Editing Colin Maclaurin (section)

==Contributions to mathematics==
[[File:Acta Eruditorum - I monete geometria, 1747 – BEIC 13417751 (cropped).jpg|thumb|Illustration of critique of ''De fluxionibus libri duo'' published in [[Acta Eruditorum]], 1747]]
Maclaurin used [[Taylor series]] to characterize maxima, minima, and points of inflection for infinitely differentiable functions in his ''Treatise of Fluxions''. Maclaurin attributed the series to [[Brook Taylor]], though the series was known before to [[Isaac Newton|Newton]] and [[James Gregory (mathematician)|Gregory]], and in special cases to [[Madhava of Sangamagrama]] in fourteenth century India.<ref>{{cite web
| publisher=Canisius College
| work=MAT 314
| url=http://www.canisius.edu/topos/rajeev.asp
| title=Neither Newton nor Leibniz&nbsp;– The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala
| access-date=9 July 2006
| url-status=dead
| archive-url=https://web.archive.org/web/20060806040307/http://www.canisius.edu/topos/rajeev.asp
| archive-date=6 August 2006
| df=dmy-all
}}</ref>
Nevertheless, Maclaurin received credit for his use of the series, and the Taylor series expanded around 0 is sometimes known as the ''Maclaurin series''.<ref name=Grabiner1997>{{cite journal
  | last=Grabiner
  | first=Judith
  |date=May 1997
  | title=Was Newton's Calculus a Dead End? The Continental Influence of Maclaurin's Treatise of Fluxions
  | journal=The American Mathematical Monthly
  | volume=104
  | issue=5
  | pages=393–410
  | publisher=Mathematical Association of America
  | format=[[PDF]]
  | doi=10.2307/2974733
  | jstor = 2974733| url=http://scholarship.claremont.edu/pitzer_fac_pub/121
  }}</ref>
[[File:Colin Maclaurin color.jpg|thumb|Colin Maclaurin (1698–1746)]]
Maclaurin also made significant contributions to the gravitation attraction of ellipsoids, a subject that furthermore attracted the attention of d'Alembert, A.-C. Clairaut, Euler, Laplace, Legendre, Poisson and Gauss.  Maclaurin showed that an oblate spheroid was a possible equilibrium in Newton's theory of gravity. The subject continues to be of scientific interest, and Nobel Laureate [[Subramanyan Chandrasekhar]] dedicated a chapter of his book ''Ellipsoidal Figures of Equilibrium'' to [[Maclaurin spheroid]]s.<ref name=Grabiner1997/> Maclaurin corresponded extensively with [[Alexis Clairaut|Clairaut]], [[Pierre Louis Maupertuis|Maupertuis]], and [[Jean-Jacques d'Ortous de Mairan|d'Ortous de Mairan]].<ref>{{cite book|editor=Mills, Stella|isbn=0906812089|lccn=81215733|title=The collected letters of Colin Maclaurin (1698–1746)|location=Nantwich, Cheshire, UK|publisher=Shiva|year=1982|postscript=; xx+496 pages, 218 letters; correspondents include Newton, Halley, [[Robert Simson|Simson]], de Moivre, Voltaire, Sir [[Hans Sloane]] & Sir [[Martin Folkes]] }}</ref><ref>{{cite journal|doi=10.1126/science.218.4567.45|title=Review of ''The Collected Letters of Colin Maclaurin'', edited by Stella Mills|year=1982 |last1=Hankins |first1=Thomas L. |author-link=Thomas L. Hankins|journal=Science |volume=218 |issue=4567 |pages=45–46 |pmid=17776705 }}</ref>

Independently from [[Euler]] and using the same methods, Maclaurin discovered the [[Euler–Maclaurin formula]].  He used it to sum powers of [[arithmetic progression]]s, derive [[Stirling's formula]], and to derive the Newton–Cotes numerical integration formulas which includes [[Simpson's rule]] as a special case.<ref name=Grabiner1997/>

Maclaurin contributed to the study of [[elliptic integral]]s, reducing many intractable integrals to problems of finding arcs for hyperbolas.  His work was continued by d'Alembert and Euler, who gave a more concise approach.<ref name=Grabiner1997/>

In his ''Treatise of Algebra'' (Ch. XII, Sect 86), published in 1748 two years after his death, Maclaurin proved a rule for solving square linear systems in the cases of 2 and 3 unknowns, and discussed the case of 4 unknowns.<ref>{{cite book
  | last = MacLaurin
  | first = Colin
  | title = A Treatise of Algebra, in Three Parts.
  | url = https://archive.org/details/atreatisealgebr03maclgoog
  | year = 1748
| publisher = Printed for A. Millar & J. Nourse
 }}</ref><ref>{{cite journal
  | last=Hedman
  | first=Bruce
  |date=November 1999
  | title=An Earlier Date for "Cramer's Rule"
  | journal=Historia Mathematica
  | volume=26
  | issue=4
  | pages=365–368
  | publisher=Academic Press Elsevier
  | doi=10.1006/hmat.1999.2247
  | doi-access=free
  }}</ref> This publication preceded by two years [[Gabriel Cramer|Cramer]]'s publication of a generalization of the rule to ''n'' unknowns, now commonly known as [[Cramer's rule]].