Cotes, Roger

(1682-1716), mathematician and astronomer

by Domenico Bertoloni Meli

© Oxford University Press 2004 All rights reserved

Cotes, Roger (1682-1716), mathematician and astronomer, was born on 10 July 1682 at Burbage, Leicestershire, the second son of Robert Cotes, rector of Burbage, and his wife, Grace, daughter of Major Farmer of Barwell, Leicestershire. Besides his elder brother, Anthony (b. 1681), Roger had a younger sister, Susanna (b. 1683). He was first educated at Leicester School, where his talent in mathematics was noticed before he was twelve. The Revd John Smith, Cotes's uncle, therefore took him to his house so that he might personally forward him in his studies. Smith, married to Cotes's aunt Hannah Cotes, had a son, Robert Smith (bap. 1689, d. 1768), whose life was to be closely linked to that of his cousin. Cotes later studied at St Paul's School, London, from where he corresponded with his uncle on mathematical matters. He was admitted as a pensioner to Trinity College, Cambridge, on 6 April 1699, and graduated BA in 1702. He was elected minor fellow in 1705, major fellow in 1706, and proceeded MA in the same year. In January 1706 he was nominated first Plumian professor of astronomy and experimental philosophy, though his election did not take place until 16 October 1707. His appointment was favoured by his influential mentor Richard Bentley, master of Trinity; by Newton's successor as Lucasian professor, William Whiston, who claimed to be in mathematics 'a child to Mr Cotes' (Whiston, 133); and by Newton himself. It was, however, opposed by the astronomer royal John Flamsteed, who favoured his former assistant John Witty. Cotes was elected fellow of the Royal Society on 30 November 1711, and was ordained deacon on 30 March and priest on 31 May 1713.

Cotes was the first occupant of the Cambridge chair established by Thomas Plume (1630-1704), archdeacon of Rochester, who bequeathed nearly £2000 to maintain a professor and erect an astronomical observatory. Plans for an observatory at Trinity had already been drafted by Bentley before Plume's bequest. The observatory was eventually housed over the king's or great gate at Trinity College, together with living quarters for the Plumian professor. Cotes lived there with his cousin Robert Smith, who worked as his assistant and later succeeded him to the Plumian chair. On his appointment Cotes raised a subscription to supplement the costs for the erection of the observatory and supervised its construction. Although it was operational in his time it was not completed until 1739, and was demolished in 1797. Between August and September 1703 a number of instruments had been destined for the observatory, and these were supplemented following Plume's endowment with a brass sextant by John Rowley worth £150 and a clock donated by Newton worth about £50, now in the master's lodge at Trinity College. Further, Cotes's design for a transit telescope is carefully described in a manuscript at Clare College, Cambridge. Bentley said the observatory was 'well stor'd with the best instruments in Europe' (Willis, 500); however, Stephen Gray, who worked there for a few months as Cotes's assistant, stated in a letter to Flamsteed: 'I saw nothing there that might deserve your notice' (Cohen, 'Neglected sources', 47).

The most notable observation by Cotes was that of the total eclipse of 22 April 1715 OS, reported by Edmond Halley in the Royal Society's Philosophical Transactions. Cotes 'had the misfortune to be opprest by too much company, so that though the heavens were very favourable, yet he missed both the times of the beginning of the eclipse and that of total darkness' (PTRS, 29, 1714, 253). In a letter to Newton, Cotes reported that his assistant Richard Waller, who had a method for determining the middle of the obscuration, 'called out to Me, Now's the Middle, though I knew not at that time what he meant' (Edleston, 182). However, Cotes was able to observe the occultation of three spots, the end of total darkness, and the end of the eclipse, as well as to produce for Newton drawings by himself and an 'ingenious Gentleman' showing a brilliant ring about a sixth of the diameter of the moon at the centre of a luminous cross. Another observation by Cotes of March 1716, this time of a 'Great Meteor', was published in 1720 (PTRS, 31, 60).

