Joseph Fourier

French mathematician and physicist

• Born: March 21, 1768
• Birthplace: Auxerre, France
• Died: May 16, 1830
• Place of death: Paris, France

In deriving and solving equations representing the flow of heat in bodies, Fourier developed analytical methods that proved to be useful in the fields of pure mathematics, applied mathematics, and theoretical physics.

Early Life

The twelfth child of master tailor Joseph Fourier and the ninth child of Édmie Fourier, Jean-Baptiste-Joseph Fourier (few-ree-ay) became an orphan at the age of nine. He was placed in the local Royal Military School run by the Benedictine Order and soon demonstrated his passion for mathematics. Fourier and many biographers after him attribute the onset of his lifelong poor health to his habit of staying up late, reading mathematical texts in the empty classrooms of the school. He completed his studies in Paris. He was denied entry into the military and decided to enter the Church and teach mathematics.

Fourier remained at the Benedictine Abbey of St. Benoit-sur-Loire from 1787 to 1789, occupied with teaching and frustrated that he had little time for mathematical research. Whether he left Paris because of the impending revolution or because he did not want to take his vows is uncertain. He returned to Auxerre and from 1789 to 1794 served as professor and taught a variety of subjects at the Royal Military School. The school was run by the Congregation of St. Maur, the only religious order excluded from the postrevolutionary decree confiscating the property of religious orders.

Fourier became involved in local politics in 1793 and was drawn deep into the whirlpool as internal unrest and external military threats turned the committees on which he served into agents of the Terror. Fourier made the mistake of defending a group of men who turned out to be enemies of Robespierre. He was arrested and nearly guillotined, spared only by the death of Robespierre. He became a student at the short-lived École Normale, mainly to have the opportunity to go to Paris and meet Pierre-Simon Laplace, Joseph-Louis Lagrange, and Gaspard Monge, the foremost mathematicians in France. In 1795, the École Polytechnique was opened, and Fourier was invited to join the faculty, but he was arrested once again, this time by the extreme reactionaries who hated him for his role in the Terror, even though he did much to moderate the excesses of the Terror in Auxerre. As with many other aspects of Fourier’s life during the Revolution, the exact reason for his release is unknown. In any case, he was released and occupied himself with teaching and administrative duties at the École Polytechnique.

In 1798, Fourier was chosen to be part of Napoleon I’s expedition to Egypt. Fourier was elected permanent secretary of the newly formed Institute of Egypt, held a succession of administrative and diplomatic posts in the French expedition, and conducted some mathematical research. Upon his return to France in 1801, Fourier was named by Napoleon to be the prefect of Isère, one of the eighty-four newly formed divisions of France. It is during his prefecture that Fourier began his life’s work.

Life’s Work

Fourier’s work in the development of an analytical theory of heat diffusion dates from the early nineteenth century, when he was in his early thirties, and after he had distinguished himself in administration of scientific and political institutions in Egypt. He had demonstrated a talent and passion for mathematics early, but he had not yet made significant contributions to the field. It was during whatever time he could spare from his administrative duties as prefect that he made his lasting contribution to physics and mathematics.

Fourier remained at Grenoble until Napoleon’s downfall in 1814. He turned a poorly managed department into a well-managed one in a short time. It is not clear why Fourier began to study the diffusion of heat, but in 1804 he began with a rather mathematically abstract derivation of heat flow in a metal plate. He conducted numerous experiments in an attempt to establish the laws regulating the flow of heat. He expanded the scope of problems addressed, polished the mathematical formalism, and infused physical concepts into the derivation of the equations that expressed heat flow. In 1807, he presented a long paper to the French Academy of Sciences, but opposition from Laplace and others prevented its publication. At issue was a fundamental disagreement over mathematical rigor and the underlying physical concepts.

Laplace’s first objection was that Fourier’s methods were not mathematically rigorous. Fourier claimed that any function could be represented by an infinite trigonometric series—a sum of an infinite number of sine and cosine functions each with a determinable coefficient. Such series were instrumental in Fourier’s formulation and solution of the problems of the diffusion of heat. Only later were Fourier’s methods shown to be strictly rigorous mathematically. The second objection concerned the method of derivation. Laplace preferred to explain phenomena by the action of central forces acting between particles of matter. Fourier, while not denying the correctness or the usefulness of that approach, took a different approach. Heat, for Fourier, was the flow of a substance and not some relation between atoms and their motions. He attempted, successfully, to account for the phenomenon of heat diffusion through mathematical analysis. His paper of 1807 languished in the archives of the Academy of Sciences, unpublished.

As a result of his work in Egypt and his position as permanent secretary of the Institute of Egypt, Fourier edited and wrote the historical introduction to the Description de l’Égypte (1809-1828; description of Egypt). He worked on this project from around 1802 until 1810.

A prize was offered in 1810 by the Academy of Sciences on the subject of heat diffusion, and Fourier slightly revised and expanded his 1807 paper to include discussion of diffusion in infinite bodies and terrestrial and radiant heat. Fourier won the prize, but he had faced no serious competition. The jury criticized the paper in much the same way as the 1807 paper had been criticized, and again Fourier’s work was not published. Eventually, after years of prodding, the work was published in 1815.

Fourier was probably not happy being virtually exiled from Paris, the scientific capital of France. He seemed destined to live out his days in Grenoble. With Napoleon’s abdication in April, 1814, Fourier provisionally retained his job as prefect during the transfer of power to Louis XVIII. He also managed to alter the route Napoleon took from Paris to exile in Elba, bypassing Grenoble, in order to avoid a confrontation between himself and Napoleon. Upon Napoleon’s return in March, 1815, Fourier prepared the defenses of the town and made a diplomatic retreat to Lyons. Fourier returned before completing the journey upon learning that Napoleon had made him prefect of the Rhône department. He was dismissed before Napoleon fell once again.

