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Roger Cotes advanced the understanding of trigonometric functionsthrough his original work on integration, functions, and the nth roots of unity. He is also remembered for his contributions to Isaac Newton's work on universal gravitation.
Cotes was born in Burbage, England, the son of the Reverend Robert Cotes and his wife, Grace. He attended the Leicester School; his uncle, the Reverend John Smith, was so impressed by the boy's ability in mathematics that he decided to oversee Cotes' early education. When Cotes left Leicester School for St. Paul's School in London, he and his uncle regularly corresponded about science and mathematics. Cotes was admitted as a scholarship student to Trinity College in Cambridge in 1699. He completed his undergraduate work in 1702 and earned a master's degree in 1706.
Looking to the Stars
Like many mathematicians of his era, Cotes was also an astronomer. The universe offered fascinating mathematical puzzles, and astronomical studies had an important practical benefit in an era when successful trade--especially for island nations like England--depended on sailing vessels that charted course by the stars.
In letters to Newton, Cotes described a heliostat telescope that used a clock mechanism to make an interior mirror revolve. He refined existing solar and planetary tables and made notes on the total solar eclipse in 1715.
Cotes was named as a fellow of Trinity College in 1705. In 1706--at the age of 24--he was appointed as the first Plumian professor of astronomy and natural philosophy at Cambridge. Cotes immediately began to solicit donations for an observatory at Trinity. Complete with living quarters, the observatory was built over King's Gate, and Cotes lived there with his cousin Robert Smith. The observatory itself was not completed in Cotes' lifetime and was demolished in 1797.
Cotes established a school of physical sciences at Trinity, and he and colleague William Whiston began a series of experiments in 1707. The details of these experiments were published after Cotes' death as Hydrostatical and Pneumatical Lectures by Roger Cotesin1738.
In 1709, Cotes threw himself into the preparation of the second edition of Newton's Philosophiae naturalis principia mathematica, in which Newton refined his theory of universal gravitation. He and Newton worked closely for more than three years on the second edition. Cotes wrote a preface for the work that defended Newton's hypothesis against competing ideas. He argued that Newton's hypothesis, based on observation, was accurate and superior to theories based on description unaccompanied by rational explanation and, certainly, to theories based on superstition and the occult. Cotes' work on this edition was a labor of love--he received 12 copies of the book as payment for more than three years of work.
The Work of a Lifetime
Cotes published only one paper on mathematics during his lifetime. His Logometriawas released in 1714, and this paper is evidence not only of Cotes' brilliance, but his persistent and organized approach to the study of mathematics. Cotes calculated the natural base for a system of logarithmsand introduced two inventive methods for calculating Briggsian logarithmsfor a number and interpolating intermediate values. He applied integration to problems related to quadratures, arc lengths areas of surfaces of revolution, and atmospheric density. In attempting to evaluate the surface area of an ellipsoid of revolution, Cotes identified not one, but two approaches to the problem, using both logarithms and arc sines to develop formulas to calculate area.
Logometria was included in a book of Cotes' papers compiled after his death by Robert Smith and published in 1722 as Harmonia mensurarum. The lengthiest section of the book includes Cotes' work in systematic integration. His work was based on a geometrical result involving n equally spaced points on a circle--now known as Cotes' theorem--that is equivalent to finding all the factors of x n -- an when n is a positive integer. Another section of Harmonia mensurarum includes miscellaneous papers on estimating errors, Newton's differential method, the descent of heavy bodies, and cycloidal motion. Cotes' solution to determining the area under a curve, in its modern version, is known as the Newton-Cotes formula.
Cotes died from a fever at the age of 33. "Had Cotes lived," Newton mourned, "we might have known something."
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