An essay on the art of decyphering. In which is inserted a discourse of Dr. Wallis. Now first publish'd from his original manuscript in the publick library at Oxford

London: printed for L. Gilliver and J. Clarke, 1737. First edition. JOHN WALLIS THE CRYPTOGRAPHER.

First edition, very rare, of the only published portion of the treatise on cryptography which Wallis composed in his last years. Wallis was the most important English mathematician before Newton, but his contributions to cryptography, which occupied him for more years than his mathematics, have been neglected by most of his biographers. "At the time of his death Wallis was not only the incumbent Savilian professor of geometry and the Keeper of the Archives of the University of Oxford, but also the first holder of the post of decipherer in the office of the Secretary of State. This was a new post which had only been created towards the end of the reign of William III, in April 1701, and which reflected the importance that monarch placed on surveillance and intelligence in the conduct of his policy at home and abroad. Although official status as decipherer came late, Wallis had in fact been employed by government officers in the breaking of codes for over sixty years since the Civil Wars, through the Commonwealth and the Protectorate to the Restoration and beyond. Nor was this a one-sided relationship. Wallis's skill as a code-breaker was of mutual benefit both to him and to his various employers. More than anything else it set him on his academic career in Oxford and enabled him to steer a remarkably steady course through the heavy seas of politics and religion in seventeenth-century England" (Beeley). Leibniz, who regarded Wallis as the leading cryptographer in Europe and made repeated attempts to persuade Wallis to reveal his methods to the wider world, saw an affinity between code-breaking and algebra, and modern scholars have identified a close relationship between Wallis's cryptographic and mathematical techniques, in particular in the 'ingenious series of interpolations' in his epoch-making Arithmetica infinitorum by which he found an infinite product expressing the exact value of 4/π and devised an algebraic formulation of the method of indivisibles which profoundly influenced Newton's invention of the calculus (see Domenico Bertoloni Meli in ODNB). In addition to a coded letter from the Duke of Buckingham (which Davys himself decodes), Davys (pp. 9-23) prints the remarkable introduction to Wallis's treatise in which Wallis argues for the importance of ciphers, in particular during civil wars, 'where the intermingling of opposite parties makes it difficult if not impossible to distinguish friends & foes' (p. 10). He outlines the history of his involvement in code-breaking, which began over supper at the residence of his spiritual charge Lady Mary Vere: a guest produced a letter intercepted from the royalist side after the capture of Chichester in December 1642, which Wallis solved in an evening. The arduous decipherment of a second, much more complex letter from Charles I's exiled secretary of state, Sir Francis Windebank, set Wallis on the path to becoming unofficial code-breaker for the parliamentarians, a position of a kind he believed to be unique in England: "For all those letters, which, during these wars, have been intercepted by either party, I do not know that there hath been any one deciphered save those that came to my hands ... As for the reasons that moved me thus to expose them to view; I shall only say this much: I did not think it worth the while to publish them in print (nor, perhaps would it be convenient so to do) & yet thought them so considerable as not to be altogether suppressed" (p. 22). Davys, according to his own account, was versed in the art of deciphering from his youth, and could produce witnesses of undoubted reputation, who had tried him with letters in cipher, generally to find his deciphering to be correct. Unlike John Falconer's Cryptomenysis Patefacta (1685), which described various kinds of ciphers and deciphering thereof, Davys focuses on numerical code, which was the mainstream cipher at the time. His aim was to inform the public that the art of deciphering had a solid basis, despite some arguments circulated at the time that deciphering was nothing but conjectures" (Tomokiyo). RBH lists no other complete copy since 1931. ESTC lists 10 locations in US. Some five copies of Wallis's original manuscript are known, all, with possibly one exception, held by either the Bodleian or the British Library.

Provenance: The Earls of Macclesfield (South Library bookplate on front paste-down, small pressure stamp to first four preliminary leaves).

