If a mathematician knows just one thing about John Pell, it probably concerns the Diophantine equation that bears his name: y2 = ax2 + 1. The attribution of this equation to Pell is due to Euler and it is often called a misattribution, because the equation actually appeared in the text Teutsche Algebra by Johann Heinrich Rahn. However, Pell had a significant role in the composition of Rahn’s book, so the attribution may in fact be quite well deserved.
The collaboration between Pell and Rahn is just one of many episodes in 17th century mathematics documented in the account of the life and work of John Pell by Malcolm and Stedall constructed to contextualize Pell’s correspondence with his patron, Sir Charles Cavendish. This impressive historical volume consists of three parts: an account of the life of John Pell by Malcolm, an account of the mathematics of John Pell by Stedall, and an annotated edition of the correspondence between the two.
The correspondence itself consists of 115 letters written between 1741 and 1751. For most of this period, Pell is a professor in Amsterdam and then in Breda, while Cavendish is living in various locations on the Continent: Wellingor, Hamburg, Paris and Antwerp. The letters are from Pell’s papers, so they consist of originals by Cavendish and drafts or fair copies by Pell. A large part of the discussion, but by no means all of it, is of a mathematical nature. In the opening letters Pell and Cavendish are trying to make a telescope using hyperbolic mirrors, according to a design by Descartes. Much of the rest of the rest of the technical discussion concerns topics of pure mathematics, primarily algebra and geometry. In many cases, they are discussing the works of important contemporary mathematicians. The names dropped in the course of these letters read like a Who’s Who of the 17th century: Descartes, Mersenne, van Schooten, Bonaventura, Torricelli, Roberval, as well as non-mathematicians, such as Hobbes.
Pell was a contemporary of Boyle, Hooke and Wren and an early member of the Royal Society, rising to the office of vice president. He left a vast number of manuscripts, collected in the British museum, but published little in his lifetime. His mathematical achievements include a particular way of constructing an algebraic proof in a three-column layout, called Pell’s Method by Wallis, which Stedall describes as being “remarkably close to a modern computer algorithm.” He also discovered the double angle formula for tangents, and greatly improved methods for calculating tables of logarithms using finite differences – a task which occupied him for a number of years. Pell was particularly crucial in disseminating the ideas Thomas Harriot, also largely unpublished, to whose work he was exposed by Thomas Aylesbury.
I enjoyed all three parts of the book, although I suspect the interest of most mathematicians would be concentrated on the second piece. Stedall’s narrative concerning Pell’s mathematics gives an excellent overview of the state of mathematics in the middle of the 17th century, the period between Viète and Newton. This section of the book will make an excellent addition to any library’s collection of history of mathematics books. Although it appears to be written with the historian of mathematics in mind, it could equally well serve as material for undergraduate research projects.
In 1939, at a time when almost no attention had yet been paid to Pell’s manuscripts and letters, H. W. Turnbull suggested that Pell “may well prove to be an unsuspected genius.” The authors of this book conclude that although he was one of the leading mathematicians of his time and was possessed of great intellectual ability, he fell just short of being a “genius.” Nevertheless, his mathematical correspondence is fascinating and is presented here along with an engaging account of his life and times.
Rob Bradley is Professor of Mathematics at Adelphi University. He is president of the CSHPM (the Canadian Society for History and Philosophy of Mathematics) and the Euler Society.