# Joseph Fourier

Joseph Fourier

Jean Baptiste Joseph Fourier
Born

March 21, 1768
Auxerre, Yonne, France

Died May 16, 1830
Residence France
Nationality French
Field Mathematician, physicist, and historian
Institutions École Normale
École Polytechnique
Alma mater École Normale
Notable students  Gustav Dirichlet
Giovanni Plana
Claude-Louis Navier
Known for Fourier transform
Religious stance Roman Catholic

Jean Baptiste Joseph Fourier (March 21, 1768 – May 16, 1830) was a French mathematician, physicist and government administrator during the reign of Napoleon who is best known for his study of heat conduction, and for using series of trigonometric functions, now called Fourier series, to solve difficult mathematical problems. Early in life, he contemplated becoming a Benedictine monk, but joined the French Revolution instead.

## Early years

Fourier was born at Auxerre in the Yonne département of France, the son of a tailor. He was orphaned at an early age. When he turned eight, he was recommended to the bishop of Auxerre, and through this introduction, he was educated in a military school run by the Benedictines of the Convent of St. Mark. By age 13, he had been introduced to higher mathematics, and it is said that his enthusiasm for the subject was such that he gathered the wax from candle ends so he could continue his studies through the night.

## Professional life

Fourier had hoped to pursue a career in the military, but was turned down on the pretext that he was not of noble birth. He then prepared for a life as a Benedictine monk. He attached himself to the Abbey of St. Benoit-sur-Loir with the purpose of officially entering the order. The rumblings of the French Revolution caused him to give this vocation up, and in its place, he accepted a chair in mathematics at the Military School of Auxerre. In 1789, he read a paper, On the Resolution of Numerical Equations of All Degrees, before the French Academy of Sciences. This paper introduced novel ways to arrive at solutions to equations where the unknown is raised to higher powers.

Fourier took an active role the French Revolution, as he was keenly attracted to its egalitarian ideals. He was at odds with the bloody turn the revolution took, and warned an acquaintance that a tribunal was seeking to pass judgment on him. For this, Fourier was briefly imprisoned, but managed to escape what would normally have been a certain death sentence at the time.

In 1795 Fourier was assigned for teacher-training at the École Normale Supérieure, established by the convention to create replacements for clerical instructors in local schools throughout France. Among the teachers at the institute were famed mathematicians Pierre-Simon Laplace and Joseph-Louis Lagrange. The style of teaching promoted at the institute was anti-autocratic, and encouraged dialog between students and teachers. After this training, Fourier assumed a chair at the École Polytechnique.

## Work in Egypt

Fourier went with Napoleon on his eastern expedition in 1798 as part of the Egyptian Institute, organized as a cultural research organization, but also designed to collect intelligence about the local culture. Fourier was assigned to the mathematics section, of which Napoleon was himself a member, and eventually assumed the post of perpetual secretary for the organization, while submitting several papers on mathematics for its proceedings. He was later made governor of Lower Egypt.

During this period, Fourier acted with great tact and diplomacy, and became a personal favorite of Napoleon. After the British victories and the capitulation of the French under General Menou in 1801, Fourier returned to France, and on January 2, 1802, was made prefect of Isère, based in Grenoble. As prefect, he acted to bring peace among warring political factions, and promoted engineering projects such as the drainage of swamps to create fertile farmland. It was while holding this office that he made his experiments on the propagation of heat. Also during this time, he saved Jean Francois Champollion, the scholar credited with deciphering the Rosetta Stone, from induction into the military by pleading on his behalf for a special exemption.

When he was deposed for the first time and exiled in Alba, Napoleon attempted to retain power by forming an army, which, to Fourier's great embarrassment, was headed for Grenoble, Fourier having recommended allegiance to the king. In March 1815, Napoleon had Fourier arrested and brought to his headquarters, where he expressed disappointment that Fourier did not support his return to power. He removed Fourier from his post in Grenoble, but a few days later, appointed him to a new position as prefect of the Rhone with an annual salary of six thousand francs. Napoleon was soon ousted again, and Fourier never collected the salary.

## Theory of Heat

In 1807, Fourier published the first account of his theory of heat, which he submitted to the French Academy of Sciences. His theory basically demonstrates the manner in which heat moves through a body if the various heat sources, initial temperatures, heat conductivity of the body's interior and the radiating characteristics of the body's surface are known. The Academy then offered an award for the further mathematical development of the theory. While his 1811 submission for this prize was recognized, the conclusions it contained were criticized by some of the leading French mathematicians of the day for lack of rigor, a characterization that Fourier protested. Others insisted that Fourier had failed to credit Biot for work completed in 1804, while another group said they had developed a superior exposition of what was basically the same material.

The controversies delayed complete recognition of his work, which he finally published in 1822 under the title The Analytic Theory of Heat. In this exposition, Fourier bases his analysis on the premise, originally proposed by Isaac Newton, that the flow of heat between two adjacent parts of a solid is proportional to the extremely small difference of their temperatures.