"Wallis describes his first code breaking in his autobiography as follows. 'About the beginning of our Civil Wars, in the year 1642, a Chaplain of [parliamentarian general] Sr. Will. Waller's, (one evening as we were sitting down to Supper at the Lady Vere's in London, with whom I then dwelt,) shewed me an intercepted Letter written in Cipher. He shewed it me as a Curiosity (and it was indeed the first thing I had ever seen written in Cipher.) And asked me between jest and earnest, whether I could make any thing of it. And he was surprised when I said (upon the first view) perhaps I might, if it proved no more but a new Alphabet. It was about ten a clock when we rose from Supper. I then withdrew to my chamber to consider of it. And by the number of different Characters therein, (not above 22 or 23:) I judged that it could not be more than a new Alphabet, and in about 2 hours time (before I went to bed) I had deciphered it; and I sent a Copy of it (so deciphered) the next morning to him from whom I had it. And this was my first attempt at Deciphering. This unexpected success, on an easy Cipher, was then looked upon as a great matter; and I was somewhile after pressed to attempt one of another Nature; which was a Letter of Mr. Secretary Windebank [dated Paris, 1 March 1641 OS], then in France to his Son in England, in a Cipher hard enough, and not unbecoming a Secretary of State. It was in Numeral Figures, extending in number to above seven hundred, with many other Characters intermixed. But not so hard as many that I have since met with. I was backward at first to attempt it, and after I had spent some time upon it, threw it by as desperate: But, after some months, resumed it again and had the good hap to master it. Being encouraged by this success, beyond expectation; I afterwards ventured on many others (some of more, some of less difficulty) and scarce missed of any, that I undertook, for many years, during our civil Wars, and afterwards. But of late years, the French Methods of Cipher are grown so intricate beyond what it was wont to be, that I have failed of many; tho' I have master'd divers of them. Of such deciphered Letters, there be copies of divers remaining in the Archives of the Bodleyan Library in Oxford; and many more in my own Custody, and with the Secretaries of State.'

"During 1644-1649, a period overlapping his deciphering work for the parliamentarians, Wallis served as a scribe to the Westminster Assembly, established by the Parliament to restructure the Church of England. His career as a mathematician started in about 1647, when he was thirty-one. In 1649, he was appointed Savilian professor at Oxford, the post he held for life. He obtained the post in place of a royalist predecessor because of his service to the parliamentarians. While Wallis's mathematical achievement was not so remarkable at the time of the appointment, he proved his merit in the following years and in 1656 published his major work Arithmetica Infinitorum.

"In London in the 1640s, scientists had started to have a weekly meeting, which later developed into the Royal Society. Among the fellows who discussed various topics was John Wilkins (1614-1672), author of Mercury or the Secret and Swift Messenger (1641). Both Wilkins and Wallis moved to Oxford in 1648-1649.

"Wallis's skill 'in unravelling and explicating secret writing hidden behind the most intricate ciphers' alongside his talent in mathematics was praised in the preface of the third edition (1652) of William Oughtred, Clavis mathematicae. Considering that Wallis was closely involved in the production of this edition, the praise might have been solicited by himself.

"In 1653, Cromwell's Protectorate government took place of the parliamentary rule and John Thurloe became head of intelligence in place of his predecessor, Thomas Scot, in July (after the dismissal of the Rump Parliament).

"In this year, Wallis deposited a collection of 227 pages of transcripts of selected letters deciphered by himself to the Bodleian Library at Oxford. The volume includes fifty-two cipher letters, with solutions and keys. ... Wallis attached a brief introduction to the collection, of which the text is printed in John Davys, An Essay on the Art of Decyphering (1737).

"Wallis observes that the Civil War brought about wide use of cipher, which had been little known to any but the secretary of princes etc.

"Of his deciphering, he says 'I have not often failed in any that I have attempted, which was of any considerable Quantity, suitable to the Difficulty of the Cipher wherein they were written.' This was as of 1653. More than forty years later (in 1697), Wallis says in his autobiography; 'of late years, the French Methods of Cipher are grown so intricate beyond what it was wont to be, that I have failed of many; tho' I have master'd divers of them.' He may refer to adoption of two-part code (in which assignment of code numbers are not in alphabetical order) and/or special symbols for deleting or repeating the preceding letter.

"The above remark of Wallis involves another insight. He recognized that a certain amount of materials in cipher according to the difficulty of the cipher is required to allow deciphering. The more difficult the cipher is, the more material is required. This is similar to the idea of 'unicity distance' in modern cryptology.

"Then follows descriptions of his first attempts of deciphering in the same strain as what he described in his autobiography.

"Wallis was aware of earlier works on the art of deciphering by Baptista Porta etc. but he found that those works, dealing with simple monoalphabetic substitution, were of no help to him. He points out there cannot be a fixed method of deciphering and quotes the maxim: consilium in areni capere ([gladiators] take the measure upon the spot).

"Wallis makes a point that it was not due to lack of skill or caution on the part of those who used the ciphers deciphered by him. He declares that if he would ever need to use a cipher, he would choose one like most of those. He points out the superiority of figure cipher (numerical code) and says if anyone could devise a more difficult cipher, it would be no more difficult to be broken 'or else will bee so extreamly tedious in use, both to him that writes by it, and to him that is to read it, that it will not admit of any tolerable Dispatch, and would certainly want many of those Conveniencies, which, in the Figure-cipher, I could easyly demonstrate.'

