Colin Maclaurin (/məˈklɔːrən/;Scottish Gaelic:Cailean MacLabhruinn;^{[pronunciation?]} February 1698 – 14 June 1746)^[1] was a Scottishmathematician who made important contributions togeometry andalgebra.^[2] He is also known for being achild prodigy and holding the record for being the youngest professor. The Maclaurin series, a special case of theTaylor series, is named after him.

Colin Maclaurin
Drawing byDavid Steuart Erskinec. 1795, from a portrait byJames Ferguson
Born	February 1698 Kilmodan,Cowal,Argyll,Scotland
Died	14 June 1746 (aged 48) Edinburgh,Scotland
Nationality	Scottish
Citizenship	Great Britain
Alma mater	University of Glasgow
Known for	Euler–Maclaurin formula Maclaurin's inequality Maclaurin series Maclaurin spheroid Maclaurin–Cauchy test Braikenridge–Maclaurin theorem Trisectrix of Maclaurin
Awards	Grand Prize of theFrench Academy of Sciences
Scientific career
Fields	Mathematician
Institutions	Marischal College,University of Aberdeen University of Edinburgh
Academic advisors	Robert Simson
Notable students	Robert Adam

Owing to changes inorthography since that time (his name was originally rendered asM'Laurine^[3]), his surname is alternatively writtenMacLaurin.^[4]

Maclaurin was born inKilmodan,Argyll. His father, John Maclaurin, minister ofGlendaruel, died when Maclaurin was in infancy, and his mother died before he reached nine years of age. He was then educated under the care of his uncle, Daniel Maclaurin, minister of Kilfinan. Achild prodigy, he entered university at age 11.

At eleven, Maclaurin, a child prodigy at the time, entered theUniversity of Glasgow. He graduatedMaster of Arts three years later by defending a thesis onthe Power of Gravity, and remained at Glasgow to studydivinity until he was 19, when he was elected professor ofmathematics in a ten-day competition atMarischal College and University in Aberdeen. This record as the world's youngest professor endured until March 2008, when the record was officially given toAlia Sabur.^[5]

In the vacations of 1719 and 1721, Maclaurin went to London, where he became acquainted withIsaac Newton,Benjamin Hoadly,Samuel Clarke,Martin Folkes, and other philosophers. He was admitted as a member of theRoyal Society.

In 1722, having provided a locum for his class at Aberdeen, he travelled on the Continent as tutor to George Hume, the son ofAlexander Hume, 2nd Earl of Marchmont. During their time inLorraine, he wrote his essay on the percussion of bodies (Demonstration des loix du choc des corps), which gained the prize of theRoyal Academy of Sciences in 1724. Upon the death of his pupil atMontpellier, Maclaurin returned to Aberdeen.

In 1725, Maclaurin was appointed deputy to the mathematical professor at theUniversity of Edinburgh, James Gregory (brother ofDavid Gregory and nephew of the esteemedJames Gregory), upon the recommendation ofIsaac Newton. On 3 November of that year Maclaurin succeeded Gregory, and went on to raise the character of that university as a school of science. Newton was so impressed with Maclaurin that he had offered to pay his salary himself.

Maclaurin usedTaylor series to characterize maxima, minima, and points of inflection for infinitely differentiable functions in hisTreatise of Fluxions. Maclaurin attributed the series toBrook Taylor, though the series was known before toNewton andGregory, and in special cases toMadhava of Sangamagrama in fourteenth century India.^[6]Nevertheless, Maclaurin received credit for his use of the series, and the Taylor series expanded around 0 is sometimes known as theMaclaurin series.^[7]

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 LaureateSubramanyan Chandrasekhar dedicated a chapter of his bookEllipsoidal Figures of Equilibrium toMaclaurin spheroids.^[7] Maclaurin corresponded extensively withClairaut,Maupertuis, andd'Ortous de Mairan.^[8][9]

Independently fromEuler and using the same methods, Maclaurin discovered theEuler–Maclaurin formula. He used it to sum powers ofarithmetic progressions, deriveStirling's formula, and to derive the Newton–Cotes numerical integration formulas which includesSimpson's rule as a special case.^[7]

Maclaurin contributed to the study ofelliptic integrals, 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.^[7]

In hisTreatise 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.^[10][11] This publication preceded by two yearsCramer's publication of a generalization of the rule ton unknowns, now commonly known asCramer's rule.

