It was 1621 when the
Cyclometricus by
Willebrord Snellius, a pupil of
Ludolph van Ceulen, was published. Snellius proved that the perimeter of the inscribed polygon converges to the circumference twice than the circumscribed polygon. As a good pupil of van Ceulen, Snellius managed to get 7 decimal places for the $\pi$ by using a 96-sided polygon. His best result, however, was 35 decimal places, which improved his master's result, 32.
The next improvement is dated 1630 by
Christoph Grienberger, the last mathematician to evaluate $\pi$ using the polygon method, while the first successful method change came out thanks to the british mathematician and astronomer
Abraham Sharp who determined 72 decimal places of $\pi$, of which 71 correct, using a series of arctangents. A few years later,
John Machin improved Sharp's result with the following formula and that allowed him to achieve the remarkable result of 100 decimal places!
\[\frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{239}\]
Machin's approach proved successful, so much so that the slovenian baron
Jurij Vega improved on two occasions the above formula obtaining a greater number of decimal digits of $\pi$, the first time in 1789 with a formula similar to Euler's one
\[\frac{\pi}{4} = 5 \arctan \frac{1}{7} + 2 \arctan \frac{3}{79}\]
then in 1794 with a Hutton-like formula
\[\frac{\pi}{4} = 2 \arctan \frac{1}{3} + \arctan \frac{1}{7}\]
The arctangent era continued with
William Rutherford
\[\frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{70} + \arctan \frac{1}{99}\]
and with
Zacharias Dase
\[\frac{\pi}{4} = \arctan \frac{1}{2} + \arctan \frac{1}{5} + \arctan \frac{1}{8}\]
Finally comes the british
William Shanks that pushing the full potential of the Machin's formula managed to get 707 decimal places, of which only 527 were correct after Ferguson's controls in 1946. Here, however, we are going in the era of mechanical calculation, prologue to computer era.
