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John Machin, (bapt. 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.

Machin's formula[2] is:

π/4 = 4 cot -1 5 - cot-1239

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 formulas are known.

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.[3]

See also

* Gresham Professor of Astronomy


1. ^ Anita McConnell, ‘Machin, John (bap. 1686?, died 1751)’, Oxford Dictionary of National Biography, Oxford University Press, 2004. Accessed 26 June 2007. DOI:10.1093/ref:odnb/17533
2. ^ Machin's Formula at MathWorld
3. ^ List of Subscribers to the Cyclopaedia at

External links

* O'Connor, John J.; Robertson, Edmund F., "John Machin", MacTutor History of Mathematics archive, University of St Andrews, .


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