Joseph Fourier

• 1. Introduction

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  Joseph Fourier fell down a flight of stairs on the 4th of May 1830. He had been living with a heart condition since his days in Egypt and Grenoble, but that fall aggravated the malady beyond what anyone had feared. Twelve days later, he died in his bed. He was sixty-two years old.

  Fourier left behind a body of work so far-reaching that today's students of mathematical physics still learn his equations in their first courses. His name is carved into the Eiffel Tower, one of only seventy-two inscribed there. His tomb in Pere Lachaise Cemetery is decorated with an Egyptian motif. That detail alone raises questions. Why would a mathematician from Burgundy be buried beneath Egyptian ornamentation? What sent a tailor's orphaned son into Napoleon's inner circle of scientific advisers? And how did a man studying the flow of heat in solid bodies stumble onto the first scientific description of the greenhouse effect?

  Those questions trace back to Auxerre, in Burgundy, where Fourier was born on the 21st of March 1768.

• 2. An Orphan's Education

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  Fourier's father was a tailor in Auxerre, and Fourier lost both parents by the age of nine. The path forward came through the Bishop of Auxerre, who recommended the boy to the Benedictine Order of the Convent of St. Mark. The monks gave him an education that would shape the rest of his life.

  By the time Fourier sought a military career, he ran into a structural wall. Commissions in the scientific corps of the army were reserved for those of good birth. Fourier was not of good birth, and so he was ineligible. He accepted a military lectureship on mathematics instead, a position that kept him close to the intellectual world he wanted but shut him out of the institutional one.

  The French Revolution changed that calculus. Fourier took a prominent part in promoting the revolution in his own district, serving on the local Revolutionary Committee. The Terror that followed, however, landed him in prison briefly. His release came in 1795, when he was appointed to the Ecole Normale. He then succeeded Joseph-Louis Lagrange at the Ecole Polytechnique, a mathematician of the first rank having handed him the keys to one of France's most prestigious scientific posts.

• 3. Napoleon's Scientific Adviser

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  In 1798, Fourier sailed to Egypt alongside Napoleon Bonaparte as a scientific adviser to the expedition. Napoleon appointed him secretary of the Institut d'Egypte, the scholarly body Napoleon had founded at Cairo with a view of weakening British influence in the East.

  When the British fleet cut the French forces off from France, Fourier's role shifted. He organized the workshops on which the French army depended for their munitions of war. He was simultaneously a mathematician contributing papers to the Cairo Institute and a logistician keeping an army supplied in a foreign land.

  The British victories eventually forced the capitulation of the French under Jacques-Francois Menou in 1801, and Fourier returned to France. Napoleon had not finished directing his life. Fourier had resumed his academic post at the Ecole Polytechnique when Napoleon noted, in a remark that effectively ended the matter, that the Prefect of the Department of Isere had recently died and that he wished to appoint citizen Fourier to the place. Faithful to Napoleon, Fourier accepted. He moved to Grenoble as Governor of the Department of Isere, where he oversaw road construction and other projects. It was in Grenoble, in the margins of an administrative career, that Fourier began to experiment on the propagation of heat. He also contributed to the monumental Description de l'Egypte, the vast scholarly record of the Egyptian expedition.

• 4. The Theory of Heat

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  On the 21st of December 1807, Fourier presented his paper On the Propagation of Heat in Solid Bodies to the Paris Institute. It would take another fifteen years before he published the full treatise, the Theorie analytique de la chaleur, in 1822.

  The book rested on Newton's law of cooling: the flow of heat between two adjacent particles is proportional to the extremely small difference of their temperatures. From that foundation, Fourier built three major contributions, one purely mathematical and two physical. His mathematical claim was that any function of a variable, whether continuous or discontinuous, can be expanded in a series of sines of multiples of the variable. The claim as stated was not entirely correct without additional conditions, but his observation that some discontinuous functions are the sum of infinite series was a genuine breakthrough. Joseph-Louis Lagrange had touched on particular cases of the same idea but had not pursued the subject. Peter Gustav Lejeune Dirichlet was the first to provide a satisfactory demonstration, with some restrictive conditions. The question of determining when such a series converges has remained a fundamental problem across the centuries since.

  The two physical contributions were equally consequential. Fourier advanced the concept of dimensional homogeneity in equations: a formula can be formally correct only if the dimensions match on either side of the equality sign. His work in this area made important contributions to dimensional analysis. He also proposed the partial differential equation for conductive diffusion of heat, now known simply as the heat equation, which is taught to every student of mathematical physics and is the most basic example of a parabolic partial differential equation.

  Fourier's treatment of heat diffusion in a cylinder inspired Joseph Liouville and Jacques Charles Francois Sturm to develop what became the Sturm-Liouville theory. Fourier also gave a thorough discussion of Bessel functions years before Friedrich Wilhelm Bessel himself took up the subject. The Theorie was translated into English fifty-six years after its publication, by Freeman in 1878, with editorial corrections. Mathematician Jean Gaston Darboux edited and republished the French edition in 1888.

• 5. Roots and Early Mathematics

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  Before heat consumed his attention, Fourier made a mark in a different corner of mathematics. In his youth he discovered an inductive proof of Descartes' rule of signs, a result governing the possible number of positive real roots a polynomial can have.

  He later worked on determining and locating the real roots of polynomials, a project he never finished. After his death, Claude-Louis Navier edited and published the unfinished work in 1831. It contained, among other original material, Fourier's theorem on polynomial real roots, which Fourier had published in 1820. Francois Budan had published an independently derived version of a very close theorem in 1807 and again in 1811. The two results are so closely related that each is a corollary of the other, and Budan's theorem is sometimes listed under Fourier's name as well. Fourier's proof was the version that appeared in most nineteenth-century textbooks on the theory of equations. A complete resolution of the underlying problem arrived in 1829, when Sturm provided a general solution.

• 6. Discovering the Greenhouse Effect

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  In the 1820s, Fourier turned to a puzzle about planetary temperatures. He calculated that an object the size of the Earth, placed at the Earth's distance from the Sun, should be considerably colder than the planet actually is if it were warmed only by incoming solar radiation. The gap between his calculation and the observed temperature was large: thirty-three degrees.

  He published articles on the problem in 1824 and 1827, examining possible sources of the additional heat. Because of that large thirty-three-degree discrepancy, Fourier ultimately made a mistaken conclusion: he believed that a significant portion of the additional warmth came from radiation originating in interstellar space. That specific inference was wrong.

  What was not wrong was his consideration of the atmosphere as a possible insulator. In his articles, Fourier drew on an experiment conducted by Horace Benedict de Saussure. De Saussure had lined a vase with blackened cork and inserted several panes of transparent glass separated by intervals of air. Midday sunlight entered through the glass. The temperature climbed higher in the interior compartments than in the outer ones. Fourier reasoned that if gases in the atmosphere could form a stable barrier analogous to those glass panes, they would produce a similar warming effect on planetary temperatures. He was careful to note that the actual mechanisms in the atmosphere included convection, which de Saussure's device did not replicate. Fourier never used the phrase "greenhouse effect"; that metaphor came later. His consideration of the idea, however, is widely recognized as the first scientific proposal of the process. In 1830, the year Fourier died, he was elected a foreign member of the Royal Swedish Academy of Sciences.