In 1706 William Jones published a work Synopsis palmariorum matheseos or, A New Introduction to the Mathematics, Containing the Principles of Arithmetic and Geometry Demonstrated in a Short and Easie Method ... Designed for ... Beginners. This contains on page 243 the following passage:-
There are various other ways of finding the lengths or areas of particular curve lines, or planes, which may very much facilitate the practice; as for instance, in the circle, the diameter is to the circumference as 1 to (16/5- 4/239) - 1/3(16/53- 4/2393) &c. = 3.14159 &c. = π. This series (among others for the same purpose, and drawn from the same principle) I received from the excellent analyst, and my much esteemed friend Mr John Machin; and by means thereof, van Ceulen's number, or that in Art. 64.38 may be examined with all desirable ease and dispatch.Jones also reports that this formula allows π be calculated:-
... to above 100 places; as computed by the accurate and ready pen of the truly ingenious Mr John Machin.No indication is given in Jones's work, however, as to how Machin discovered his series expansion for π so when de Moivre wrote to Johann Bernoulli on 8 July 1706 telling him about Machin's series for π he suggested that Johann Bernoulli might tell Jakob Hermann about Machin's unproved result. He did so and Hermann quickly discovered a proof that Machin's series converges to π. He produced techniques that show other similar series also converge rapidly to π and he wrote on 21 August 1706 to Leibniz giving details. Two years later, on 6 July 1708, de Moivre wrote again to Johann Bernoulli about Machin's series, on this occasion giving two different proofs that it converged to π.
On 30 November 1710 Machin was elected a Fellow of the Royal Society. Keill had published a paper in the Transactions of the Royal Society of London which accused Leibniz of plagiarism and Leibniz wrote to the Royal Society complaining strongly. Keill repeated his accusations in a letter to Leibniz saying that two letters from Newton, sent to Leibniz through Oldenburg, must have given him the principles of the calculus. Leibniz wrote again to the Royal Society asking them to correct the wrong done to him by Keill's claims. In response to this letter the Royal Society set up a committee to pronounce on the priority dispute. It was totally biased, not asking Leibniz to give his version of the events and having a membership consisting of Newton's friends. Machin, as well as Keill and Taylor, sat on the committee and, of course, found in favour of Newton.
We have already mentioned that Taylor was a friend of Machin and that the two corresponded about mathematical questions. Taylor wrote to Machin on 26 July 1712 stating what we now call Taylor's theorem. Taylor wrote in this letter that a comment made by Machin during a coffeehouse conversation had given him the idea. Machin had explained to Taylor in Child's Coffeehouse how to use Newton's series to solve Kepler's problem and also how Halley's method finds roots of polynomial equations. This was the spark which led Taylor to one of two versions of his theorem which he published three years later.
On 16 May 1713 Machin was appointed as Professor of Astronomy at Gresham College, London. He succeed Dr Torriano and went on to hold the chair until his death 38 years later. For nearly 30 of these years he acted as Secretary to the Royal Society, being appointed in 1718 and holding the post until 1747.
Bernard Cohen and Anne Whitman write of the third edition of Newton's Principia :-
While Newton was planning for a third edition, he received two independent solutions of the problem of the motion of the nodes of the moon's orbit, one by John Machin, the other by Henry Pemberton. He chose to include in the third edition the presentation by Machin rather than the one by Pemberton.A Scholium was added to Book 3, Proposition 33 of the third edition of the Principia which begins:-
J Machin, Professor of Astronomy, and Henry Pemberton, M.D., have independently found the motion of the nodes by yet another method. Some mention of the latter's method have been made elsewhere. And the papers (which I have seen) of both men contained two propositions, which agreed with each other. Here I shall present Mr Machin's paper, since it was the first to come into my hands.However, in , Machin's paper The Laws of the Moon's Motion According to Gravity which was included in Andrew Motte's 1729 English translation of Newton's Principia is described at "a poor performance". Machin's Quadrature of the Circle appeared as an appendix to Maseres' A Dissertation on the Use of the Negative Sign in Algebra: Containing a Demonstration of the Rules Usually Given Concerning it, published in London in 1758. In Scriptores Logarithmic (1796) Maseres writes:-
Mr William Jones's Synopsis palmariorum matheseo which was published in the year 1706 is the only book in which as I believe the series of Mr Machin had ever made its appearance before the publication of my Dissertation on the Use of the Negative Sign in Algebra in the year 1758. And the short and obscure manner in which it is stated in the foregoing passage of Mr Jones's book left the mathematicians in England in the dark as to the method in which it was obtained.One other publication by Machin is worth noting, namely The solution of Kepler's problem which was published in the Philosophical Proceedings of the Royal Society in 1738. However this work, like most of his contributions to astronomy, is not highly rated.
Clerke writes in  about another astronomy project:-
A large work on the lunar theory taken in hand by [Machin] in 1717 never saw the light, but a mass of his manuscripts is preserved by the Royal Astronomical Society; and writing to Jones in 1727, he asserted his claim to the parliamentary reward of £10,000 for amending the lunar tables.Machin's work on the series for has proved of lasting importance, but most of his other contributions are not of the same high standard. Despite this he had a considerable influence of the development of mathematics in England and, in his time, :-
... enjoyed a high mathematical reputation.
Article by: J J O'Connor and E F Robertson