In 1709 Cotes became heavily involved in the work for which he is best remembered, namely the revisions for the second edition of Newton's Philosophia naturalis principia mathematica, the first being out of print. Although Newton originally thought this would be a relatively speedy matter, it was not until 1713 that the 750 copies of the revised edition appeared with the imprint of Cambridge University Press, recently revived by Bentley. Cotes stated: 'I never think the time lost when we stay for his further corrections and improvements' (Edleston, 209). Indeed, virtually all aspects of Newton's work were thoroughly and painstakingly examined, including definitions and stylistic matters. Among the topics in which Cotes became most involved were the theory of tides, the theory of the moon's motion, and the determination of cometary paths. The last topic was of great importance, because the regularity of cometary motion in all directions and inclinations to the plane of the ecliptic was a powerful argument against the existence of vortices and fluids carrying the planets. Newton was lucky to have an editor such as Cotes, though their relations became strained during their collaboration: in his final version of the preface he failed to thank the Plumian professor. A preliminary draft, however, did contain a small tribute to him (Correspondence of Isaac Newton, 5.114).

After Cotes had proposed to sign his name to a preface composed by Newton, he was authorized by Newton and Bentley to sign his own preface, a lengthy and important essay outlining his own version of Newton's method. In a preliminary draft Cotes had stated that gravity was essential to bodies. However, Newton's ally Samuel Clarke, whose advice Cotes had sought before publication, suggested a number of corrections and warned him against such a statement. Cotes agreed that 'it would have furnish'd matter for Cavilling' (Edleston, 158), and claimed instead that gravity was one of the primary qualities of bodies, together with extension, mobility, and impenetrability. His preface was retained in the third edition of Newton's Principia (1726), was translated into English by Andrew Motte, and is now easily available in the revised edition by F. Cajori. In it Cotes outlined three methods for studying natural philosophy. The first, used by Aristotle and the Peripatetics, relied on 'occult', or hidden, qualities, and was entirely based on giving names to things. The second method was based on sounder assumptions, namely that all matter is homogeneous. Its adherents, however, though they rejected the proliferation of empty words, arbitrarily imagined occult fluids agitating with occult motions and pervading the pores of bodies, thus relying on fallacious hypotheses and chimeras: therefore their works were an ingenious romance. Although Cotes does not mention Descartes and the Cartesians, it is not difficult to identify them as his targets. Indeed, portions of the preface were specifically aimed at Leibniz and his 'Tentamen de motuum coelestium causis', though it was agreed by Cotes and Newton that Leibniz's name would not be mentioned. The third method relied on experimental philosophy. Its adherents assumed no arbitrary first principles and did not rely on hypotheses. From select phenomena they found the forces and their modes of operation, and then from the forces they were able to show the constitution of the remaining phenomena. In this fashion they were able to establish universal gravity. Needless to say, this was the method adopted by Newton and the Newtonians.

Cotes also published 'Logometria' (PTRS, 29, 1714, 5-47), a mathematical essay dedicated to Halley and announced by Cotes in a letter to Newton as containing 'a new sort of Constructions in Geometry which appear to me very easy, simple & general' (Edleston, 117). Although the problems there treated were not new, Cotes was able to provide interesting solutions. His essay was reprinted as the first part of the Harmonia mensurarum (1722), a posthumous work edited by his literary executor, Robert Smith, which established Cotes as probably the most talented British mathematician of the generation after Newton. Though the Harmonia provides not simply solutions but also methods, it is not for the faint-hearted, containing dozens of beautifully printed pages of mathematics, often without a single word of text. The review in the Philosophical Transactions, possibly by William Jones (PTRS, 32, 1722, 139-50), provides an extensive and helpful contemporary assessment. Cotes's work deals with the 'harmony' between measures of angles (trigonometric quantities) and measures of ratios (logarithms), which he tries to treat conjointly with a single notation. In his work he shows great interest and skill in the problem of quadratures, developing new methods often inspired by and related to his revision of Newton's Principia. His skill in this area can be traced back to his first surviving letter to Newton, when he was bold enough to correct the latter on two points (August 1709). His tables of quadratures in the Harmonia were more than an ingenious set of techniques, because the 'harmony of measures' he had found was of deep mathematical import. Smith was able to include the statement of one of Cotes's last achievements, the factorization theorem, which, thanks to a reference found in a letter of 5 May 1716 from Cotes to William Jones, he managed to rescue from among the papers left by Cotes at his death.