Fourier’s scientific work began again after 1815. One of his former pupils was now a prefect and appointed Fourier director of the Bureau of Statistics for the Seine department, which included Paris. He now had a modest income and few demands on his time. Fourier was named to the Academy of Sciences in 1817. During the next five years, he actively participated in the affairs of the Academy, sitting on commissions, writing reports, and conducting his own research. His administrative duties increased in 1822, when he was elected to the powerful position of permanent secretary of the mathematical section of the Academy. His Théorie analytique de la chaleur (1822; The Analytical Theory of Heat , 1878) differs only slightly from his 1810 essay. The papers he wrote in his later years contained little that was new. He led a satisfying academic life in his last years, but his health began to deteriorate. His rheumatism had returned, he had trouble breathing, and he was sensitive to cold. Fourier died from a heart attack in May, 1830.

Significance

The core of Joseph Fourier’s scientific work is The Analytical Theory of Heat. This work is basically a textbook describing the application of theorems from pure mathematics applied to the problem of the diffusion of heat in bodies. Fourier was able to express the distribution of heat inside and on the surface of a variety of bodies, both at equilibrium and when the distribution was changing because of heat loss or gain.

Fourier significantly influenced three different fields: pure mathematics, applied mathematics, and theoretical physics. In pure mathematics, Fourier’s most lasting influence has been the definition of a mathematical function. He realized that any mathematical function can be represented by a trigonometric series, no matter how difficult to manipulate the function may appear. Some scholars single out this concept as the stepping-stone to the work of pure mathematicians later in the century, which resulted in the modern definition of a function. Additional influences are that of a clarification of a notational issue involving integral calculus and properties of infinite trigonometric series.

Applied mathematics has been influenced to a great extent by Fourier’s use of trigonometric series and techniques of integration. The class of problems that Fourier series and Fourier integrals can solve extends far beyond diffusion of heat. His methods form the foundation of applied mathematics techniques taught to undergraduates. Some mathematicians before him had used trigonometric series in the solutions of problems, but the clarity, scope, and rigor that he brought to the field were significant.

Fourier’s influence in theoretical physics is more subdued, perhaps because of the completeness of his results. There was little room for others to extend the physical aspects of Fourier’s work—his results did not need extending. Other branches of physics appear to have been influenced by his approach, and a direct influence on the issue of determining the age of the earth by calculating its heat loss has been documented.

Bibliography

Bell, Eric T. The Development of Mathematics. 2d ed. New York: McGraw-Hill, 1945. Presents a narrative history of the decisive epochs in the development of mathematics without becoming overly technical. The majority of references to Fourier appear in chapter 13.

‗‗‗‗‗‗‗. Men of Mathematics. New York: Simon & Schuster, 1986. First published in 1937, this book remains one of the best accounts of the history of mathematics. The contributions of Fourier are discussed in chapter 12, “Friends of an Emperor.”

Fourier, Joseph. The Analytical Theory of Heat. Translated by Alexander Freeman. 1878. Reprint. Mineola, N.Y.: Dover, 2003. Fourier’s preliminary discourse to his most famous work explains in clear terms what he is attempting in the work. Devoid of technical matters, this book offers the reader a glimpse of why Fourier has achieved the status he has.

Fox, Robert. “The Rise and Fall of Laplacian Physics.” Historical Studies in the Physical Sciences 4 (1974): 89-136. This paper presents a description of the research program of Laplace, which dominated French science at one of its most successful periods, from 1805 to 1815. Fourier led the revolt against this program.

Friedman, Robert Marc. “The Creation of a New Science: Joseph Fourier’s Analytical Theory of Heat.” Historical Studies in the Physical Sciences 8 (1977): 73-100. This paper concentrates on conceptual and physical issues rather than the mathematical aspects stressed in most older works. Also discusses how Fourier’s philosophy of science compared to that of his contemporaries.

Grattan-Guinness, Ivor, with J. R. Ravetz. Joseph Fourier, 1768-1830: A Survey of His Life and Work, Based on a Critical Edition of His Monograph on the Propagation of Heat, Presented to the Institute de France in 1807. Cambridge, Mass.: MIT Press, 1972. Intertwines a close study of Fourier’s life and work with a critical edition of his 1807 monograph. The 1807 monograph is in French, but everything else is in English. Contains a bibliography of Fourier’s writings, a list of translations of his works, and a secondary bibliography.

Herivel, John. Joseph Fourier: The Man and the Physicist. Oxford, England: Clarendon Press, 1975. Although this work does not claim to be the definitive biography of Fourier, it goes much further than any other work written in English. Although the book is almost devoid of technical detail, the prospective reader would benefit from a knowledge of the history of France from 1789 to 1830.

James, Ioan. Remarkable Physicists: From Galileo to Yukawa. New York: Cambridge University Press, 2004. This collection of brief biographies of famous physicists contains a five-page biography of Fourier. Written in a nontechnical style for readers with limited knowledge of science.

Purrington, Robert D. Physics in the Nineteenth Century. New Brunswick, N.J.: Rutgers University Press, 1997. Fourier is mentioned in several places, and his work is placed in a historical context, in this survey of nineteenth century physics; references to Fourier are listed in the index. Chapter 4, “Heat and Thermodynamics,” contains information about the Fourier series.