Get $\pi$ decimals manually

Question

I know that there was a guy that could get 100 decimal digits of $\pi$ before computers were able to get thousands.

How did the guy do that?

Answer 1

According to Wikipedia, John Machin combined the formula $$\frac{\pi}{4}=4\cot^{-1}5-\cot^{-1}239$$ with the Taylor series expansion for the inverse tangent in order to compute $\pi$ to 100 decimal places.

A previous record was due to Abraham Sharp who used an arcsine series to find 72 decimal digits.

Answer 2

This page on the chronology of pi contains many useful notes on how the pre-computer era calculations of $\pi$ were performed.

Answer 3

See also Pi: A Source Book.