Did anybody use numerical analysis before modern computers were introduced?

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I was recently introduced to Numerical analysis to solve PDEs and I was wondering wether this kind of approach was also used barehand to reach approximate conclusions in the same way it is now done by employing calculators.

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Gauss calculated the orbit of Ceres

"... for it is now clearly shown that the orbit of a heavenly body may be determined quite nearly from good observations embracing only a few days; and this without any hypothetical assumption."

https://sites.math.rutgers.edu/~cherlin/History/Papers1999/weiss.html

Perhaps this answers your question.