Colin Maclaurin - Contributions To Mathematics

Contributions To Mathematics

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 Taylor, though the series was known before to Newton and Gregory, and in special cases to Madhava of Sangamagrama in fourteenth century India. Nevertheless, Maclaurin received credit for his use of the series, and the Taylor series expanded around 0 is sometimes known as the Maclaurin series (Grabiner 1997).

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 spheroids. (Grabiner 1997)

Independently from Euler and using the same methods, Maclaurin discovered the Euler–Maclaurin formula. He used it to sum powers of arithmetic progressions, derive Stirling's formula, and to derive the Newton-Cotes numerical integration formulas which includes Simpson's rule as a special case. (Grabiner 1997)

Maclaurin contributed to the study of elliptic 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.(Grabiner 1997)

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. This publication preceded by two years Cramer's publication of a generalization of the rule to n unknowns, now commonly known as Cramer's rule.