# Dictionary of National Biography, 1885-1900/Cotes, Roger

​COTES, ROGER (1682–1716), mathematician, was the second son of the Rev. Robert Cotes, rector of Burbage in Leicestershire, where he was born 10 July 1682. His mother, Grace, daughter of Major Farmer of Barwell in the same county, was connected with the noble family of the De Greys. Before the age of twelve he discovered, while at Leicester school, so marked an aptitude for mathematics, that his uncle, the Rev. John Smith, took him to his house in Lincolnshire, that he might personally forward his studies. Removed to St. Paul's School, London, he made rapid progress in classics under Dr. Gale, then head-master, while keeping up a scientific correspondence with his uncle, portions of which have been preserved and published (Correspondence of Newton and Cotes, p. 190 et seq.) He was admitted a pensioner of Trinity College, Cambridge, 6 April 1699, was chosen fellow at Michaelmas 1705, and acted as tutor to his relatives, the sons of the Marquis, afterwards Duke, of Kent. In the following year he proceeded M.A., having taken a degree of B.A. in 1702. While still an undergraduate, his extraordinary proficiency in science had attracted the notice of Dr. Bentley, the master of his college. Bentley introduced him to Newton and Whiston, whose testimonials in his favour, combined with Bentley's influence, procured his election, in January 1706, to the new professorship of astronomy and natural philosophy founded by Dr. Plume, archdeacon of Rochester, then recently dead. Whiston, who, as occupant of the Lucasian chair, was one of the electors, thus describes his share in the transaction: ‘I said that I pretended myself to be not much inferior in mathematics to the other candidate's master, Dr. Harris, but confessed that I was a child to Mr. Cotes; so the votes were unanimous for him’ (Whiston, Memoirs, p. 133).
The project of founding, with his co-operation, a first-class astronomical observatory in Trinity College was now eagerly embraced by Bentley. He raised a subscription for its erection over the King's Gate, and obtained a college order, assigning the chambers there in perpetuity to the Plumian professor. Here, accordingly, during the remaining decade of his life, Cotes dwelt with his cousin, Robert Smith, whom he chose as his assistant; and here his lectures were delivered. He did not live to see the observatory finished, and it was demolished in 1797. A brass sextant of five feet radius, constructed by Rowley at a cost of 150l., was part of its equipment; Newton contributed a fine pendulum clock; and a transit instrument was in hand early in 1708 (Corr. of Newton and Cotes, p. 198). The total solar eclipse of 22 April (O.S.) 1715 furnished Cotes with the opportunity of making his only recorded astronomical observation, relative to which Halley communicated the following particulars to the Royal Society:—
‘The Rev. Mr. Roger Cotes at Cambridge had the misfortune to be opprest by too much company, so that, though the heavens were very favourable, yet he missed both the time of the beginning of the eclipse and that of total darkness. But he observed the occultations of the three spots … also the end of total darkness, and the exact end of the eclipse’ (Phil. Trans. xxix. 253).
His description and drawing, however, of the sun's corona, transmitted 12 May to Newton, amply compensate some technical shortcomings. A brilliant ring, about one-sixth the moon's diameter, was perceived by him superposed upon a luminous cross, the longer and brighter branches of which lay very nearly in the plane of the ecliptic. The light of the shorter (polar) arms was so faint as not to be constantly visible (Corr. of Newton and Cotes, pp. 181–4). This is precisely the type of corona seen in 1867 and 1878, and associated therefore with epochs of sunspot minimum. But spots were numerous in 1715, so that Cotes's observation goes far to disprove the supposed connection.
