John Wallis

I guess that, if most mathematicians were asked about John Wallis, they would say that he had an infinite product formula for π. I knew, too, that he was a cryptanalyst in the mid-seventeenth century: I got this from Iain Pears’ novel An Instance of the Fingerpost. But, after a meeting of the British Society for the History of Mathematics, I know rather more now.

Wallis was not trained as a mathematician; his Cambridge degree was in theology. The event that changed his life happened in 1642, at the start of the English Civil War, when he was given an encrypted letter and challenged to decrypt it. It was a simple substitution cipher, and he had succeeded by the next morning. The next letter he was given was in a much more complicated code, a substitution cipher with codewords and homophones (I think), having hundreds of characters; it took him months, but he learned on the job, and was later recognised as one of Europe’s leading cryptanalysts. (Later in his life, Leibniz, with whom he was in correspondence on many topics, repeatedly urged him to publish his methods, lest they be lost.)

In 1649, when the Royalists at Oxford University were purged by the Parliamentarians, Wallis was given the post of Savilian Professor of Geometry. He held the job for 54 years, despite the regime change at the Restoration in 1660 (when his cipher-breaking talents were used against the French). Along with Isaac Barrow and James Gregory, he had the misfortune to belong to the generation before Isaac Newton, with whom he is always compared unfavourably.

Wallis’s interests were wide, but in everything he was self-taught (or at least maintained this position). Benjamin Wardhaugh described his ventures into music theory, the “canonics of music”, which was one of the topics that the Savilian Professor was obliged to teach under Sir Henry Savile’s statutes. His work here consisted almost entirely in reacting to other people’s publications.

He appeared to take no interest in music performance, not even to test whether the way string players fingered their instruments was in accordance with his theory of pitch. This was slightly odd, since the experimental method was gaining ground at the time. We were shown pictures of an instrument called a “trumpet marine”, which consisted of a single bowed string (with one half-length sympathetic string), which could be touched at various points to produce harmonics. Music theorists regarded it as a piece of experimental equipment rather than a musical instrument, but I don’t know if Wallis ever worked with it at all.

In cryptography, he also responded to challenges, intercepted letters sent to him by the Government. Towards the end of his life he worked on letters intercepted in Germany relating to the War of the Spanish Succession. He apparently never attempted to produce a cryptosystem of his own. At around this time, French ciphers were becoming more complicated (maybe the Vigenère cipher?); I do not know whether Wallis attacked these, or if so, with what success.

One of his greatest pieces of work may also have been inspired by a similar challenge. Wallis and Hobbes were enemies: Hobbes revealed Wallis’s code-cracking, and also presumed to teach mathematics to the Savilian professor (he squared the circle and proved the parallel postulate). Wallis’s response was to examine the Euclidean parallel postulate himself. Like many other mathematicians at the time, he claimed a proof; but, unlike almost all the others, Wallis did prove something of worth. He showed that the parallel postulate is equivalent to the existence of similarity transformations (asserting, for example, that a triangle of arbitrary size similar to a given triangle exists). Wallis’s proof was valid, though subsequently refined by others. He also gave several reasons, some of them metaphysical, why the similarity principle should be true. These were not accepted (though later mathematicians tried other proofs). But in the view of Vincenzo de Risi, who discussed this in some detail, the most important consequence of Wallis’s proof was that, for the first time, the parallel postulate was not seen as a property of particular figures, but of space; this opened the way to the construction of non-euclidean geometry.

In the final talk, Jesper Lützen began with an introduction to Wallis’s famous product formula, which came from his attemtps at integrating the function (1−x1/k)p. He was able to succeed when k and p are integers, but the interesting case of the circle comes when p and k are equal to 1/2. Examining half-integral values in general, and deriving a recurrence relation and inequalities, led Wallis to the formula. I was very much reminded of the classical derivation of the constant in Stirling’s formula.

His book on Infinitesimal Analysis was read by both Newton and Leibniz. Indeed, Newton got the idea for the Binomial Theorem from reading Wallis.

But the irrationals appearing in the formula before the limit is taken led Wallis to think that he had proved the impossibility of squaring the circle (which was not clearly understood in analytic terms, but seemed to be roughly speaking the assertion that π is not algebraic). He realised before the end of the paper the flaws in his proof.

But later this led him into a controversy. James Gregory produced his own “proof” that the circle cannot be squared, which relied on showing that there is no general algebraic formula for the area of a sector of a circle (and applying this to a right-angled sector). Huygens flatly rejected Gregory’s proof, and Oldenberg (secretary of the Royal Society) asked various people including Wallis to comment. Wallis also didn’t believe Gregory’s proof. According to Lützen, the protagonists were very muddled over the logic of the statements, and in particular seemed to be thinking about assertions quantified in different ways (with the quantifiers exchanged, or a universal changed to an existential).

Their successors Newton and Leibniz got further, proving, more or less, that the sine function is transcendental. But the final resolution had to wait until the nineteenth century.

An enlightening day, and thanks to the BSHM for putting it on, and for Raymond Flood, Benjamin Wardhaugh, Philip Beeley, Seigmund Probst, Vincenzo de Risi, and Jesper Lützen for their talks.


About Peter Cameron

I count all the things that need to be counted.
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