Origin of the Fourier Transform (1878)

Question

I located Joseph Fourier's book, Analytial Theory of Heat (1878), but at first glance it looks like it is all about heat. What did Fourier call the Fourier transform? When did he first use it?

Answer 1

Fourier derived what we now refer to as the Fourier sine and Fourier cosine transforms, while studying the heat in a semi-infinite rod.

You'll notice that the above is the composition of (what we now refer to as) the Fourier sine transform with itself. Fourier derived this as a limit of the discrete sine series for a finite rod, as the length of the rod tends to infinity. He argument was not particularly rigorous, even though the result is correct. He did the same with the cosine transform as well.

I've copied this from the free Google book which is an English translation of Fourier's original 1811 manuscript written in French: https://www.google.com/books/edition/The_Analytical_Theory_of_Heat/No8IAAAAMAAJ?hl=en&gbpv=1

Fourier's manuscript of 1811 was not published until 1822 because of its controversial nature, and only then because Fourier had gained such prominence that he was able to force the issue. Fourier had a lot of brilliant, original, and correct work in his manuscript. His argument for the limit of Fourier series leading to the Fourier integrals is not accepted as correct, even though his final results are correct. But at least this will show you how the Fourier sine and cosine transforms and their inverses arose.