How was the Fourier Transform created? The Fourier Transform is a very useful and ingenious thing. But how was it initiated?
How did Joseph Fourier composed the Fourier Transform formula and the idea of a transformation between periodic signals and their frequency?
The FT is very useful in frequency analysis. But was it invented to be useful for this, or did it exist before?

Fourier in his Theorie Analytique de la Chaleur says: "The equations of heat conduction like those of sound or small oscillations for liquids belong to one of the most recently discovered breanches of science which it is important to extend." This passage is quoted at length in Körner's Fourier Analysis.
He goes on to advocate a "calculus" that yields quantitative results for such problems. This involved finding functions that could approximate waves (and other functions), which could serve as solutions of differential equations.  
Oscillations and frequency analysis have always been part of this problem.
Fourier's "Memoir on Heat Transmission in Solids" dates to 1807 and is (per Wiki) is considered an important breakthrough. The key insight was that a wide range of functions could be approximated using trigonometric series. Gauss (also per Wikipedia) was the first to use ("discover") the FFT (discrete fast Fourier transform) in studying astronomy in 1805.
While I can't vouch for the Wiki article I think it's a good start.
Edit: Willie Wong's idea that the (continuous) FT was Fourier's invention (and surprisingly that idea of series of this type are not so much Fourier's invention) seems to be supported by an Overflow item here. The author of that post cites a biography of Fourier in support of his claim. 
Anon's evanescent comment about orthogonality is certainly part of the answer to this question. Gauss probably gets credit for this idea. He seems to have been first in almost everything else--why not this as well?
According to papers cited in comments, Gauss's version of the FFT was not published in his lifetime. 
