Machin's Formula for Pi

Theorem

 * $\dfrac \pi 4 = 4 \arctan \dfrac 1 5 - \arctan \dfrac 1 {239} \approx 0 \cdotp 78539 \, 81633 \, 9744 \ldots$

The calculation of $\pi$ (pi can then proceed using the Gregory Series:
 * $\arctan \dfrac 1 x = \dfrac 1 x - \dfrac 1 {3 x^3} + \dfrac 1 {5 x^5} - \dfrac 1 {7 x^7} + \dfrac 1 {9 x^9} - \cdots$

which is valid for $x \le 1$.

Proof
Let $\tan \alpha = \dfrac 1 5$.

Then: