Not in this case but sometimes mathematicians of old did have access to “computers”, ie savants who can do large calculations in their heads easily. Gauss used the services of such a person: https://hsm.stackexchange.com/questions/15609/autistic-assis...

"The History of Approximation Theory" by Karl-Georg Steffens is a great reference for historical contexts.

For Chebyshev, who devoted his life to the construction of various 'mechanisms' [1][2], his motivation was to determine the parameters of mechanisms (that minimizes the maximal error of the approximation on the whole interval).

[1] https://en.wikipedia.org/wiki/Mechanism_(engineering)

[2] https://tcheb.ru/

Ballistics, celestial bodies, even a ton of competitions between mathematicians. You can trace down analysis concepts to Archimedes easily, but by Descartes (rolling tangent) and eventually Newton / Leibniz who formalized calculus there was a lot of stuff happening. E.g. Descartes was contemporary with Galileo. So applied math and the theoretical part was under natural philosophy that eventually became physics.

I was reading a book from the early 1900's, and it referenced using computers to calculate some complex algorithms. Threw me for a loop, and I finally realized the author was talking about people. Apparently it was a thing to send long computations to a room/building full of people and get the answer back.

I was playing around with an idea that there might be some insight into Fermat primes by looking at products of complex solutions of polynomials of the form x^{2^n)+1=0, and Chebyshev polynomials came up as I was looking at the exact values of those roots.¹² As I recall, looking at finding half angle sines and cosines of increasing fractions, I ended up seeing Chebyshev polynomials emerging from the results.

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1. I may have this confused with my similar investigations into Mersenne primes and x^p-1=0.

2. My hypothesis that I could find factors of a Fermat number with n > 5 by multiplying the roots together and setting x = 2 failed on writing a program to actually check the result, but looking back on my memories of doing this, I may have made an error.

Anything that can't be solved analytically, I suppose. That's a pretty huge range of problems!

I second that. I bet such a documentary may even yield insights how to make current scientific breakthroughs

You might enjoy the book "The Language of Mathematics: Making the Invisible Visible"

There are lots of fun characterizations of them. The most significant one is that they are the polynomials that minimize the uniform norm on a given set -- classically, this set was the closed interval [-1, 1] or sometimes [-2, 2]. This means they are the polynomials that attain the smallest maximum absolute value on the set or interval of interest. This minimax property is essential for their utility and ubiquity throughout pure and applied math

The minimax property is one of the ways you can generalize to other kinds of sets, allowing you to talk about the Chebyshev polynomial of, say, an arc or a Jordan curve or a union of intervals, and many of the properties enjoyed by the classical Chebyshev polynomials end up carrying over as well, but faaaaar less is known about these generalizations. In many cases all you can hope for are asymptotics, and you need much more in the way of machinery and sophisticated tools to prove anything -- even to compute explicit formulas for the coefficients.

The classical case is nice though because they can be explored fruitfully without very much background