In the beginning of 1709 Bentley at length persuaded Newton, by the offer of assistance from Cotes, to consent to a reissue of the ‘Principia.’ It was not, however, until September that a corrected copy of the work was placed in the hands of the new editor, when the remarkable correspondence between him and Newton ensued, preserved in the original in the library of Trinity College, and published by Mr. Edleston in 1850. It must be admitted that the younger man's patience was often severely tried by Newton's long cogitations over the various points submitted to him; but it proved imperturbable. ‘I am very desirous,’ he wrote to Sir William Jones, 30 Sept. 1711, ‘to have the edition of Sir Isaac Newton's “Principia” finished, but I never think the time lost when we stay for his further corrections and improvements’ (Corr. of Newton and Cotes, p. 209). Of all his contemporaries, Cotes possessed the strongest and clearest grasp of the momentous principles enunciated by his author. He suggested many rectifications and improvements, for the most part adopted by Newton. The frequently interrupted process of printing occupied some three and a half years. Cotes's preface, an able defence of the Newtonian ​system against Cartesian and other objectors, was dated 12 May 1713; the impression at the University Press was finished about the middle of June. The reception of the work was most flattering to the editor. His preface was retained, in the original Latin, in the edition of 1726, and was anglicised in Andrew Motte's English version of the ‘Principia’ in 1729. Bentley was profoundly gratified at the encomium upon himself contained in it; and spoke of Cotes, in a letter to Bateman, as ‘one of the finest young men in Europe’ (Monk, Life of Bentley, p. 266).
Cotes was chosen a member of the Royal Society in 1711; he took orders in 1713. His sole independent appearance as an author during his lifetime was in an essay styled ‘Logometria,’ inscribed to Halley, and communicated to the Royal Society in 1713 by the advice of Newton (Phil. Trans. xxix. 5). It treated of measures of ratios, contained directions for constructing Briggs's canon of logarithms, and exemplified its use for the solution of such problems as the quadrature of the hyperbola, the descent of bodies in a resisting medium, and the density of the atmosphere at any given height. Designs of further publication, timidly entertained, were destined to prove abortive. Cotes died 5 June 1716, of a violent fever, in the thirty-fourth year of his age. ‘Had Cotes lived,’ Newton exclaimed, ‘we might have known something!’ And he was no less loved than admired, attractive manners combining with beauty of person and an amiable disposition to endear him to all with whom he came in contact. He was buried in the chapel of Trinity College, the restoration of which he had actively superintended; and the monument erected to his memory by his cousin and successor, Robert Smith, was adorned with an epitaph composed by Bentley under the influence of genuine sorrow. The master was not only attached to him as a friend, but valued him as one of his most zealous adherents; and had entertained the highest expectations of his career. Its premature close was felt in his college as a calamity the keen sense of which the lapse of a century failed to obliterate.
Robert Smith undertook the office of his literary executor. His papers were found in a state of baffling confusion. The resulting volume, dedicated to Dr. Richard Mead, bore the title ‘Harmonia Mensurarum, sive Analysis et Synthesis per Rationum et Angulorum Mensuras promotæ: Accedunt alia Opuscula Mathematica per Rogerum Cotesium. Edidit et auxit Rob. Smith,’ Cambridge, 1722. The first part included a reprint from the ‘Philosophical Transactions’ of the ‘Logometria,’ with extensive developments and applications of the fluxional calculus. The beautiful property of the circle known as ‘Cotes's Theorem’ was here first made known. Two months before his death Cotes had written to Sir W. Jones, ‘that geometers had not yet promoted the inverse method of fluxions, by conic areas, or by measures of ratios and angles, so far as it is capable of being promoted by these methods. There is an infinite field still reserved, which it has been my fortune to find an entrance into’ (Phil. Trans. xxxii. 146), adding instances of fluxional expressions which he had found the means of reducing. Upon this letter Dr. Brook Taylor based a challenge to foreign mathematicians, successfully met by John Bernoulli in 1719; and by it Smith was incited to a search among Cotes's tumbled manuscripts for some record of the discovery it indicated. His diligence rescued the theorem in question from oblivion. It was generalised by Demoivre in 1730 (Miscellanea Analytica, p. 17), and provided by Dr. Brinkley in 1797 with a general demonstration deduced from the circle only (Trans. R. Irish Acad. vii. 151).
