# John Machin

John Machin

John Machin
Born c. 1686
England
Died 9 June 1751 (aged 70–71)
London, England
Nationality English
Fields Mathematician and astronomer
Institutions Gresham College
Notable students Brook Taylor
Known for Machin-like formula

John Machin (bapt. c. 1686 – June 9, 1751),[1] a professor of astronomy at Gresham College, London, is best known for developing a quickly converging series for Pi in 1706 and using it to compute Pi to 100 decimal places.

## History

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

In 1728, he was listed as one of the subscribers to the Cyclopaedia of Ephraim Chambers.[2]

## Formula

Machin's formula[3] (for which the derivation is straightforward) is:

$\frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{239}$

The benefit of the new formula, a variation on the Gregory/Leibniz series (Pi/4 = arctan 1), was that it had a significantly increased rate of convergence, which made it a much more practical method of calculation.

To compute Pi to 100 decimal places, he combined his formula with the Taylor series expansion for the inverse tangent. (Brook Taylor was Machin's contemporary in Cambridge University.) Machin's formula remained the primary tool of Pi-hunters for centuries (well into the computer era).

Several other Machin-like formulae are known.