# Joseph Fourier

Profound study of nature is the most fertile source of mathematical discoveries.

Jean Baptiste Joseph Fourier (March 21, 1768May 16, 1830) was a French mathematician and physicist who is best known for initiating the investigation of Fourier series and their application to problems of heat flow. The Fourier transform is also named in his honor.

## Quotes

This difficult science is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind.

### The Analytical Theory of Heat (1878)

As translated by Alexander Freeman - Full text online
• The analytical equations, unknown to the ancient geometers, which Descartes was the first to introduce into the study of curves and surfaces, are not restricted to the properties of figures, and to those properties which are the object of rational mechanics; they extend to all general phenomena. There cannot be a language more universal and more simple, more free from errors and from obscurities, that is to say more worthy to express the invariable relations of natural things.
Considered from this point of view, mathematical analysis is as extensive as nature itself; it defines all perceptible relations, measures times, spaces, forces, temperatures; this difficult science is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind.
Its chief attribute is clearness; it has no marks to express confused notions. It brings together phenomena the most diverse, and discovers the hidden analogies which unite them.
• Preliminary Discourse, p.7 Note: often quoted as Mathematics [or mathematical analysis] compares the most diverse phenomena and discovers the secret analogies that unite them.
• Primary causes are unknown to us; but are subject to simple and constant laws, which may be discovered by observation, the study of them being the object of natural philosophy.
Heat, like gravity, penetrates every substance of the universe, its rays occupy all parts of space. The object of our work is to set forth the mathematical laws which this element obeys. The theory of heat will hereafter form one of the most important branches of general physics.
• Ch. 1, p. 1
• If we consider further the manifold relations of this mathematical theory to civil uses and the technical arts, we shall recognize completely the extent of its applications. It is evident that it includes an entire series of distinct phenomena, and that the study of it cannot be omitted without losing a notable part of the science of nature.
The principles of the theory are derived, as are those of rational mechanics, from a very small number of primary facts, the causes of which are not considered by geometers, but which they admit as the results of common observations confirmed by all experiment.
• Ch. 1, p. 6
• Profound study of nature is the most fertile source of mathematical discoveries.
• Ch. 1, p. 7

• He carried on his elaborate investigations on the propagation of heat in solid bodies, published in 1822 in his work entitled La Theorie Analytique de la Chaleur. This work marks an epoch in the history of mathematical physics. "Fourier's series" constitutes its gem. By this research a long controversy was brought to a close, and the fact established that any arbitrary function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807 before the French Academy. The trigonometric series ${\displaystyle \sum _{n=0}^{n=\infty }(a_{n}\sin nx+b_{n}\cos nx)}$ represents the function ${\displaystyle \phi (x)}$ for every value of ${\displaystyle x}$ if the coefficients ${\displaystyle a_{n}={\frac {1}{\pi }}\int _{-\pi }^{\pi }\phi (x)\sin nx\,dx}$, and ${\displaystyle b_{n}}$ be equal to a similar integral. The weak point in Fourier's analysis lies in his failure to prove generally that the trigonometric series actually converges to the value of the function.