The second part of the volume comprised, under the heading ‘Opera Miscellanea,’ 1. ‘Æstimatio Errorum in mixta Mathesi per variationes Partium Trianguli plani et sphærici.’ The object of this tract was to point out the best way of arriving at the most probable mean result of astronomical observations. It is remarkable for a partial anticipation of the ‘method of least squares,’ as well as for the first employment of the system of assigning different weights to observations (p. 22, see also A. De Morgan, Penny Cycl. xiii. 379). It was reprinted at Lemgo in 1768, and its formulæ included in Lalande's ‘Traité d'Astronomie.’ 2. ‘De Methodo Differentiali Newtoniana’ professes to be an extension of the method explained in the third book of the ‘Principia,’ for drawing a parabolic curve through any given number of points. 3. ‘Canonotechnia’ treats of the construction of tables by the method of differences. Its substance was translated into French by Lacaille in 1741 (Mem. Ac. des Sciences, 1741, p. 238). Three short papers, ‘De Descensu Gravium,’ ‘De Motu Pendulorum in Cycloide,’ and ‘De Motu Projectilium,’ followed, besides copious editorial notes.
Cotes's ‘Harmonia Mensurarum’ was, Professor De Morgan says, ‘the earliest work in which decided progress was made in the application of logarithms and of the properties of the circle to the calculus of fluents’ (Penny Cycl. viii. 87). But though highly ​praised, it was little read. The style was concise even to obscurity. A requisite and excellent commentary was, however, furnished by Dr. Walmesley in 1753 (Analyse des Mesures, des Rapports, et des Angles). Cotes's ‘theorem of harmonic means,’ discovered by Smith among his papers, and communicated to Maclaurin, was made the basis of the latter's treatise, ‘De linearum geometricarum proprietatibus generalibus’ (London, 1720).
Smith announced his intention of publishing further papers by Cotes on arithmetic, the resolution of equations, dioptrics, and the nature of curves, but it remained unfulfilled. Only in his own work on optics he founded a chapter (ch. v. book ii.) on a ‘noble and beautiful theorem,’ stated to have been the last invention of his lamented relative. He edited, moreover, in 1738, his ‘Hydrostatical and Pneumatical Lectures,’ issued for the third time in 1775, and translated into French by Lemonnier in 1740 under the title ‘Leçons de Physique Expérimentale.’ The course of experiments for which they were composed, begun at Cambridge by Cotes and Whiston conjointly, 5 May 1707, was among the earliest of its kind given in England. Twelve lectures were written by each of the partners, and were repeated by Whiston and Hauksbee in London, and, in part, by Smith at Cambridge. The publication of Cotes's set was finally compelled by the prospect of a surreptitious edition. Whiston considered his own so inferior that he could never prevail upon himself to print them.
A ‘Description of the Great Meteor,’ a brilliant aurora, ‘which was on the 6th of March 1716 sent in a letter from the late Rev. Mr. Roger Cotes to Robert Dannye, D.D., rector of Spofferth in Yorkshire,’ was included in the ‘Philosophical Transactions’ for 1720 (xxxi. 66). Cotes's zeal for practical astronomy only waited opportunity for full development. He remodelled Flamsteed's and Cassini's solar and planetary tables, and had undertaken to construct tables of the moon on Newtonian principles; while his description of a heliostat-telescope furnished with a mirror revolving by clockwork (Corr. of Newton and Cotes, p. 198) showed that he had already in 1708 (independently, it is probable, of Hooke's project of 1674), anticipated the system of equatorial mounting.
[Biog. Brit. (Kippis); Phil. Trans. Abridg. vi. 77 (1809); Gen. Dict. iv. 441 (1736); Nichols's Lit. Anecd. ii. 126; Nichols's Leicestershire, iv. 35, 472; Knight's Life of Colet, p. 429; Monk's Life of Bentley, passim; Whiston's Memoirs, pp. 133–5; Edleston's Correspondence of Newton and Cotes; Rigaud's Correspondence of Scientific Men, i. 257–70; Smith's Pref. to Harmonia Mensurarum; Cole's Athenæ Cantab. Add. MS. 5865, f. 53; Hutton's Mathematical Dict. (1815), Introduction to Math. Tables, p. 112, and Math. Tracts, i. 437; Montucla's Hist. des Mathématiques, iii. 149; Suter's Gesch. der math. Wissenschaften, ii. 133; Nouvelles Annales de Math. ix. 195 (1850); Delambre, Hist. de l'Astronomie au xviiie Siècle, p. 449; Marie's Hist. des Math. vii. 222.]
A. M. C.