Roger Cotes (1682-1716) was appointed first Plumian Professor of Astronomy and Experimental Philosophy in the University of Cambridgeat the age of twenty-five. From 1709 until 1713 he was much occupied with editing the second edition of Newton’s Principia. He died in 1716 shortly before his thirty-fourth birthday.
His posthumously published works reveal wide-ranging mathematical interests, a thorough acquaintance with contemporary mathematical literature, and an outstanding ability to develop techniques and applications which frequently reveal the underlying unity of apparently diverse strands of mathematical thought. The brevity of his text and frequent absence of proof gave his work a reputation for difficulty: Logometria, the one paper he published independently during his lifetime certainly suffers from these defects, the result of reticence and modesty rather than any fundamental lack of clarity. His concern to improve on the method of integration “by cunic areas” led him to a growing understanding of the part played by complex numbers in the relationship between hyperbola areas, and elliptic and circular areas; and also to a statement equivalent to i θ =1n(cos θ + i sin θ). He attempted to develop this relationship by using a common notation for the two types of integral and his discovery of the Cotes property of the circle enabled this work to be substantially extended. In the illustrative examples Cotes displayed brilliantly his gift of developing, applying and to some extent unifying existing mathematical knowledge – here the fluxional calculus and coordinate geometry. His work in applications of the calculus represents a definite advance.
In his papers on numerical methods Cotes repeated (unknowingly) some of Newton’s work, and in the Canonotechnia developed it further, anticipating work by Stirling and others.
Cotes’ astronomical interests resulted in the paper Aestimatic Errorum, a paper which attracted quite considerable interest, particularly in France.
The new ideas in Cotes’ work (eg methods of integration iθ=1n(cosθ+i sinθ), interpolation formulae, methods of approximate numerical integration) were rapidly developed during the eighteenth century, for the most part without any published references to Cotes. Eighteenth century comment on Cotes is usually very complimentary, but detailed note of his work, apart from Aestimatic Errorum is rare. Cotes’ influence on Mathematicians and on the course of Mathematical development must therefore be regarded as slight.
Had Cotes lived, the growing confidence and reputation of this modest, likeable and much respected man must have resulted in fruitful exchanges with Taylor, Maclaurin, Stirling, De Moivre and possibly Jean Bermoulli and Euler and we might indeed “have known something”.
The Scope of the Work
Logometria (part I) is translated and a commentary provided. The Mathematical achievement in the rest of the Harmonia Mensurarum is studied in detail, summarised and discussed: contemporary commentaries are considered. Mathematical aspects of Cotes’ work as an “experimental Philosopher” are also discussed, together with relevant correspondence and some other mathematical papers which have come to light.
Cotes’ work as editor of the second edition of Newton’s Principia is not studied in this thesis.