"He was rather optimistic about security of cipher. While he does not boast that no other person could do what he had done, he observes that letters may not be intercepted; intercepted letters may not come to a capable person; the person may not be induced to make a try, etc.

"Indeed, figure cipher would continue to be the mainstream in diplomatic ciphers for many years. Considering, however, Wilkins' silence in Mercury on figure cipher (1641), Thickness's naive confidence in the harmonic alphabet (1772), and Blair's proposal of intricate new schemes which are basically simple substitution (1819), Wallis had a shrewd insight, derived from his deciphering experience" (Tomokiyo).

"Wallis's exploits as a decipherer failed to achieve the recognition which he evidently so dearly sought. It was as a mathematician, and to a lesser extent as a linguist and theologian, that his name ranked among the most esteemed scholars in the republic of letters. But there was one notable exception among Europe's learned who did not pass over Wallis's exploits in silence and this was Gottfried Wilhelm Leibniz. Recognizing that minds that excel in algebra, 'the pinnacle of calculus', as he calls it, also excel in deciphering, Leibniz already at the outset of his mathematical career, during his momentous stay in Paris 1672-1676, compares the search for the rule of a series, or of an array or a 'table', with the search for the key of a cipher. After noting that one series might be part of another and that in such cases it is necessary to choose that which is simplest and best accommodates the data, he goes on to describe this as being the doctrine of discovery or hypothesis, 'of which no one up to now has dealt with accurately'. But for at least part of this doctrine, namely that concerning the construction or solving of ciphers, Leibniz makes clear that he already has someone in mind for the job: 'I should like this to be accurately dealt with by Wallis', he writes.

"Leibniz remained fascinated by the potential of the art of deciphering throughout his life, both for scientific and political reasons, and particularly in later years often entreated Wallis to reveal something of his methods. He addressed this topic also in his anonymous review of Wallis's Treatise of Algebra for the Acta eruditorum. Referring to Jacques Auguste de Thou's testimony to François Viète's skill as a decipherer, Leibniz suggests that Wallis would equal or even surpass the praises poured on the French mathematician if he were to provide posterity with a specimen of his own accomplishments. Apart from those fortunate enough to visit the Bodleian Library, scholars had nothing else to go on ...

"Wallis was quite evidently torn between two opposing and indeed irreconcilable forces. On the one side he sought acclaim within the Republic of Letters as one who was able to break even the most difficult codes. His various efforts to provide some sort of documentation of the deciphering work he had done are a clear reflection of this aspiration. On the other side he recognized that if he were too forthcoming he would severely compromise himself as well as those who employed him. Anything more than the visible display of his achievements would have necessitated providing insight into the precise nature of the methods and strategies he employed. To have done this would have meant not only potentially destroying the uniqueness of his skill and therefore also undermining what was hitherto considered to be an unassailable if informal position in the apparatus of state, it would also have given those who devised the codes over which he toiled, and more often than not succeeded in breaking, precisely the kind of information they sought. The structures of political intelligence were not only concerned with concealment of information, but also with discovering just how much the other side was capable of finding out and if necessary modifying the code so as to make it even more impenetrable. On more than one occasion Wallis complained to secretary of state Daniel Finch, second earl of Nottingham, that the French 'change their Ciphers often'. He even suggested that they 'study every time to make them harder then before'. Wallis's indispensability, his status, and his reputation inevitably rested on the inscrutability of his methods.

"The academic career of John Wallis was more than anything else founded upon his remarkable ability to decipher encoded political and military intelligence. Discovered during the height of the Civil Wars, this 'jewell for a Princes use' was employed to advantage by parliamentary forces fearful of adversaries inside and outside government. It was principally Wallis's support in thwarting the Presbyterian plot of 1649-51 which led to his being offered a Savilian professorship at Oxford. His expertise as decoder stood him in good stead, too, after the Restoration and was perhaps the single most important factor ensuring that he remained in post at the University right up until the end of his life" (Beeley).

Beeley, 'Breaking the Code: John Wallis and the Politics of Concealment, G.W. Leibniz und der Gelehrtenhabitus, Li & Noreik, eds. (2016), pp. 49-82. Tomokiyo, Cryptiana: Articles on Historical Cryptography (;/a>). Macclesfield 602 (this copy); Sotheby's, 4 November, 2004, £3,360 (= $6,182).

4to (254 x 205 mm), pp. [4], iii, [1], 58, [1, advertisements], woodcut initials and head- and tailpieces, tables and codes in the text, uncut. Contemporary, probably original, marbled boards with a paper backstrip (the spine paper with 2 or 3 small breaks). Preserved in a green cloth clamshell case. A fine copy.