Around 1706, John Machin found the arc tangent formula .

He expanded it using Gregory's series to compute to over 100 digits. The convergence of this series is much more rapid than that for the simple Gregory–Leibniz series: .

Let and be the series for and . Manchin's formula is then .

Split and into four series according to the signs of their coefficients:

,

,

,

.

Machin used values truncated to 103 decimals, which means that only terms greater than are calculated.

To make sure that all terms of and with indices are less than , it is necessary that . For , this means .