a Fourier Series when the period of the periodic function tends to infinity. The Fourier Transform is an analytic tool derived from the coefficients of the integral expansion of a Fourier Series. He pioneered the use of Fourier Series and Integrals because he needed them to solve a range of problems related to the flow of heat. Fourier was not the first person to realize that certain trigonometric expressions could be used to represent certain functions but he did develop the systematic use of such expressions to represent arbitrary functions.

## BIOGRAPHY

The son of a tailor, Joseph Fourier was a member of a large family. Both of his parents died by the time he was nine. His education began at a local, church-run, military school, where he quickly showed talent in his studies and especially in mathematics. His school persuaded him to train as a priest. While preparing to take holy orders he taught his fellow novices mathematics. Fourier may well have entered the priest- hood, but due to the French Revolution new priests were banned from taking holy orders. Instead he returned to his home town of Auxerre and taught at the military school. His friend and mathematics teacher, Bonard, encouraged him to develop his mathematical research, and at the end of 1789 Fourier travelled to Paris to report on this research to the Academie des Sciences.

Inspired by what he had experienced in Paris, Fourier joined the local Popular party on his return to Auxerre. In later years Fourier's involvement in local politics would lead to his arrest. He was arrested twice but each time was granted clemency.

France lost many of its teachers during the first years of the Revolution. One of the solutions to the shortage of teachers was the establishment of the Ecole Normale in Paris. Fourier, as a teacher and an active member of the Popular Society in Auxerre, was invited to attend in 1795. His attendance at the short- lived Ecole gave him the opportunity to meet and study with the brightest French scientists. Fourier's own talent gained him a position as assistant to the lecturers at the Ecole Normale.

The next phase of his career was sparked by his association with one of the lecturers, Gaspard Monge. When the French ruling council, the Directorate, ordered a campaign in Egypt, Monge was invited to participate. Fourier was included in Monge's Legion of Culture, which was to accompany the troops of the young general Napoleon Bonaparte (even then a national hero due to his successful campaigns in Italy).

The Egyptian campaign failed, but Fourier along with his fellow scientists managed to return to France. The general Fourier had accompanied to Egypt was now First Consul. Fourier had intended to return to Paris but Napoleon appointed Fourier as prefect of Isère. The prefecture gave Fourier the resources he needed to begin research into heat propagation but thwarted his ambition to be near the capital.

## THEORIES OF HEAT PROPAGATION

The prevailing theory of heat, popularized by Simeon-Denis Poisson, Antoine Lavoisier and others, was a theory of heat as a substance, "caloric." Different materials were said to contain different quantities of caloric. Fourier had been interested in the phenomenon of heat from as early as 1802. Fourier's approach was pragmatic; he studied only the flow of heat and did not trouble himself with the vexing question of what the heat actually was.

The results from 1802–1803 were not satisfactory, since his model did not include any terms that described why heat was conducted at all. It was only in 1804, when Jean-Baptiste Biot, a friend of Poisson, visited Fourier in Grenoble that progress was made. Fourier realized that Biot's approach to heat propagation could be generalized and renewed his efforts. By December of 1807 Fourier was reading a long memoir on "the propagation of heat in solids" before the Class of the Institut de France.

The last section of the 1807 memoir was a description of the various experiments which Fourier had undertaken. It concentrated on heat diffusion between discrete masses and certain special cases of continuous bodies (bar, ring, sphere, cylinder, rectangular prism, and cube). The memoir was never published, since one of the examiners, Lagrange, denounced his use of the Fourier Series to express the initial temperature distribution. Fourier was not able to persuade the examiners that it was acceptable to use the Fourier Series to express a function which had a completely different form.

In 1810, the Institut de France announced that the Grand Prize in Mathematics for the following year was to be on "the propagation of heat in solid bodies." Fourier's essay reiterated the derivations from his earlier works, while correcting many of the errors. In 1812, he was awarded the prize and the sizeable honorarium that came with it.

Though he won the prize he did not win the outright acclaim of his referees. They accepted that Fourier had formulated heat flow correctly but felt that his methods were not without their difficulties. The use of the Fourier Series was still controversial. It was only when he had returned to Paris for good (around 1818) that he could get his work published in his seminal book, The Analytical Theory of Heat.

## THE FOURIER LEGACY

Fourier was not without rivals, notably Biot and Poisson, but his work and the resulting book greatly influenced the later generations of mathematicians and physicists.

Fourier formulated the theory of heat flow in such a way that it could be solved and then went on to thoroughly investigate the necessary analytical tools for solving the problem. So thorough was his research that he left few problems in the analysis of heat flow for later physicists to investigate and little controversy once the case for the rigour of the mathematics was resolved. To the physical sciences Fourier left a practical theory of heat flow which agreed with experiment. He invented and demonstrated the usefulness of the Fourier Series and the Fourier integral—major tools of every physical scientist. His book may be seen both as a record of his pioneering work on heat propagation and as a mathematical primer for physicists. Lord Kelvin described the Theory of Heatas "a great mathematical poem."

Fourier's own attitude to his work is illustrated by the "Preliminary Discourse" he wrote to introduce his book. As a confirmed positivist, he stated that whatever the causes of physical phenomena, the laws governing them are simple and fixed—and so could be discovered by observation and experiment. He was at pains to point out that his work had application to subjects outside the physical sciences, especially to the economy and to the arts. He had intended to write a companion volume to his Theory of Heat that would cover his experimental work, problems of terrestrial heat, and practical matters (such as the efficient heating of houses); but it was never completed.

His talents were many: an intuitive grasp of mathematics, a remarkable memory and an original approach. Fourier was a man of great common sense, a utilitarian, and a positivist.

*David A. Keston*

*See also*: Heat and Heating.

## BIBLIOGRAPHY

Bracewell, R. N. (1989). "The Fourier Transform." Scientific American 6:62.

Grattam-Guiness, I. (1972). Joseph Fourier (1768–1830): A Survey of His Life and Work. Cambridge, MA: MIT Press.

Herivel, J. (1975). Joseph Fourier: The Man and the Physicist. New York, NY: Clarendon Press.

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