John Machin (bapt. c. 1686 – June 9, 1751) was a professor of
astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
at
Gresham College
Gresham College is an institution of higher learning located at Barnard's Inn Hall off Holborn in Central London, England. It does not enroll students or award degrees. It was founded in 1596 under the will of Sir Thomas Gresham, and hosts ove ...
, 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
Newton
Newton most commonly refers to:
* Isaac Newton (1642–1726/1727), English scientist
* Newton (unit), SI unit of force named after Isaac Newton
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* ''Newton'' (film), a 2017 Indian film
* Newton ( ...
in 1712.
On 16 May 1713 he succeeded
Alexander Torriano as professor of astronomy in
Gresham College
Gresham College is an institution of higher learning located at Barnard's Inn Hall off Holborn in Central London, England. It does not enroll students or award degrees. It was founded in 1596 under the will of Sir Thomas Gresham, and hosts ove ...
, 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
(Whatever shines should be observed)
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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:
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* Derivation proceeding, a proc ...
is straightforward) is:
:
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.
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.) 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.
See also
* Gresham Professor of Astronomy
The Professor of Astronomy at Gresham College, London, gives free educational lectures to the general public. The college was founded for this purpose in 1597, when it appointed seven professors; this has since increased to ten and in addition the ...
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