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