Besides Gauss, many others including Runge (yes, that Runge) and Burkhardt (yes, the one on Einstein's committee) independently discovered the FFT well before the 1950s. Like so much of Gauss's work, his work on the FFT was unpublished during his lifetime.
Probably it was the conjunction of the algorithm and the emerging power of the digital computer that caused the Cooley-Tukey paper to take off at that historical moment.
http://www.cis.rit.edu/class/simg716/Gauss_History_FFT.pdf
Besides Gauss, many others including Runge (yes, that Runge) and Burkhardt (yes, the one on Einstein's committee) independently discovered the FFT well before the 1950s. Like so much of Gauss's work, his work on the FFT was unpublished during his lifetime.
Probably it was the conjunction of the algorithm and the emerging power of the digital computer that caused the Cooley-Tukey paper to take off at that historical moment.