In his 1822 work, Fourier pioneered the application of what are commonly known as Fourier series to the problems of heat transfer. A Fourier series is a series whose terms are composed of trigonometric functions. Fourier showed that most functions can be represented by such a series.

While questions still remain about his precise contribution to mathematics, there can be no doubt that his theory of heat and the mathematical tools he used to describe it were extremely influential to later scientists. While Fourier solved many of the problems of heat flow in a solid, and derived equations for its description, later researchers, including Georg Ohm and William Thomson (Lord Kelvin), applied his analysis to describe electrical phenomena such as the distribution of electrical fields and the flow of electric current in a conductor.

## Later years

In 1817, Fourier was nominated for membership in the French Academy of Sciences, but his political history blocked his election. Circumstances were improved in 1822, when he accepted a post in the physics section of the Academy. He soon after was appointed perpetual secretary of the Institute of France, and in 1827 he was elected to the Academy's membership.

In his later years, Fourier, who took up residence in Paris, suffered from rheumatism. To combat the affliction, he kept his living quarters heated even in summer. In his last days, he suffered from shortness of breath attributed to heart disease. His poor health was aggravated by a fall sustained on May 4, 1830. He refused treatment, but on the day of his death, May 16, called a physician to assist him, shortly thereafter succumbing to his illness.

## Contributions in Algebra

Fourier left an unfinished work on determinate equations which was edited by Claude-Louis Navier and published in 1831. This work contains much original matter—in particular, there is a demonstration of Fourier's theorem on the position of the roots of an algebraic equation. Joseph Louis Lagrange had shown how the roots of an algebraic equation might be separated by means of another equation whose roots were the squares of the differences of the roots of the original equation. François Budan, in 1807 and 1811, had enunciated the theorem generally known by the name of Fourier, but the demonstration was not altogether satisfactory. Fourier's proof is the same as that usually given in textbooks on the theory of equations. The final solution of the problem was given in 1829 by Jacques Charles François Sturm.

## Other work

Fourier is credited with the discovery in his essay in 1827 that gases in the atmosphere might increase the surface temperature of the Earth.[1] This was the effect that would later be called the greenhouse effect. He established the concept of planetary energy balance—that planets obtain energy from a number of sources that cause temperature increase. Planets also lose energy by infrared radiation (that Fourier called "chaleur obscure" or "dark heat") with the rate increasing with temperature. A balance is reached between heat gain and heat loss; the atmosphere shifts the balance toward the higher temperatures by slowing the heat loss. Although Fourier understood that rate of infrared radiation increases with temperature, the Stefan-Boltzmann law which gives the exact form of this dependency (a fourth-power law) was discovered fifty years later.

Fourier recognized that Earth primarily gets energy from solar radiation, to which the atmosphere is transparent, and that geothermal heat doesn't contribute much to the energy balance. However, he mistakenly believed that there is a significant contribution of radiation from interplanetary space. He believed the temperature of interplanetary space was 50 or 60 degrees below freezing.

Fourier referred to an experiment by M. de Saussure, who exposed a black box to sunlight. When a thin sheet of glass is put on top of the box, the temperature inside of the box increases.[2] Infrared radiation was discovered by William Herschel twenty five years later.

## Example of a Fourier Series

In Fourier's most famous work, Memoir on the Propagation of Heat in Solid Bodies, one finds the following mathematical expression, which is a simple example of what is now called a Fourier series:

${\displaystyle \varphi (y)=a\cos {\frac {\pi y}{2}}+a'\cos 3{\frac {\pi y}{2}}+a''\cos 5{\frac {\pi y}{2}}+\cdots .}$

The dots at the end of the series mean that the terms of the series continue indefinitely in the same pattern shown.

Although other mathematicians had used trigonometric series, Fourier appears to have been the first to realize that they could be used to represent functions of an arbitrary nature, including discontinuous functions.

## Footnotes

1. Spencer Weart, “The Carbon Dioxide Greenhouse Effect,” Center for History of Physics of the American Institute of Physics. Retrieved September 17, 2007.
2. Joseph Fourier, “Memoire sur les temperatures du globe terrestre et des espaces planetaires,” Trans. W. M. Connolley. Retrieved September 18, 2007.

## References

• Arago, F. 1846. Eloge Historique de Joseph Fourier (summarized with additional commentary). North British Review 4:380-412.
• Arago, F. 1859. Biographies of Distinguished Scientific Men. Trans. W. Smyth, B. Powell, R. Grant. Boston: Ticknor and Fields.
• Asimov, Isaac. 1982. Asimov's Biographical Encyclopedia of Science and Technology, 2nd ed. New York: Doubleday. ISBN 0385177712
• Fourier, J.-B.J. 1827. Mémoires de l'Académie Royale des Sciences de l'Institut de France VII: 570-604.
• Gillispie, Charles Coulston. 1975. Dictionary of Scientific Biography. New York: Scribner. ISBN 0684101211
• Patton, A. A. 1863. A History of the Egyptian Revolution. London: Trubner.
• Porter, Roy, and Marilyn Ogilvie, eds. 2000. The Biographical Dictionary of Scientists, 3rd ed. New York: Oxford University Press. ISBN 0195216636