# The Most Influential

# People in History

## Jean-Baptiste-Joseph Fourier

/ FOO-ryeh /

1768–1830

French Mathematician and Egyptologist

###### Mathematics Ranking 33rd of 46

Issued by Liechtenstein in 2014. Fourier series formula at bottom.

Jean-Baptiste-Joseph Fourier exerted a strong influence on mathematical physics through his The Analytical Theory of Heat (Théorie analytique de la chaleur) published in 1822. He showed how the conduction of heat in solid bodies may be analyzed in terms of infinite mathematical series now called by his name, the Fourier series. Far transcending the particular subject of heat conduction, his work stimulated research in mathematical physics, encompassing many natural occurrences such as sunspots, tides, and the weather. With the exception of calculus, the Fourier series have had a greater impact on the development of mathematical physics than almost any other branch of mathematics.(1) His work also had a great influence on the theory of functions of a real variable, one of the main branches of modern mathematics.(2) Fourier was also a noted Egyptologist leading the expedition in 1799 that discovered the Rosetta stone which led to the decipherment of Egyptian hieroglyphics.

Bio

Fourier, the son of a tailor, first attended the local military school conducted by Benedictine monks. He showed such proficiency in mathematics in his early years that he later became a teacher in mathematics at the same school. The ideals of the French Revolution then carried him into politics, and more than once his life was in danger. When the École Normale was founded in 1794 in Paris, he was among its first students, and, in 1795, he became a teacher there. The same year, after the École Polytechnique was opened, he joined its faculty and became a colleague of Gaspard Monge and other mathematicians.

In early 1798, the French government named Napoleon Bonaparte to head a military expedition into Egypt. Napoleon asked Monge and the chemist Claude-Louis Berthollet to choose the scientific members of the expedition. In May, Monge chose his pupil and colleague Fourier. Three months later, Fourier was named perpetual secretary of the Institute d’Égypte. Napoleon took the expedition to Egypt in late July 1798 and then left in August 1799 to return to Paris to seize power as First Consul. A few months before he left, Napoleon named Fourier leader of one of two scientific expeditions that spent the autumn of 1799 investigating the monuments and inscriptions in Upper Egypt. Fourier’s expedition team discovered the Rosetta stone in July 1799. Jean Francois Champollion’s decipherment of the Rosetta stone in 1822 suddenly allowed researchers to translate the hieroglyphics found in Egyptian antiquities.

Fourier remained in Egypt with the expedition for two more years until the Anglo-French peace treaty of 1801 forced the French to withdraw (and relinquish the Rosetta stone to the English). The Rosetta stone now resides in the British Museum in London, England. Fourier virtually governed half of Egypt around 1800.

After his return to France, Fourier was charged with the publication of the enormous mass of Egyptian materials. This became the Description de l’Égypte, to which he also wrote a lengthy historical preface on the ancient civilization of Egypt. He was also appointed an administrator for the national government of the Isère department, a position he held from 1802 to 1814, with his headquarters at Grenoble. He showed great administrative ability, as in directing the drainage of swamps, while continuing his Egyptological and mathematical work. In 1809 Napoleon made him a baron. Following Napoleon’s fall from power in 1815, Fourier was appointed director of the Statistical Bureau of the Seine, allowing him a period of quiet academic life in Paris. In 1817 he was elected to the Académie des Sciences, of which, in 1822, he became perpetual secretary (executive director). Because of his work in Egyptology he was elected in 1826 to the Académie Française and the Académie de Médecine.

Fourier Series

The story of the Fourier series is somewhat complicated because the idea of solving certain types of equations, particularly partial differential equations, in terms of sines and cosines, goes back to d’Alembert, Euler, and Daniel Bernoulli.

Mathematicians as far back as the mid-eighteenth century thought of using sines and cosines because they were trying to describe the way a string vibrates (e. g., a violin string) when it is plucked. If we plot the distortion of the string when we first pull upward on it, we can express the shape of the string in an x, y Cartesian coordinate system as y = f(x), where y is the height of any point x (the x coordinate of the point) of the string above the horizontal. If we let go of the string, every point on it moves, each with its own speed, toward the horizontal so that the string becomes distorted. The function y = f(x) thus becomes distorted and this distortion changes with time. Hence, the function f(x) changes from moment to moment as well as from point to point. We must therefore write this function as y = f(x, t) so that y is a function of two changing variables x and t. The equation of motion that represents the motion of each point on the string is called a partial differential equation and its solution f(x, t) shows how the shape changes from moment to moment.

Before Fourier began his analysis of the mathematics of a vibrating string, Bernoulli and Euler noted that the mathematical forms of such vibrations are similar to the graphs of the sine and cosine of the variable x and so they tried to represent the function of the vibrating string by the sum of products of sines and cosines without much success. Lagrange went further than that and almost arrived at Fourier’s great discovery.

Fourier did not begin his great synthesis purely as a mathematical exercise but as the solution of a physical problem—the conduction of heat in a solid. Heat flows in a solid because of the temperature difference from point to point and from moment to moment so that the equation describing the way the temperature changes is a partial differential equation for the temperature. Fourier saw that the solution of this equation can be expressed as a product of a function of time and a function of position x. He then showed that the function of x can be written as an infinite series of sines and cosines of x.

Although Fourier developed his trigonometric series to solve physical problems such as heat conduction, his most important contribution to the general mathematical theory of trigonometric series was to show that any part of a function can be expanded into a Fourier series.(3) Thus if the function y = f(x) is plotted in the x, y plane and a piece of it between two values of x, x1, and x2, is considered separately, this piece can be represented as the sum of an infinite number of sines and cosines within this x strip between x1 and x2.

The importance of the Fourier series for physical phenomena can be understood from the simplification it introduces in analyzing sounds. No matter how complex a sound may be, it can be expressed as a sum of pure sounds (simple notes) which are just sine and cosine vibrations (single frequencies), each of which is called a harmonic. Thus, the Fourier series is really a harmonic analysis of any phenomena that can be represented as a functional relationship between two different entities.(4)

The Fourier series led to another important concept—the Fourier integral, which is even more important in physics than the Fourier series.(5) This integral is a simple and elegant mathematical procedure that enables us to transform the emphasis from one frame of reference to another. In considering the motion of a particle, for example, we may describe this motion in a spatial frame of reference—that is, in terms of the spatial path of the particle in a three-dimensional Cartesian coordinate system x, y, and z, or in what physicists call a momentum frame. A momentum frame is a three-dimensional frame, each point of which represents the three quantities mvx, mvy, and mvz (called px, py, and pz). The symbol m is the mass of the moving particle and vx, vy, and vz are the particle’s velocities in the x, y and z directions, where the position of the particle in space is at the point x, y, z. The quantities px, py, and pz are the components of the particle’s momentum at the point x, y, z. The Fourier integral shifts our analysis from the coordinate space of the particle to its momentum space. This is very useful in studying the dynamics of a particle, particularly in quantum mechanics.

Footnotes:

(1) Lloyd Motz and Jefferson Hane Weaver, The Story of Mathematics (New York, 1993), p. 223.

(2) Dirk Jan Struik, “Joseph Fourier,” Encyclopaedia Britannica, May 12, 2019, https://www.britannica.com/biography/Joseph-Baron-Fourier.

(3) Motz and Weaver, p. 224.

(4) Ibid., p. 224.

(5) Ibid., p. 224.

Key Reference:

Who Is Fourier? – A Mathematical Adventure originally published in Japanese, translated into English by Alan Gleason, 1995.

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