The Harmonia also contains three opuscula mathematica of considerable interest. In the first, Aestimatio errorum in mixta mathesis, Cotes studied the proportions among the least contemporary variations of the sides and angles of plane and spherical triangles. His work had important applications to astronomy and aroused considerable interest, especially among French astronomers such as J. J. de Lalande, N. L. de Lacaille, who translated Cotes's tract, and J. B. J. Delambre. The Aestimatio errorum ends with a brief discussion of how to find the most probable place of an object by weighing the errors of a number of slightly different observations. That discussion gained Cotes a place in the history of eighteenth-century error theory. The second opusculum, De methodo differentiali Newtoniana, develops the method of interpolation of curves described by Newton in the Principia (book 3, lemma 5), which was of considerable significance for cometography. Lastly, Canonotechnia is a further development of Newton's methods of interpolation and approximate integration which can be found in his Methodus differentialis (1711). Cotes's book ends with three brief applications and an explicatory note by Robert Smith.

Smith was also responsible for another posthumous edition of Cotes's works, Hydrostatical and Pneumatical Lectures, which went through three English editions between 1738 and 1775 and was translated into French by Lemonnier (Paris, 1742). The course of lectures for which they had been written was begun by Cotes and Whiston in 1707, though Smith did not include the lectures prepared by Whiston. Smith's publication, which was prompted by the prospect of an unauthorized edition, provides an interesting picture of natural philosophy teaching at Cambridge in the eighteenth century.

Cotes died unexpectedly on 5 June 1716 of a 'Fever attended with a violent Diarrhoea and constant Delirium' (Edleston, lxxvi). He was buried on 9 June in Trinity College chapel, and Richard Bentley composed an epitaph inscribed on his memorial. In 1758 Robert Smith, by then master of Trinity College, arranged for the erection of a bust in his cousin's memory by Peter Scheemakers, which is now in the Wren Library. On the page reproducing Bentley's epitaph in his own copy of the Harmonia mensurarum, Smith wrote: 'Sir Isaac Newton, speaking of Mr. Cotes, said "If he had lived we might have known something"' (Gowing, 141).


Correspondence of Sir Isaac Newton and Professor Cotes, ed. J. Edleston (1850)
R. Gowing, Roger Cotes: natural philosopher (1983)
'An account of the book, intituled Harmonia mensurarum', PTRS, 32 (1722-3), 139-50
The correspondence of Isaac Newton, ed. H. W. Turnbull and others, 5-6 (1975-6)
D. J. Price, 'The early observatory instruments of Trinity College, Cambridge', Annals of Science, 8 (1952), 1-12
A. Koyré, Newtonian studies (1965), 273-82
I. B. Cohen, Introduction to Newton's 'Principia' (1971)
I. B. Cohen, 'Neglected sources for the life of Stephen Gray', Isis, 45 (1954), 41-50
R. A. Chipman, 'The manuscript letters of Stephen Gray, FRS, 1666/7-1736', Isis, 49 (1958), 414-33
S. P. Rigaud and S. J. Rigaud, eds., Correspondence of scientific men of the seventeenth century, 1 (1841), 257-70
W. Whiston, Memoirs of the life and writings of Mr William Whiston: containing memoirs of several of his friends also (1749)
R. Willis, The architectural history of the University of Cambridge, and of the colleges of Cambridge and Eton, ed. J. W. Clark, 4 vols. (1886)
Isaac Newton's Philosophiae naturalis principia mathematica, ed. A. Koyré, I. B. Cohen, and A. Whitman, 2 (1972), 817-26
parish register (baptism), 25 July 1682, Burbage, Leicestershire

CUL, Cambridge lectures on hydrostatics and pneumatics [fair copy]
Trinity Cam.
Whipple Museum, Cambridge, astronomical instruments |  Clare College, Cambridge, Morgan MSS

P. Scheemakers, marble bust, Trinity Cam., Wren Library; repro. in D. McKitterick, The making of the Wren Library (1995), 124

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