## Théorie Analytique de la Chaleur.

Paris: Firmin Didot, 1822.

First edition of the first mathematical study of heat diffusion, the first major mathematization of a branch of physics outside mechanics. “This work marks an epoch in the history of both pure and applied mathematics. It is the source of all modern methods in mathematical physics ... The gem of Fourier’s great book is ‘Fourier series’” (Cajori, *A History of Mathematics*, p. 270). “In this groundbreaking study, arguing that previous theories of mechanics advanced by such outstanding scientists as Archimedes, Galileo, Newton and their successors did not explain the laws of heat, Fourier set out to study the mathematical laws governing heat diffusion and proposed that an infinite mathematical series may be used to analyse the conduction of heat in solids: this is now known as the ‘Fourier Series’. His work paved the way for modern mathematical physics” (Introduction to the 2009 reprint by Cambridge University Press).

❧Dibner 154; Evans 37; Sparrow 68; *Landmark Writings in Western Mathematics* 26; Norman 824; *En Français dans le Texte* 232.

“There is no doubt that today this book stands as one of the most daring, innovative, and influential works of the nineteenth century on mathematical physics. The methods that Fourier used to deal with heat problems were those of a true pioneer because he had to work with concepts that were not yet properly formulated. He worked with discontinuous functions when others dealt with continuous ones, used integral as an area when integral as an anti-derivative was popular, and talked about the convergence of a series of functions before there was a definition of convergence … such methods were to prove fruitful in other disciplines such as electromagnetism, acoustics and hydrodynamics. It was the success of Fourier’s work in applications that made necessary a redefinition of the concept of function, the introduction of a definition of convergence, a reexamination of the concept of integral, and the ideas of uniform continuity and uniform convergence. It also provided motivation for the discovery of the theory of sets, was in the background of ideas leading to measure theory, and contained the germ of the theory of distributions” (González-Velasco, p. 428).

“There had been a long 18th-century debate about trigonometric series in connection with solutions to the wave equation and the shape of a vibrating string. On the one hand it seemed reasonable that a string could have any continuous initial shape — that was Euler’s view — on the other hand the equation could only be solved by functions to which the calculus applied … Fourier proposed to reopen the debate by boldly asserting that any solution to the heat equation, which he was the first to derive, could be written as an infinite sum of sines and cosines for the simple reason that any function could be written that way. This is a dramatic claim, and it was still more so in his day, because the consensus was that however broadly a function might be defined all the functions that arise in practice are finite sums of familiar ones: polynomials, sines, cosines, exponentials and logarithms, nth roots, and the like. They could also be infinite power series, and indeed infinite trigonometric series, but nonetheless they had the usual sorts of properties, such as smoothly varying graphs. No-one said so in so many words, but it is clear that the expectation was there, and Fourier in particular simply assumed that every function is continuous, as is clear from his account of the coefficients of a Fourier series … One of the dramas introduced by Fourier’s series was that they readily flout all these expectations … at various stages in the 19th century they provided fresh, and disturbing, examples of just what functions could do. Contrary to what Fourier himself believed, if Cauchy’s work began the exploration of what rigorous mathematics can do, Fourier series can indicate just what theory is up against …

“The *Théorie Analytique de la Chaleur*, in which Fourier presented his ideas, was written in several stages. He submitted a version to the Paris Academy of Sciences in 1807, but although Laplace, Lacroix and Monge were in favour of publishing it, Lagrange blocked publication, apparently because its treatment of trigonometric series differed markedly from the way he, Lagrange, had stipulated in the 1750s. Another chance came in 1810, when the Academy of Sciences announced a prize competition on heat diffusion. Fourier submitted a revised memoir, which won, but was criticised for a lack of rigour and generality. Fourier thought the criticism unfair, but revised it again, and the resulting book came out in 1822 (after Lagrange's death and when Fourier’s standing was rising in the Academy).

“Joseph Fourier was born in Auxerre, France in 1768. He was orphaned at the age of 9 and placed in the town’s military school where he learned mathematics and a sense of civic responsibility. He was nearly guillotined at the height of the Terror in 1794, but the sentence was withdrawn and Fourier was able to go to the Ecole Normale. In 1795 he was appointed an assistant lecturer at the Ecole Polytechnique, working under Lagrange and Monge, and in 1798 Monge, a prominent supporter of Napoleon, selected Fourier to go on the French expedition to Egypt. After the British defeated them there, Fourier returned to France in 1801, hoping to resume his work at the Ecole Polytechnique, but Napoleon had been impressed by his organisational talents and sent him instead to be the prefect of Governor of the Department of Isere.' He was so successful here that Napoleon made him a Baron in 1808, and in1809 he finished his contribution to the *Description d'Egypt*, a massive account and glorification of ancient Egypt based on the surveys that French engineers had made of Egyptian pyramids and other remains.

“The eventual defeat of Napoleon was the lowest point of Fourier’s life, but in 1816 he obtained a position as Director of the Bureau of Statistics for the Department of the Seine, a position which left him good time for research. His political enemies now in power delayed his appointment to the reformed Academy of Sciences for a year but he eventually rose to become the permanent secretary of the Academy in 1822 and to be elected to the Académie Française in 1827. He died in 1830 as the result of complications from an illness caught in Egypt” (Gray, pp. 13-15).

González-Velasco, ‘Connections in mathematical analysis: the case of Fourier series,’ American Mathematical Monthly 99 (1992), 427-41. Gray, *The real and the complex: A history of analysis in the 19th century*, 2015.

4to (258 x 202 mm), pp. [4], [i], ii-xxii, 639, [1: blank] and two engraved plates, contemporary half calf over marbled boards, paper flaw to lower right corner of first gathering, far outside the the printing area. A very fine and clean copy with no restoration.

Item #3841

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Price:
$32,000.00
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