I've always grappled with the topic of anything Fourier during my undergrad days. Until recently when revisiting why I learned what I did, I discovered how Fourier's desire to understand the flow of heat through a solid body led to the creation/discovery of Fourier Series and correspondingly the Fourier Transform.
However, I've never been able to make the mental leap from heat equation to creation of Fourier Series.
Are there any good sources (references, books, videos (most preferable) etc.,) that provide a walkthrough to help "discover/create" the findings of Fourier by yourself?
This is purely an intellectual exercise out of curiosity - it may not be worth it, but I'm curious to be able to understand it "naturally" vs. being told it is so.
My expected "thought experiment setup" is a thin rod with thermometers at regular intervals and using that to derive what Fourier did - not sure if this is even possible but the idea is to go from the simplest abstraction to the actual concept at hand. It's okay if things are mixed with modern calculus to help understand it better since at the time it was still in flux...