## Fourier’s memoir 200 years later

Exactly 200 years ago, in 1807, Fourier submitted to the Académie des Sciences his famous memoir on his theory of heat propagation. It was controversial at the time and was finally published only in 1822. Since Gallica has a digital copy of that book, I’ve spent a little time reading it.

The style is in fact not that dated, especially for a physics book where one has to discuss at length one’s motivations, which Fourier does very carefully. It’s fascinating to see all the nowadays famous equations pop up gradually. In the second chapter, the heat equation first appears on page 123. This is followed by a table describing the dimensional analysis of the various quantities he introduced.

In chapter 3 he turns to heat propagation in a parallelepiped of infinite length, actually a semi-infinite axially symmetric cylinder whose base A is a source of constant heat while the cylindrical boudary has a constant temperature (i.e. Dirichlet boundary conditions, although Dirichlet was two years old at the time 🙂 ). The length coordinate is $x$ while the radial one is $y$, see his (fig. 7). And all the action regarding Fourier series starts here: he looks for the stationnary solution to this problem and finds that one is led to solve (in his notations) $\frac{d^2v}{dx^2}+\frac{d^2v}{dy^2}=0\ (*)$.

He then first looks for elementary solutions by using separation of variables, and finds (still using his notations) that $v=Fx.fy= e^{-mx}.\cos(my)$. He excludes the case $m < 0$ as heat must decay, and finds that to satisfy the boundary conditions $m$ must belong to 1, 3, 5, 7… (Is that the first occurence of quantization in the history of science? Very probably not, since at least Laplace is meant to have considered such Laplace equation earlier, but I don’t know about the precise reference.)

Fourier then adds “One will easily form a more general value of $v$ by adding more similar terms to the previous ones, and one shall get $v=ae^{-x}\cos(y)+be^{-3x}\cos{3y}+ce^{-5x}\cos(5y)+de^{-7x}\cos(7y)+\dots+etc.$ It is obvious that this function $v$ satisfies (*) and the boundary conditions.” After discussing that the constraint of having $v=1$ on A will allow to determine the values of $a, b, c, d...$ he ends this section by “we shall see later on that the motion always decomposes in a multiplicity of elementary motions, all of which occur as if they were alone“.

I’ve stopped reading this book at that point due to lack of time, but it’s clearly something I’d like to come back to in the summer. One thing I would also like to know about is the story of that memoir of Galois which was sent to Fourier in 1830. Apparently Fourier died that year and the memoir was “lost”. This has always puzzled me: does that mean Fourier’s children have thrown it away with some of their dad’s correspondance (and if so what more treasures where lost that way?), or that Fourier himself had already done so?