“If he had lived, we might have known something”

The title of this post is Newton’s rather surprising comment on hearing of the early death of the Cambridge mathematician Roger Cotes at the age of 33 in 1716. I say rather surprising, as Newton was not known for paying compliments to his mathematical colleagues, rather the opposite. Newton’s compliment is a good measure of the extraordinary mathematical talents of his deceased associate.


Roger Cotes This bust was commissioned by Robert Smith and sculpted posthumously by Peter Scheemakers in 1758. Source: Trinity College Cambridge

Cotes the son of rector from Burbage in Leicestershire born 10th July 1682 is a good subject for this blog for at least three different reasons. Firstly he is like many of the mathematicians portrayed here relatively obscure although he made a couple of significant contributions to the history of science. Secondly one of those contributions, which I’ll explain below, is a good demonstration that Newton was not a ‘lone’ genius, as he is all too often presented. Lastly he just scraped past the fate of Thomas Harriot, of being forgotten, having published almost nothing during his all too brief life, he had the luck that his mathematical papers were edited and published shortly after his death by his cousin Robert Smith thus ensuring that he wasn’t forgotten, at least by the mathematical community.

Cotes was recognised as a mathematical prodigy before he was twelve years old. He was taken under the wing of his uncle, Robert Smith’s father, and sent to St Paul’s School in London from whence he proceeded to Trinity College Cambridge, Newton’s college, in 1699. Newton whilst still nominally Lucasian Professor had already departed for London and the Royal Mint. Cotes graduated BA in 1702. Was elected minor fellow in 1705 and major fellow in 1706 the same year he graduated MA. His mathematical talent was recognised on all sides and in the same year he was nominated, as the first Plumian Professor of Astronomy and Natural Philosophy, still not 23 years old. However he was only elected to this position on 16th October 1707. It should be noted that the newly created Plumian Chair was only the second chair for the mathematical sciences in Cambridge following the creation of the Lucasian Chair in 1663. In comparison, for example, Krakow University in Poland, the first humanist university outside of Northern Italy, already had two dedicated chairs for the mathematical sciences in the middle of the sixteenth century. This illustrates how much England was lagging behind the continent in its promotion of the mathematical sciences in the Early Modern Period.

Cotes election to the Plumian chair was supported by Richard Bentley, Master of Trinity, and by William Whiston, in the meantime Newton’s successor as Lucasian professor, who claimed to be “a child to Mr Cotes” in mathematics but was opposed by John Flamsteed, the Astronomer Royal, who wanted his former assistant John Witty to be appointed. In the end Flamsteed would be proven right, as Cotes was shown to be a more than somewhat mediocre astronomer.

Cotes’ principle claim to fame is closely connected to Newton and his magnum opus the Principia. Newton gave the task of publishing a second edition of his masterpiece to Richard Bentley, who now took on the role filled by Edmond Halley with the first edition. Now Bentley who was a child prodigy, a brilliant linguist and a groundbreaking philologist was anything but a mathematician and he delegated the task of correcting the Principia to his protégé, Cotes in 1709. Newton by now an old man and no longer particularly interested in mathematical physics had intended that the second edition should basically be a reprint of the first with a few minor cosmetic corrections. Cotes was of a different opinion and succeeded in waking the older man’s pride and convincing him to undertake a complete and thorough revision of the complete work. This task would occupy Cotes for the next four years. As well as completely reworking important aspects of Books II and III this revision produced two highly significant documents in the history of science, Newton’s General Scholium at the end of Book III, a general conclusion missing from the first edition, and Cotes’ own preface to the book. Cotes’ preface starts with a comparison of the scientific methodologies of Aristotle, the supporters of the mechanical philosophy, where here Descartes and Leibniz are meant but not named, and Newton. He of course come down in favour of Newton’s approach and then proceeds to that which Newton has always avoided a discussion of the nature of gravity introducing into the debate, for the first time, the concept of action at a distance and gravity as a property of all bodies. The second edition of Principia can be regarded as the definitive edition and is very much a Newton Cotes co-production.

