**John Machin** was an English mathematician and astronomer best known for the formulas he invented for calculating $\pi$.

- We know that he acted as a private tutor to Brook Taylor teaching him mathematics in 1701, two years before Taylor entered St John's College Cambridge.
- We also know that Machin was friendly with Keill, who taught at Oxford, and with de Moivre who like Machin was a private tutor of mathematics at this time.
- 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.
- 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.
- 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.
- On 16 May 1713 Machin was appointed as Professor of Astronomy at Gresham College, London.
- He chose to include in the third edition the presentation by Machin rather than the one by Pemberton.
- 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.
- 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.
- 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.

Born 1680, England. Died 9 June 1751, London, England.

View full biography at MacTutor

Astronomy, Origin England, Number Theory, Special Numbers And Numerals

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive