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Mathematical Treasure: Simpson on Fluxions

Frank J. Swetz (The Pennsylvania State University)

Thomas Simpson (1710-1761) was a self-trained English mathematician, learning his first mathematics from a used copy of Cocker’s Arithmetic, which he bought from a traveling peddler. Simpson’s ability in mathematics earned him a membership in The Royal Society and eventually, in 1743, a professorship at the Royal Military Academy at Woolwich. His early career was spent as an actuary for insurance companies. He wrote many texts, particularly on probability and the computation of annuities. In 1737, he published A New Treatise on Fluxions, which was considered quite advanced for its time.


Thirteen years later in 1750, he released his more comprehensive The Doctrine and Application of Fluxions. Better appreciated than its predecessor, the book was considered the best reference on Newton’s calculus of the 18th century. The title page shown above is from the 1805 edition and contains a biographical tribute to its author, Thomas Simpson.

Also noted for numerical methods of integration, Thomas Simpson is best remembered today as the creator of “Simpson’s Rule.”

Page one, above, provides some basic definitions.


Pages 140-141 provide examples of Simpson’s presentation as well as some applied problems involving the method of fluxions.

The images above are presented courtesy of the Pennsylvania State University Library.

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Simpson on Fluxions," Convergence (May 2015)