OEuvres de Fourier. Edited by Jeon Gaston Darboux.Reprint of the 1888 original. Cambridge University Press, Cambridge, 2013

A useful reprint of the selected mathematical works of Joseph Fourier (1768-1830) has been made. The original edition of 1888-90 of the Oeuvres de Fourier was part of the project of the French Ministry of Education (Ministère de l’Instruction Publique) to make available the works of eminent scientists. The aforementioned followed the publication of the works of Laplace, Lavoisier, Fresnel and Cauchy. The first volume is entirely made up of the reprint of the "Théorie analytique de la Chaleur" (Paris, Didot, 1822) in which the editors limited themselves only to correction of typographic errors due to a few inaccuracies in the calculations. Fourier’s greatest work collected and co-ordinated his research in the field of the propagation of heat, begun in 1807 with his celebrated memoir in which he introduced the partial differential equation which describes heat diffusion in solid bodies. The mathematical tool for the study of heat diffusion are trigonometric series, which had already appeared in the mid eighteenth century in the study of the equation of a vibrating string by d’Alembert, Euler, Daniel Bernoulli and Lagrange, which Riemann then named Fourier series. The possibility to represent a function by means of a trigonometric series had, however, been rejected by Euler and Lagrange, and Fourier’s memoir of 1807 remained unedited for a long time since it was not considered rigorous enough. In the second decade of the nineteenth century, however, the heat equation discovered by Fourier received the first signs of recognition. The study of this equation with further mathematical developments converged in the "Théorie analytique" in 1822. The rigorous proof of the convergence of the Fourier series became a subject of investigation carried out by important mathematicians of the nineteenth century: from Dirichlet, to Riemann, to Cantor, and became the main field of the development of real analysis, starting from the integration theory. The "Théorie analytique de la Chaleur" also includes many physical considerations and extends the study of the heat equation to infinite three-dimensional solid bodies. Having established the principles and general equations, the method is progressively applied to the propagation of heat in an infinite rectangular plate (chapter III), then, in the following chapters, in solid bodies (annulus, sphere, cylinder and prism of infinite length, and cube), and in chapter IX in a homogeneous unlimited mass in all directions. This required another mathematical tool, the Fourier transform, which had already been introduced in his previous memoirs of 1811, 1817, 1818, but had also been investigated by Laplace, and even as far back as a memoir by Clairaut of 1754 on the apparent orbit of the Sun. Fourier’s main work was translated in English, with additional notes, in 1878 by A. Freeman (The Analytical Theory of Heat, Cambridge University Press). For the second volume see : MR3470071 The second volume of the Oeuvres de Fourier collects selected memoirs of Fourier, ordered in the same way as the edition of the Oeuvres de Lagrange (1867-1892), edited starting from 1885 once more by Darboux himself, not in chronological order but grouped according to collections, journals or series in which they had originally been included. Those on the analytic theory of heat constitute completion of the treatise "Théorie analytique de la chaleur," and, in particular concern the temperature of the Earth and the planetary spaces. Another series of works is linked to the theory of numeric equations. The volume commences from five memoirs printed in the volumes of the "Mémoires de l’Académie des Sciences de l’Institut de France" over the period from 1826 to 1831. There follow seven memoirs or extracts published in the "Bulletin de la Société Philomatique" (between 1808 and 1826), among which one on the approximation of the roots of an algebraic equation and a memoir on the vibration of elastic laminae and the motion of waves, then four memoirs taken from the "Annales de Chimie et de Physique" (in the period from 1817 to 1828) in which the physical and experimental aspects of radiant heat are mostly developed. Volume II proceeds with the memoir on statics which contains the demonstration of the principle of virtual velocities and the theory of moments ("Journal de l’Ecole Polytechnique", 1798), and two memoirs on the mean results deduced from a large number of observations and the measurement errors (1826, 1828). Finally, there is a memoir on the heat transfer in fluids (published posthumously in the "Mémoires de l’Académie des Sciences" in 1833), and a report on tontines (1821), as a supplement to the first section. The studies on tontines and observational errors recall Fourier’s activity in the Restoration Period as an employee of the office of statistics of the Paris district. Fourier’s name is connected to another field of mathematics: a generalization of Descartes’ rule of signs to determine the position of the roots of an algebraic equation. Fourier’s results, dating back to the first years of his mathematical research, and already present in his lessons on analysis at the École Polytechnique, were published only in part posthumously by Navier ("Analyse des équations déterminées", Première partie, Paris, Didot, 1830). They did not appear in the "Oeuvres", except for the memoir published in the "Bulletin de la Société Philomatique" in 1820 containing the so-called Fourier-Budan theorem. Several other scientific and literary writings by Fourier were not printed in the "Oeuvres". Fourier co-ordinated the great work on Egypt, "Description de l’Egypte", for which he wrote the substantial "Preface" and the "Recherches sur les sciences et le gouvernement de l’Egypte". Like d’Alembert, he was a member of the Académie Française. Moreover, as the secretary of the Académie des Sciences, he wrote reports and eulogies. The two volumes do not include Fourier’s correspondence, which is interesting also from a political point of view, particularly with reference to the mission in Egypt (1798) and the period when he was Prefect of Isère (Grenoble). Most of Fourier’s manuscripts are preserved in the Bibliothèque Nationale in Paris (mss Fonds français nn. 22501-22529). The two volumes do not contain a biography of Fourier, for which the editor refers to the eulogy written by Arago and the "Discours" by Cousin; however, volume II presents a brief but interesting introduction ('avertissement') and a chronological list of Fourier’s published scientific works. Fourier’s works were the subject of in-depth studies in the last century starting with Ivor Grattan-Guinness ("Joseph Fourier 1768-1830", The MIT Press, Cambridge, Mass.-London, 1972, MR0419139) and John Herivel ("Joseph Fourier. The Man and the Physicist", Clarendon Press, Oxford, 1975). The funereal monument to Fourier is to be found in Père Lachaise Cemetery in Paris; his protégé, Champollion, who was the first to decipher the ancient Egyptian hieroglyphic writing, was buried next to him.