John Machin, (1680-9 June 1751), was a professor of astronomy at Gresham College, London, is known to have developed one of the best known ways to develop a convergent series for the calculation of π in 1706 and that he later used to expand π with about 100 decimal places.

The Fórmula of Machin: π 4 = 4 arctan ⁡ 1 5 & # x2212; arctan ⁡ 1 239 {\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}}

The benefit of this new formula is that it is a variation of the Leibniz series (π / 4 = arctan 1), in order to increase the ratio of its convergence, and in this way you can make calculations that with a smaller number of steps reach the value of π.

To calculate π up to 100 decimal places, Machin combined his formula with the Taylor series development of the arc-tangent function. (Brook Taylor was Machin's contemporary at the University of Cambridge.) Machin's formula maintained its hegemony to calculate π decimals for centuries, even until the beginning of the computer age.

John Machin served as secretary to the Royal Society from 1718 to 1747. He was also a member of the commission that decided the dispute between Leibniz and Newton in 1712.

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