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Ptolemy invented a system to describe the periodic motion of the planets by epicycles. Fourier did something similar for periodic motion in mechanics. Every such motion can be thought of as constructed of out of different kinds of epicycles, namely sine functions with different amplitudes and frequencies.

Was Fourier inspired by Ptolemy?

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    $\begingroup$ Fourier was not interested in periodic motion. The spread of heat in solids is very far from periodic, he was using sines and cosines in the spatial variables, not time, and over intervals where periodicity was irrelevant. A relation is only visible in hindsight. $\endgroup$ – Conifold Jul 18 at 22:09
  • $\begingroup$ @Conifold I think the spread of hear is pretty periodic. It spreads in one long period, the spreading going to zero at infinity. $\endgroup$ – Deschele Schilder Jul 18 at 22:13
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Fourier was not inspired by Ptolemy or by mechanics. His work where he developed Fourier series and invented Fourier transform was about heat. He never mentions Ptolemy (or any astronomer). The variable in his Fourier series is always a space variable, not the time.

Ptolemy and other astronomers were concerned not only with periodic motions but with what we would call almost periodic motions, $\sum c_k e^{i\lambda_k t}$, where $\lambda_k$ are in general non-commensurable. Later (in 20th century) such series were called 'non-harmonic Fourier series", or almost periodic functions, and Fourier analysis was generalized to include them.

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