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

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.


Filed under History of Mathematics

22 responses to “Cromwell’s Code Breaker.

  1. Great article! I’m currently working on Wallis for my PhD research, and it’s always nice to connect with other people who are interested in him.

    • Been to your blog and read what’s there. Are you prepared to reveal more detail on the subject of your doctoral thesis?

      • Sure. I’m studying Wallis from the perspective of science and religion. I’ve found some deep methodological and epistemological connections between his work in physics and his work in theology, which you can see, for instance, in Wallis’s comparison of the Holy Trinity to the dimensions of a cube.

        What did you think of my blog?

  2. Ha, ha, ha! “Lost in the immense shadow of Isaac Newton…” Rabelais could not have penned a more ironic phrase, even when he was lampooning Herr Trippa aka Agrippa. Abracadabra.

  3. Wallis was educated at Felsted School in Essex and then at Emmanuel College in the University of Cambridge.

    • He was schooled at three different places including Felsted school, however you are quite correct Emmanuel College is indeed Cambridge and not Oxford. Now corrected, thank you. Dreadful slip of the keyboard given the rivalry 😉

  4. Dave Pritchard

    As somebody who doesn’t know seventeenth-century mathematics very well, my immediate association with John Wallis is the hatchet job in Aubrey’s “Brief Lives”, which alleges that Wallis “steals flowers from others to adorn his own cap”. Presumably there’s a back story to this claim of plagiarism: is Aubrey simply reflecting his contemporaries’ general personal dislike of Wallis, or is there a more specific piece of politics behind it? I’d be interested to know…

    • If you don’t mind, I’d like to weigh in on this one. As I see it, Aubrey’s charge against Wallis reflects a few things. Although standards of intellectual property and academic honesty are embryonic at best in the seventeenth century, there are charges of “plagiary” or “stealing feathers from others’ caps” throughout scholarly discourse. Firstly, these accusations form part of an ad hominem argument, popular since at least the Renaissance, to suggest that one’s opponent is slow-witted. The idea is that everyone borrows material from other sources, and frequently without acknowledging those sources, but if you can’t do so in an original way that makes those sources hard to detect it’s because you lack the wit to do so.

      Secondly, in the seventeenth century in particular, accusations of plagiary reflect scholars’ obsession with priority for inventions and discoveries. Disputes over who first invented or discovered something are just about ubiquitous in the seventeenth-century Republic of Letters, and few people were involved in as many as John Wallis. The early Fellows of the Royal Society were involved in efforts to establish a standard by which the first inventor or discoverer could be determined; some, like Christiaan Huygens, preferred to conceal their discoveries in the form of a cipher in a document with a particular date, the solution to which they could reveal if someone challenged their priority. Others, like Henry Oldenburg, preferred to keep records within the Royal Society’s register books, making the Society the unofficial arbiter of priority disputes. Ultimately, though, the Republic of Letters couldn’t agree on a standard, and individuals generally insisted on a way of determining priority that would favour their own case, so priority disputes were rampant. When insisting on one’s own standard for establishing priority, it was very common to call accuse one’s opponent of plagiary.

      Finally, there were personal rivalries, which could both cause and be caused by such polemical disputes. In this case, John Aubrey happened to be a good friend of Hobbes, Wallis’s perpetual rival. He was also an admirer of William Holder, an English cleric and musical theorist, who was Wallis’s opponents in a heated priority dispute. Wallis and Holder both developed methods for teaching people who had been deaf their whole lives to speak. Their methods were both articulatory: they would use a model of a mouth to demonstrate how to make particular sounds by moving the mouth, lips, and throat. Holder claimed to be the first to succeed in applying such a method, in 1659-1660 in the case of Alexander Popham, the son of a nobleman and Civil War general. Wallis argued that Holder had not, in fact, been successful with Popham, which made Wallis’s success with another patient in 1661, Daniel Whaley, the first successful application of the method. The dispute was carried out in a series of polemical publications and, as far as I can tell, each of the two disputants emerged thinking he had proved his case. I think Aubrey’s accusation of ‘stealing feathers’ has to be understood in the context not only of his friendship for Hobbes, but also his admiration for Holder. Wallis had antagonized both of these men, and Aubrey’s response was typical: accuse him of dishonesty and lack of wit by charging him with plagiary.

      • Thank you Adam for your extensive ruminations on Aubrey’s accusations of plagiarism against Wallis to which I have little to add. However I thought we could add Aubrey’s actual words to the debate for anybody who might come along to read it.

        ‘Tis certain that he is a person of real worth, and may stand with much glory upon his own basis, needing not (to) be beholding to any man for fame, of which he is so extremely greedy, that he steals flowers from others to adorn his own cap, — e.g. he lies at watch, at Sir Christopher Wren’s discourse, Mr Robert Hooke’s, Dr William Holder, etc; puts down their notions in his note book, and then prints it, without owning the authors. This frequently, of which they complain.

        But though he does an injury to the inventors, he does good to learning, in publishing such curious notions, which the author (especially Sir Christopher Wren) might never have the leisure to publish himself.

        I personally find the second paragraph interesting as Aubrey seems to come to Wallis’ defence, at least to a certain extent.

      • laura

        Super interesting. I’ve been really curious in the concept of plagiarism charges in the 17th century and what those charges reflect. In particular the stream of charges against Descartes — do they reflect something real on Descartes’ part (maybe in the case of Beeckman?), intellectual jealousy? nationalism? All of the above? It seems easier to understand in the cases of Wallis and Newton. But both Leibniz and Huygens, while “Cartesians”, adopt what reads to me like fairly passive-aggressive attitudes toward Descartes, and both of them either directly or indirectly accuse him of plagiarism of various kinds. Huygens constantly refers to Descartes as “the first man to contemplate these issues [the characteristics of light, planet motion, preservation of momentum, whatever] seriously” yet also worked hard to distance himself from Descartes, and was instrumental in spreading the accusation that Descartes stole the law of refraction from Snell, an idea which later picked up a lot of steam (though I don’t think modern historians accept it). Leibniz also established or repeated various plagiarism charges against Descartes. The attitude toward Descartes stands in contrast to the general respect with which slightly earlier figures like Galileo and Kepler are treated. Is this because Descartes is considered a contemporary or rival? It seems a strange explanation considering Descartes only outlived Galileo by 8 years and was publishing his major works at the same time.

      • Certainly in the case of both Wallis and Newton nationalism played an important role but we still don’t quite know why Newton developed such a hatred of Descartes.

        I personally think it’s wrong to call Leibniz a Cartesian. However you are right in his attitude to the plagiarism chages against Descartes. It was Leibniz who turned Newton’s somewhat indirect implication about de Dominis into a direct plagiarism charge.

        Huygens is indeed a strange case. He is the only mathematical physicist who can truly be called Cartesian but in his own work whilst openly praising the master he silently dumped almost all of his principles. In terms of Snel (and the English spelling with two ls is wrong) don’t forget that Snel and Huygens were both Dutch.

  5. Dave Pritchard

    Thanks for the very detailed answer! I guess the Wallis/Hobbes mudslinging is notorious (to the extent that even I’d heard of it), but I’d not come across the Wallis/Holder priority dispute before.

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  12. An for those of us who write about infinity, don’t forget his contribution of the lemniscate.

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