Cotes’ posthumously published mathematical papers contain a lot of very high class but also highly technical mathematics to which I’m not going to subject my readers. However there is one of his results that I think should be better known, as the credit for it goes to another. In fact a possible alternative title for this post would have been, “It’s not Euler’s Formula it’s Cotes’”.

It comes up fairly often that mathematicians and mathematical scientists are asked what their favourite theorem or formula is. Almost invariably the winner of such poles polls is what is known technically as Euler’s Identity

e + 1 = 0


Now this is just one result with x = π of Euler’s Formular:

cosx  + isinx = eix

Where i is the square root of -1, e is Euler’s Number the base of natural logarithms and x is an angle measured in radians. This formula can also be expressed as a natural logarithm thus:

ln[cosx + isinx] = ix

and it is in this form that it can be found in Cotes’ posthumous mathematical papers.

As one mathematics’ author expresses it:

This identity can be seen as an expression of the correspondence between circular and hyperbolic measures, between exponential and trigonometric measures, and between orthogonal and polar measures, not to mention between real and complex measures, all of which seemed to be within Cotes’ grasp.

Put simply, for the non-mathematical readers, this formula is one of the most important fundamental relationships in analysis.

Cotes died unexpectedly on 5th June 1716 of a, “Fever attended with a violent Diarrhoea and constant Delirium”. Despite his important contributions to Newton’s Principia Cotes is largely forgotten even by mathematicians and their ilk so the next time somebody waxes lyrical about Euler’s Formula you can gentle point out to them that it should actually be called Cotes’ Formula.


Roger Cotes Memorial Stone Trinity College Cambridge



Rogerus Roberti filius COTES

CollegI [sic] hujus S Trinitatis socius
Astronomiae et Experimantalis
Philosophiae Professor Plumianus;
qui immatura Morte praereptus,
Pauca quidem ingenij sui Pignora reliquit;
Sed egregia, sed admiranda,
ex in accessis Matheseos Penetralibus
Felici sol[l]ertia tum primum eruta:
post magnum illum Newtonum
Societatis hujus Spes altera, et decus gemellum:
Cui ad Summam Doctrinae Laudem
Omnes Morum Virtutumque Dotes
In cumulum accesserunt:
Eo magis spectabiles amabilesque,
Quod in formoso Corpore Gratiores venirent.
Natus Burbagij in Agro Leicestriensi Iul. X MDCLXXXII.
Obiit Iun. V. MDCCXVI.

Here is buried Roger Cotes son of Robert Cotes, Fellow of Trinity College and Plumian Professor of Astronomy and Experimental Philosophy.  Snatched away by an early death, he left few tokens of his genius; but these few are of no common kind, exciting admiration for their having been first unearthed from the very heart of learning by his successful intelligence.  After the great Newton he was the second hope of the College, and its twin ornament.  Every possible quality of character and virtuous behaviour crowned his scientific fame; and these qualities were the more splendid and attractive because they were lodged in a handsome body.  He was born in Burbage in Leicestershire on 10th July 1682 and died on 5th June 1716.


Filed under History of Astronomy, History of Mathematics, History of Physics, Newton

8 responses to ““If he had lived, we might have known something”

  1. Amen, and well done. Cotes has been one of my favorites since I read of him in HS.

  2. Markk

    Well his name is still remembered by many students (whether they want to or not) if he is the Cotes of the Newton-Cotes method of numeric integration, which I believe is true.

  3. I had just been thinking of him, when I saw it was (arguably) his birthday. I’d first run across him when I was supplying biographical footnotes to people as I mentioned them in class, and Newton’s comment is an irresistible hook.

    MacTutor’s biography credits him with the use of radians as angle measure. I’ve been trying to learn well enough that it’s almost never that simple, but, after all, there do have to be the people who develop a useful idea to the point it becomes widely accessible, and his logarithm form of the Euler identity certainly looks like the sort of thing that’d convince people radians were especially important.

  4. Jim Cliborn

    Nice post, thank you for the insights!

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  7. Ot

    Thus, Euler read Cotes writings and plagiarized the formula?

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