Cromwell’s Code Breaker.

People who aren’t deeply cognisant with the history of seventeenth century mathematics might be forgiven for thinking that Isaac Newton was the only significant English mathematician in this century of scientific change. This is far from the truth, a fairly large group of English mathematician, now largely under the radar, made significant contributions to the discipline throughout the century. Newton, personally, listed William Oughtred, Christopher Wren and John Wallis who was born on 23 November 1616 as, in his opinion, the most important English mathematician of the century. John Wallis’ career as a mathematician was so extraordinary that one could write a novel about it; in fact somebody did. Iain Pears’ An Instance of the Fingerpost features John Wallis as one of its central characters. Why? John Wallis was a one-man Parliamentary Bletchley Park during the English Civil War, using his extraordinary mathematical talents to decipher the intercepted coded missives of the Royalist forces; he rose to fame and fortune as Cromwell’s code breaker.

John Wallis by Sir Godfrey Kneller

Wallis was the third of five children of the Reverend John Wallis minister of Ashford in Kent. He received an excellent all round education that was however completely devoid of mathematics. His only contact with mathematics, as a child, was through a book that his older brother brought home from school. Already knowledgeable in Latin, Greek and Hebrew, he went up to Emmanuel College, Oxford Cambridge in 1632 where he appears to have studied everything except mathematics. In his autobiography he wrote:

But did forward prosecute it [mathematics], as a pleasing diversion at spare hours, as books of Arithmetick or others Mathematical fell occasionally in my way, without any to direct me, what books to read, or what to seek, or in what methode to proceed. For Mathematicks were not, at that time, looked upon as Accademical Learning; but the business of Traders, Merchants, Sea-men, Carpenters, land-measurers, or the like; or perhaps some Almanak-makers in London: And of more than 200 students at that time in our College, I do not know of any two that had more of Mathematicks than myself, which was but very little; having never made it my serious studie (otherwise than as a pleasant diversion) till some little time before I was designed for a Professor in it.

This brief passage says an incredible amount about the level of mathematical education in England in the middle of the seventeenth century. A state of affairs that is confirmed in the writings of many other seventeenth century English mathematicians. Even at the beginning of the eighteenth century John Arbuthnot complained that mathematics was not taught in a single English school.

Wallis graduated BA in 1637 and MA in 1640 and upon leaving university was ordained into the priesthood. He served in various positions mostly as a private chaplain. Following the death of his mother in 1643 he inherited a substantial estate and became privately wealthy, removing the necessity to work for a living although he continued to do all of his long life.  It was at a supper party in 1642 that Wallis was first shown an encrypted letter and his talent for deciphering came to the fore; a talent that was then exploited throughout the Civil War by the Parliamentarian Forces. His detractors would later accuse him of having deciphered private letters of the Royal Family, a charge that he strenuously denied.

Some time around 1647 Wallis chanced upon a copy of William Oughtred’s Clavis Mathematicae, which according to his own account he devoured in a couple of weeks. In the absence of any real formal mathematical training within the education system, budding mathematicians were forced to teach themselves or to seek the services of a private mathematical tutor of whom Oughtred, who deserves and will receive his own blog post, was by far and away the best. His Clavis Mathematicae was justifiably considered the best algebra textbook available in Europe at the time. Oughtred tutored many of the leading English mathematicians of the period and, although he taught himself from Oughtred’s book, Wallis considered himself one of Oughtred’s pupils. Later he would dedicate one of his most important books to Oughtred and also edit and publish his posthumous writings.

In 1649 Wallis was appointed Savilian Professor of Mathematics at Oxford, the previous incumbent having been removed because of his Royalist sympathies. This was an extraordinary move, as at this point in his life Wallis had received no formal mathematical training what so ever and published no mathematical works. However he was to go on for the next forty years as one of the most able incumbents this honourable chair of mathematics has ever had. Using the facilities of the university library Wallis taught himself the whole of the then mathematical curriculum and over the years published major works on a very wide range of mathematical topics.

Wallis’ work for the Parliamentarian forces and his very obvious political appointment to the Savilian Chair might have caused him major problems at the Restoration, if it had not been for his stand at the tail of Charles I. He openly opposed the execution of the King and even signed a petition against it. The result of these actions was that Charles II confirmed his appointment to the Savilian Chair and even appointed him a Royal chaplain as well as nominating him for a committee to revise the book of common prayer.

Wallis was highly active in several of the groups that would go on to form the Royal Society of which he was a founding member. In fact his autobiography contains one of the accounts on which our knowledge of the prehistory of the Royal Society is based.

Amongst his numerous mathematical publications his most important were his Treatise on Algebra and his Arthmetica infinitorum. The former contains a detailed history of the topic as well as presenting the most complete study of the subject up till that time. The later is one of the most important works on analysis before Leibniz and Newton pulled the strands of the subject together to create the calculus. Newton, who was mostly reluctant to acknowledge any of his sources, openly admitted his debt to Wallis’ masterpiece.

Wallis was fiercely nationalist in his science, editing and promoting the posthumous works of other seventeenth century English mathematicians most notably Oughtred, Harriot and Horrocks. He even accused René Descartes of having plagiarised Harriot’s algebra; an accusation that has never been entirely disproved. (Descartes suffered badly within the European mathematical community being accused of having plagiarised the law of refraction from Snel, the mechanical philosophy from Beeckman and having Newton imply indirectly that he plagiarised the correct explanation of the rainbow from Marco Antonio de Dominis.) Descartes was not the only major European philosopher to suffer the rough edge of Wallis’ tongue. He started a major dispute with Thomas Hobbes in 1655 over Hobbes’ claim to have successfully squared the circle. The dispute rumbled on with the two heavy weight Oxford scholars firing off vitriolic pamphlets at each other at regular intervals until Hobbes’ death in 1679.

Wallis was not only the leading English mathematician of the age but he also translated and published Greek scientific works as well as writing and published extensively on a wide range of other subjects including, logic, grammar and linguistics and theology. A large and robust man with an immense intellect and a forthright manner he was both respected and loathed by his contemporaries.

Lost in the vast shadow cast by Isaac Newton, John Wallis is a towering figure of seventeenth century English intellectual history, who deserves to be much better known than he is.