CALCULUS Understanding Its Concepts and Methods

Brook Taylor (1685--1731), Colin Maclaurin (1698--1746) --- Historical Sketches

It seems only natural to write about these two prolific mathematicians in the same Historical Sketch because they are both best known for their work on representations of functions by infinite series. These representations, known as Taylor series and Maclaurin series, are part of the standard fare of every calculus course.

Brook Taylor was born in 1685 into wealthy, cultured, but strict English family of Edmonton, Middlesex, England, now a part of Greater London, and was educated by private tutors up to the time he enrolled in St. John's College of Cambridge University. At Cambridge, he studied law but also became engaged with mathematics, and he graduated with a law degree in 1709. But by this time, he had already written a mathematics research paper and was becoming known in the mathematical world. Indeed, he was both elected to the Royal Society in 1712 and appointed to a committee to adjudicate the dispute between Leibniz and Newton concerning calculus priority. In 1714, he was elected Secretary to the Royal Society and held that position for four years, a period that was his most productive in terms of mathematical output.

Taylor is given credit for conceiving the concept of the calculus of finite differences, the tool of integration by parts, and of course the Taylor series representation of functions. He suffered from problems of health, and he died at the age of 46.

Colin Maclaurin** **was born in 1698 into a minister's family
residing in the village of Kilmodan, in Argyllshire, a county of western
Scotland. He was a child prodigy who at the age of eleven became a student at
the University of Glasgow. Then at the age of nineteen, he was elected
Professor of Mathematics at Marischal College in Aberdeen, and two years after
that, he was elected to the Royal Society of London and came to know
Isaac Newton. In fact, it was Newton who
recommended to the University of Edinburgh that Maclaurin be made a professor
of mathematics there, and he was, in 1725. In 1740, he shared a French Academy
of Sciences prize with Leonhard Euler and
Daniel Bernoulli. In 1742, Maclaurin
wrote a two-volume treatise defending Newton's mathematics as a rebuttal to
Bishop George Berkeley, who had claimed
that Newton used faulty reasoning. That is the book in which the special case
of Taylor series appears, now called Maclaurin series.

When in 1745 organized supporters of King James II marched on Edinburgh, Maclaurin engaged himself in defending the city, after which he escaped to England. This effort destroyed his health, and he died at age 48.

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Taylor and Maclaurin polynomials

Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.