We have learned in class about the Newton-Cotes rules for numerical integration. We all know much about the accomplishments of Sir Isaac Newton and the impact that his work has had on science and mathematics. His colleague, Roger Cotes, was no slouch either.

Roger Cotes was born in Burbage, Leicestershire, England in 1682. By the early age of 12, his teachers at Leicester School had recognized his extraordinary talent for mathematics, and his uncle was very influential in tutoring Cotes so that he had every chance to develop these talents. Later, Cotes graduated from Cambridge University with a bachelor’s degree at age 19 and continued his studies there. At age 25, he was appointed the University’s first Plumian Professor of Astronomy and Experimental Philosophy. Astronomy was always one of Cotes’s major areas of interest, and this interest led him into doing work with Newton to design a heliostat telescope with a mirror revolving by clockwork. Among other works involving astronomy, he recomputed the solar and planetary tables of J.D. Cassini and Flamsteed, and he intended to create tables of the moon’s motion, based on Newtonian principles. Finally, in 1707 he formed a school of physical sciences at Trinity in partnership with William Whiston.

From 1709-1713 Cotes was heavily involved in editing the second edition of Newton’s famous Principia, a book that explained Newton’s theories on universal gravitation. He did not simply proof-read the work; rather, he very carefully studied it, “…gently but persistently arguing points with Newton.” Newton and Cotes spent nearly three and half years collaborating on the work, in which they fully deduce, through Newton’s principles, the theory of the moon and the equinoxes, as well as, the theory of comets and their orbits. The total number of works printed for this addition was limited to 750 copies. As reward to Cotes, he was given a share of the profits and 12 copies of his own. Cotes’ original contribution to the work involved a preface with supported the scientific superiority of Newton’s principles over the then popular idea of vortices presented by Descartes. Cotes concluded that “the Newton’s law of gravitation was confirmed by observation of celestial phenomenon and did not depend on unexplained occult forces which Cartesian critics alleged.”

Apart from his work with Newton, Cotes was a greatly influential mathematician himself. Along with making substantial advances in calculus, interpolation processes, and logarithmic calculations, he is said to have “invented” the radian measure of angles and to have found the continued fraction representation of e. In another rather curious accomplishment, it is a mystery to many involving how Cotes arrived at the approximation…

Sadly, Roger Cotes died of a violent fever at the age of 32. Many have said that his mathematical abilities put him second only to Newton from his generation in England. Just after his death, Newton remarked, “If he had lived we would have known something.”

References:
http://www.hardycalculus.com/html/sketches/cotes.htm
http://en.wikipedia.org/wiki/Roger_Cotes
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cotes.html
http://numericalmethods.eng.usf.edu/anecdotes/cotes.html
http://en.wikipedia.org/wiki/Newton-Cotes_formulas