In 1733, Maclaurin married Anne Stewart, the daughter of Walter Stewart, theSolicitor General for Scotland, by whom he had seven children. His eldest sonJohn Maclaurin studied law, was aSenator of the College of Justice, and became Lord Dreghorn; he was also joint founder of theRoyal Society of Edinburgh.^[12]

Maclaurin actively opposed theJacobite rising of 1745 and superintended the operations necessary for the defence of Edinburgh against the Highland army. Maclaurin compiled a diary of his exertions against the Jacobites, both within and without the city.^[13] When the Highland army entered the city, however, he fled toYork, where he was invited to stay by theArchbishop of York.

On his journey south, Maclaurin fell from his horse, and the fatigue, anxiety, and cold to which he was exposed on that occasion laid the foundations ofdropsy. He returned to Edinburgh after theJacobite army marched south, but died soon after his return.

He is buried atGreyfriars Kirkyard,Edinburgh. The simple table stone is inscribed simply "C. M. Nat MDCXCVIII Ob MDCCXLVI" and stands close to the south-west corner of the church but is supplemented by a more wordy memorial on the outer wall of the church.

The mathematician and formerMIT PresidentRichard Cockburn Maclaurin was from the same family.

The Maclaurin Society (MacSoc), the Mathematics and Statistics Society at Glasgow University, is named in his honour.

Colin MacLaurin Road within Edinburgh University'sKing's Buildings complex is named in his honour.

Some of his important works are:

• Geometria Organica – 1720
• De Linearum Geometricarum Proprietatibus – 1720
• Treatise on Fluxions – 1742 (763 pages in two volumes. The first systematic exposition of Newton's methods.)
• Treatise of Algebra – 1748 (two years after his death.)
• Account of Newton's Discoveries – Incomplete upon his death and published in 1748^[14]
• Account of Sir Isaac Newton's philosophical discoveries (in French). Paris: Laurent Durand. 1749.

Colin Maclaurin was the name used for the new Mathematics and Actuarial Mathematics and Statistics Building atHeriot-Watt University, Edinburgh.

•  
  French edition of theTreatise of algebra (1748)
•  
  French edition of theAccount of Sir Isaac Newton's philosophical discoveries (1749)

• Braikenridge–Maclaurin theorem
• Trisectrix of Maclaurin
• Cayley's sextic
• Cramer's paradox
• Hesse configuration
• Sinusoidal spiral

• Anderson, William,The Scottish Nation, Edinburgh, 1867, vol.VII, p. 37.
• Ball, W. W. Rouse (1908).A Short Account of the History of Mathematics (4th ed.). pp. 384–387. Retrieved20 January 2008.
• "Overview of Colin Maclaurin".Gazetteer for Scotland.University of Edinburgh School of GeoSciences. Retrieved20 January 2008.
• Friedman, Erich."Colin Maclaurin".Periodic Table of Mathematicians.Stetson University. Archived fromthe original on 14 February 2008. Retrieved20 January 2008.
• O'Connor, John J.;Robertson, Edmund F.,"Colin Maclaurin",MacTutor History of Mathematics Archive,University of St Andrews
• Sageng, Erik, 2005, "A treatise on fluxions" inGrattan-Guinness, I., ed.,Landmark Writings in Western Mathematics. Elsevier: 143–58.
• Tweddle, Ian (November 1998)."The prickly genius—Colin Maclaurin (1698–1746)"(PDF).The Mathematical Gazette.82 (495).Leicester:Mathematical Association:373–378.doi:10.2307/3619883.JSTOR 3619883. Archived fromthe original(PDF) on 31 October 2007. Retrieved20 January 2008.

• Bruce A. Hedman, "Colin Maclaurin's quaint word problems," College Mathematics Journal 31 (2000), 286–288.
• Bruneau, Olivier (2011).Colin Maclaurin, l'obstination mathématicienne d'un newtonien. Presses Universitaires de Nancy.
• Sageng, Erik (2006) [2004]. "MacLaurin, Colin".Oxford Dictionary of National Biography (online ed.). Oxford University Press.doi:10.1093/ref:odnb/17643.(Subscription orUK public library membership required.)