Roger matriculated at Trinity College, Cambridge, on 6 April 1699 as a pensioner, meaning that he did not have a scholarship and paid for his own keep in College. He graduated with a B.A. in 1702 and remained at Cambridge where he was elected to a fellowship in 1705. In January 1706 he was nominated to be the first Plumian Professor of Astronomy and Experimental Philosophy. This was a remarkable achievement for Cotes who, at that time, was on 23 years of age. His exceptional abilities had been fully appreciated, however, by many at Cambridge such as William Whiston with whom he had quickly formed a friendship. Both Newton and Whiston recommended Cotes for the Chair, as did Richard Bentley who was master of Trinity College. There were some, however, who opposed his appointment, the most high profile of whom was Flamsteed, the astronomer royal. By the time that Cotes was formally elected as Plumian Professor on 16 October 1707 he had, in the previous year, been elected to a more prestigious fellowship as well as being awarded his M.A. Meli gives the background to the establishment of the chair in :-
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.It is not entirely clear how successful Cotes was in his role as an observational astronomer. In the first place there are somewhat contradictory accounts of the quality of the instruments in the Cambridge observatory. Cotes designed a transit telescope to add to a collection of instruments which had been purchased or donated. For example Newton donated a clock which still survives at Trinity College. Bentley, the master of Trinity College we mentioned above, claimed that the Observatory had "the best instruments in Europe" but an assistant who worked there wrote to Flamsteed saying "I saw nothing there that might deserve your notice". The truth is probably somewhere in between, since it would be natural for the master of Trinity to boast of the facilities, while the assistant, who only worked there for a short time, was probably trying to please Flamsteed. In terms of the observations that Cotes made, perhaps the most significant was the total eclipse on 22 April 1715. However, Halley describes this event in a paper in the Philosophical Transactions of the Royal Society where he says that 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.Cotes himself wrote a letter to Newton concerning the eclipse in which he explained that his assistant had discovered a method to determine the mid-point of the eclipse and he :-
... called out to me, "Now's the middle", though I knew not at that time what he meant.None of this speaks very highly of Cotes' dedication as an observer, but nevertheless he did note some important facts concerning this eclipse and other astronomical events. However, his mathematical abilities put him second only to Newton from his generation in England. Before going on to look at his mathematical contributions let us note that he was elected a fellow of the Royal Society on 30 November 1711, was ordained a deacon on 30 March 1713, and was ordained a priest on 31 May 1713.
From 1709 until 1713 much of Cotes' time was taken up editing the second edition of Newton's Principia. He did not simply proof-read the work, rather he conscientiously studied it, gently but persistently arguing points with Newton. For example in  a discussion is considered which took place between Cotes and Newton in 1711 concerning the velocity of water flowing from a hole in a cylindrical vessel. During the discussion they gave various approximations to the fourth root of 2 which is approximately 1.189207115. Newton gave the following rational approximations (we add decimal values to see their accuracy)
13/11 = 1.181818182
25/21 = 1.190476190
Cotes only published one paper in his lifetime, namely Logometria, published in the Philosophical Transactions of the Royal Society for March 1714, which he dedicated to Halley. It contains (in the words that Cotes used himself in a letter to Newton ):-
... a new sort of construction in geometry which appear to me very easy, simple and general.In this Cotes explained gave a method of finding rational approximations as convergents of continued fractions, and the author of  suggests that this explains how he found the approximation 44/37 to the fourth root of 2 which we mentioned above.
Cotes was particularly pleased with his rectification of the logarithmic curve as he made clear in a letter to his friend William Jones in 1712. In particular his work on logarithms led him to study the curve r = a/θ which he named the reciprocal spiral. Cotes extended the work of Varignon when he rectified the Archimedean spiral and the parabola of Apollonius, a problem first proposed by Fermat, showing that both have the same integral. His work here was based on the formula
Cotes discovered an important theorem on the nth roots of unity, gave the continued fraction expansion of e, invented radian measure of angles, anticipated the method of least squares, published graphs of tangents and secants, and discovered a method of integrating rational fractions with binomial denominators. His substantial advances in the theory of logarithms, the integral calculus, in numerical methods particularly interpolation and table construction of integrals for eighteen classes of algebraic functions led Newton to say:-
... if he had lived we might have known something.According to Edleston , Cotes died of a:-
... fever attended with a violent diarrhoea and constant delirium.He was buried four days later in the chapel of Trinity College.
Some of the work which Cotes hoped to publish with Cambridge University Press was published eventually by Thomas Simpson in The Doctrine and Application of Fluxions (2 Vols, London, 1750). Robert Smith edited Cotes' major posthumous work, the Harmonia mensurarum which appeared in 1722. It is fitting at this point to explain who Robert Smith was, and how he interacted with Cotes. In fact he was the son of Cotes' uncle, the Reverend John Smith, and had become friends with Cotes when his uncle had him live in his house as a boy. Later Robert Smith was Cotes' assistant when he was Plumian Professor, and eventually succeeded him as Plumian professor. It was Smith who, many years after Cotes' death, when he was master of Trinity College, had a bust of Cotes erected. This bust, which is shown above, is now in the Wren library.
Let us return to Cotes' posthumous work the Harmonia mensurarum . As well as reprinting Logometria it contains the three mathematical works:
2. De methodo differentiali Newtoniana.
In 1738, 22 years after Cotes died, Smith published the lectures which Cotes had given on experimental physics Hydrostatical and pneumatical lectures.
Article by: J J O'Connor and E F Robertson