# 6th July

On this day in 1708, de Moivre wrote to Johann Bernoulli about Machin's series: $(\frac{16}{5} - \frac{4}{239}) - \frac{1}{3} (\frac{16}{5^3} - \frac{4}{239^3})\, ... = 3.14159 ... = \pi$, on this occasion giving two different proofs that it converges to π.

This series follows from the identity $\frac {\pi }{4}=4\tan^{-1} {\frac {1}{5}}-\tan^{-1} {\frac {1}{239}}$.

This series follows from the identity $\frac {\pi }{4}=4\tan^{-1} {\frac {1}{5}}-\tan^{-1} {\frac {1}{239}}$.