John Machin

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

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

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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.^[2]

On 16 May 1713 he succeeded Alexander Torriano as professor of astronomy in Gresham College, and held the post until his death, which occurred in London on 9 June 1751. Machin enjoyed a high mathematical reputation. His ingenious quadrature of the circle was investigated by Hutton, and in 1706 Machin computed the value of π by Halley's method to one hundred decimal places. A mass of his manuscripts is preserved by the Royal Astronomical Society; and writing to William Jones in 1727, he asserted his claim to the parliamentary reward of £10,000 for amending the lunar tables.^[2]

In 1728, he was listed as one of the subscribers to the Cyclopaedia of Ephraim Chambers.^[3]

Formula

Machin's formula^[4] (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 ( $π$ /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 π 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.

References

mathematical constant $π$
Part of a series of articles on the
3.1415926535897932384626433...
Uses
Area of a circle Circumference Use in other formulae
Properties
Irrationality Transcendence
Value
Less than 22/7 Approximations Madhava's correction term Memorization
People
Archimedes Liu Hui Zu Chongzhi Aryabhata Madhava Jamshīd al-Kāshī Ludolph van Ceulen François Viète Seki Takakazu Takebe Kenko William Jones John Machin William Shanks Srinivasa Ramanujan John Wrench Chudnovsky brothers Yasumasa Kanada
History
Chronology A History of Pi
In culture
Indiana pi bill Pi Day
Related topics
Squaring the circle Basel problem Six nines in $π$ Other topics related to $π$
v t e

^ 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
^ ^a ^b Clerke 1893.
^ List of Subscribers to the Cyclopaedia at library.wisc.edu
^ Machin's Formula at MathWorld

Attribution

This article incorporates text from a publication now in the public domain: Clerke, Agnes Mary (1893). "Machin, John (d.1751)". In Lee, Sidney (ed.). Dictionary of National Biography. Vol. 35. London: Smith, Elder & Co.