I was reading a chapter of Fourier's seminal work "Analytic Theory of Heat". The third chapter of this book was translated by the famous Stephen Hawking in his book "God created the integers". I fail to find a reference for how Fourier determine the coefficients of the Fourier series. Fourier, in my opinion, should be ranked as one the greatest mathematicians in the 19th century for he laid a great foundation on the development of trigonometric series, an essential area of modern mathematics.
Can you summarize his findings and point out which part of his work contains his derivation? I am greatly interested in how he was able to determine the coefficients of Fourier series, a task that Daniel Bernoulli failed to achieve.
I hope my shallow knowledge of mathematics do not offend and repel you from this question.