John Machin (bapt. c. 1686 – June 9, 1751) was a professor of

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, 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.

History

John Machin served as secretary of the

Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...

from 1718 to 1747. He was also a member of the commission which decided the Calculus priority dispute between

Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...

and

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in 1712. On 16 May 1713 he succeeded Alexander Torriano as professor of astronomy in

, 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

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; and writing to William Jones in 1727, he asserted his claim to the parliamentary reward of £10,000 for amending the lunar tables. In 1728, he was listed as one of the subscribers to the '' Cyclopaedia'' of

Ephraim Chambers Ephraim Chambers ( – 15 May 1740) was an English writer and encyclopaedist, who is primarily known for producing the '' Cyclopaedia, or a Universal Dictionary of Arts and Sciences''. Biography Chambers was born in Milton near Kendal, Westmor ...

Formula

Machin's formula''Machin's Formula'' at MathWorld
/ref> (for which the

derivation Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...

is straightforward) is: :

\frac = 4 \arctan \frac - \arctan \frac

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 In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...

expansion for the inverse tangent. ( Brook Taylor was Machin's contemporary in

Cambridge University , mottoeng = Literal: From here, light and sacred draughts. Non literal: From this place, we gain enlightenment and precious knowledge. , established = , other_name = The Chancellor, Masters and Schola ...

.) Machin's formula remained the primary tool of pi-hunters for centuries (well into the computer era). Several other

Machin-like formula In mathematics, Machin-like formulae are a popular technique for computing to a large number of digits. They are generalizations of John Machin's formula from 1706: :\frac = 4 \arctan \frac - \arctan \frac which he used to compute to 100 d ...

e are known.

References

Attribution *

External links

* * {{DEFAULTSORT:Machin, John 1752 deaths 18th-century English mathematicians Fellows of the Royal Society 18th-century British scientists Freemasons of the Premier Grand Lodge of England

History

Formula

See also

References

External links