The analytical theory of heat and Fourier series

1822 (1807) Energy and motion theory

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An analytical theory of heat conduction: a partial differential equation together with a general way of expanding an arbitrary initial temperature distribution into a trigonometric series.

It was built in Théorie analytique de la chaleur (1822) by the Frenchman Joseph Fourier. The theory deliberately rests on no hypothesis about what heat is: it relates only measurable quantities (heat flux and temperature gradient), and for that reason it survived the collapse of the caloric theory.

Its mathematical form became a template for other sciences: William Thomson carried it over to electrostatics, Ohm to current in a conductor, Fick to diffusion.

Why this date

The date is that of the first publication of the book: Paris, 1822. The memoir on the propagation of heat was submitted to the Institut in December 1807, refereed by Lagrange, Laplace, Monge and Lacroix, and was not printed. The reworked text won the Academy prize of 1811 but appeared as a separate edition only in 1824, so the book of 1822 serves as the anchor date.

What it grew from

• Differential and integral calculus

  influenced · Confidence: medium

  Fourier worked with the means of the analysis that grew out of the calculus of Newton and Leibniz and directly continued the work of Bernoulli, Euler and Lagrange on trigonometric series, although the heat conduction equation required the later apparatus of partial differential equations.

  Link sources

  • T. N. Narasimhan, Fourier's heat conduction equation: history, influence, and connections, Reviews of Geophysics 37 (1), 151-172 (1999)

    Fourier was clearly aware of the earlier work of Bernoulli, Euler, and Lagrange relating to solutions in the form of trigonometric series. He was also aware that Euler, D'Alembert and Lagrange viewed trigonometric series with great suspicion.

  • Joseph Fourier, MacTutor History of Mathematics Archive, University of St Andrews (1997) Verified 20 Sep 2026

    The first objection, made by Lagrange and Laplace in 1808, was to Fourier's expansions of functions as trigonometrical series, what we now call Fourier series.

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

• Caloric (calorique) as a weightless conserved fluid

  influenced · Confidence: medium

  Fourier began from a formulation in the spirit of the Laplacian caloric school (Biot's 1804 work on a heated bar), but then replaced the hypothesis about the nature of heat with a law relating measurable heat flux and temperature gradient.

  Link sources

  • T. N. Narasimhan, Fourier's heat conduction equation: history, influence, and connections, Reviews of Geophysics 37 (1), 151-172 (1999) Verified 25 Sep 2026

    Just as Biot before him, Fourier initially formulated the heat conduction problem as an n-body problem, stemming from the Laplacian philosophy of action at a distance.

  • T. N. Narasimhan, Fourier's heat conduction equation: history, influence, and connections, Reviews of Geophysics 37 (1), 151-172 (1999) Verified 25 Sep 2026

    When Fourier commenced his work on heat conduction at the turn of the nineteenth century, the nature of heat was still unresolved. Those of the mechanistic school, including Biot, believed that heat was a permeating fluid.

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

What grew out of it

• The two laws of thermodynamics (Clausius 1850, Thomson 1851)

  influenced · Confidence: medium

  Fourier was William Thomson's entry into the theory of heat, and it was the notion of conduction, which Thomson had been working on through Fourier in the 1840s, that raised the question of the work lost when heat is merely conducted, the question that led to the dynamical theory of heat.

  Link sources

  • T. N. Narasimhan, Fourier's heat conduction equation: history, influences and connections, Proceedings of the Indian Academy of Sciences (Earth and Planetary Sciences) 108 (3), 117-148 (1999)

    William Thomson (1824-1907), also known as Lord Kelvin, was greatly influenced by Fourier's work. Thomson's first two articles, written at ages 16 and 17, were in defense of Fourier's mathematical approach.

  • Crosbie W. Smith, William Thomson and the Creation of Thermodynamics: 1840-1855, Archive for History of Exact Sciences 16 (3), 231-288 (1977) Verified 20 Sep 2026

    attention was now being focussed on conduction which of course was the very concept with which THOMSON had been dealing during the 1840's in his work on FOURIER.

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

• Maxwell's electromagnetic theory of light

  influenced · Confidence: medium

  The mathematical form of Fourier's heat conduction equations, through Thomson's 1842 analogy between the flow of heat and attraction, gave Maxwell the template for a field description of electricity.

  Link sources

  • James Clerk Maxwell, On Faraday's Lines of Force, Transactions of the Cambridge Philosophical Society (1856) Verified 20 Sep 2026

    Yet we find that the mathematical laws of the uniform motion of heat in homogeneous media are identical in form with those of attractions varying inversely as the square of the distance.

  • T. N. Narasimhan, Fourier's heat conduction equation: history, influences and connections, Proceedings of the Indian Academy of Sciences (Earth and Planetary Sciences) 108 (3), 117-148 (1999)

    Later, he demonstrated the similarities between the mathematical structures of Fourier's heat conduction equation and the equations of electrostatics stemming from the works of Laplace and Poisson (Thomson 1842).

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

Node sources

• Joseph Fourier, Théorie analytique de la chaleur, Discours préliminaire (Paris, Firmin Didot) (1822) Verified 5 Oct 2026

  Les causes primordiales ne nous sont point connues; mais elles sont assujetties à des lois simples et constantes, que l'on peut découvrir par l'observation, et dont l'étude est l'objet de la philosophie naturelle.

• T. N. Narasimhan, Fourier's heat conduction equation: history, influence, and connections, Reviews of Geophysics 37 (1), 151-172 (1999)

  His observational approach (like that of Lavoisier and Laplace before him), was independent of what the fundamental nature of heat was.

• T. N. Narasimhan, Fourier's heat conduction equation: history, influence, and connections, Reviews of Geophysics 37 (1), 151-172 (1999)

  Fourier submitted his manuscript to the French Academy in December 1807. As was the practice, the secretary of the Academy appointed a committee of reviewers consisting of four of the most renowned mathematicians of the time, Laplace, Lagrange, Monge, and Lacroix.

"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

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