John Wallis and Cryptanalysis

The English mathematician John Wallis (1616-1703) was one of the pioneers in the art of deciphering and deciphered many letters in cipher throughout his career.

Table of Contents

Codebreaking during the English Civil War

Collection of Deciphered Papers Deposited in a Public Library


Accused for Deciphering the King's Letters Taken at Naseby

Codebreaking during the Reign of Charles II

Codebreaking after the Glorious Revolution

Personal Ciphers

Inquiry from Leibniz

Wallis' Successor

List of Letters


Codebreaking during the English Civil War

From late 1642 to 1653, Wallis deciphered many letters for the parliamentarians. 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 jeast 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; see another article for a cipher, probably not the same one, used by Windebank], 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 seaven hundred, with many other Characters intermixed. But not so bard 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.
(Wallis' Autobiography (1697); Peter Langtoft's chronicle p. clviii)

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' 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 to 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. (Beeley (2016) p.65)

Collection of Deciphered Papers Deposited in a Public Library

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 inculdes fifty-two cipher letters, with solutions and keys. The introduction (at least of the auctioned copy (see below)) is dated 1 July 1653. The latest date in the collection was 4 April 1653 for a letter in French from The Hague. (Wallis included three letters undeciphered in order to let the reader see the difficulty of deciphering work. One of them, a letter from the Duke of Buckingham, was deciphered by Davys.) 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. (For use of ciphers by secretaries of state before the English Civil War, see, e.g., another article.)

Of his deciphering, he says "I have not often failed in any that I have attempted, which was of any considerable Quantity, sutable 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 (see above); "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 (see the second specimen reproduced here).

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 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) (see another article), Thickness' naivce confidence in the harmonic alphabet (1772) (cf. another article in Japanese), and Blair's proposal of intricate new schemes which are basically simple substitution (1819) (see another article), Wallis had a shrewd insight, derived from his deciphering experience.

Transcripts of Letters

The Bodleian Library has two copies of this volume (MS e Musaeo 203 and MS Eng. misc. e.475). (Jacqueline A. Stedall (2002), A Discourse Concerning Algebra: English Algebra to 1685 (Google); catalogue)) A third transcript, titled "A Collection of Letters and other Papers, intercepted in Cipher, during the late warres in England. Deciphered by John Wallis D. D. Professor of Geometry in the University of Oxford. Being a Transcript of a like Collection presented by him to the Bodleyan Library, in the University of Oxford", was sold at an auction for £29,000 in 2020 (Dominic Winter Auctioneers). It was passed on through William Wallis, great-great-grandson of Wallis, and may well have been Wallis's own copy. It even includes a fifty-third letter that is not in the copy in the Bodleian Library. According to the auction site, it includes the following:

• The first letter in the volume is the one which Wallis deciphered for the first time after dinner at Lady Vere's residence in London ("A letter intercepted by Sir William Waller, after ye taking of Chichester at the beginning of the warres").

• The next is Windebank's letter, the tough one mentioned in the autobiography (having more than 700 figures) ("An intercepted letter, from secretary Francis Windebanck to his sonne").

• Nearly half are letters of 1650 from royalist agents in Breda and conern the pact between Charles II in exile and the Scottish Covenanters.

• "An intercepted letter from the King, while he was at Newcastle, to the Prince". (The images of p.29, p.30 available. This is the cipher presented as "Cipher with the Prince of Wales (February 1647)" in another article.)

• a letter of Prince Charles (later Charles II) requesting money from Lawrence Loe for a journey to Calais (Saint Germain, 1 July 1648) (Charles sailed from Calais to Holland to take control of some dozen ships that defected to the royalist cause.)

• letters from Earl of Lauderdale (Wikipedia) to Countess of Carlisle (Wikipedia) and others. (The image of p.43 (8 July 1648) available. See another article.)

• letters to the Marquess of Ormond (1649-50).

• "A letter to Prince Rupert, concerning divers of the Royalists in France".

• a letter in French from Graymond, French agent in Edinburgh, to Cardinal Mazarin. (Possibly, the "Cipher intended for the Cardinall" of the image of p.81. It appears to be the same as the cipher I reconstructed from a letter of P. de Graymond, Edinburgh, 10/1 May 1649 presented in another article.)

• a letter in French from Graymond to Jean de Montereul.

• a fifty-third letter from Pet[er] Townesend, "A Letter from Flanders intercepted in May 1658"


John Thurloe

That Wallis served under the spymaster John Thurloe during the Protectorate regime (1653-1659) is taken for granted by several authors ( Wikipedia's description (accessed on 11 November 2012) that Wallis established a codebreaking department under Thurloe seems to be ungrounded). Oxford DNB (s.v. Thurloe) only mentions Samuel Morland and Isaac Dorislaus as assisting in deciphering royalist codes, while one author, in describing Thurloe's inheritance to the Restoration government, says Dorislaus, occasionally assisted by Morland, was employed in opening and copying letters in the Post Office and mentions Wallis as a "cipher expert" (Alan Marshall, Intelligence and Espionage in the Reign of Charles II, 1660-1685, p.80, 83-84, 23, 93).

Thurloe State Papers includes intercepted cipher letters, among which An inercepted letter from Paris, 8 July 1653 (NS) is deciphered. The handwriting may reveal the decipherer. (Ciphers of the royalists Kingston, John Barwick, and General Massey (see another article) are also deciphered but these, not derived from Thurloe's papers as noted in the Preface of vol. 1, were probably deciphered by the proper recipients.) On the other hand, Dutch ciphers, seemingly simple substitution ciphers, remain undeciphered. (As an example, see Beverning and Vande Perre to the Dutch ambassador Boreel at Paris, 1 September 1653 (NS)). Besides, An intercepted letter of du Gard in a letter to White, Brussels, 10 June 1656 NS, An intercepted letter, Brussels, 12 August 1656 NS, An intercepted letter, from Jo. Waddall, 22 August 1656, etc. remain undeciphered. It is wondered if Wallis might not have done better.

After all, the description that Wallis returned under Thomas Scot after the dismissal of Thurloe (Underdown, p. 295) appears to be correct.

Thomas Scot

In May 1659, upon reinstatement of the Long Parliament, intelligence activities were reverted to Thomas Scot, who had been replaced by Thurloe in 1653. Wallis served Scot at this period. According to Scot's confession, he obtained information on negotiations between the King and the Presbyterians in England more from intercepted letters than from his correspondents.

[the intercepted letters] were every word & syllable in Cypher, and decyphered by a learned gentleman incomparably able that way, Dor Wallis of Oxford (who never concerned himself in the matter, but only in ye art & ingenuity); it is a jewell for a Princes vse & service in that kind.
(C.H.Firth, 'Thomas Scot's Account of His Actions as Intelligencer during the Commonwealth', English Historical Review (1897), p.121 (Internet Archive))

As an example, in early 1660, letters of Barwick, Rumbold (both were royalists in England) etc. were intercepted by the garrison of Dunkirk and deciphered by Wallis.

although every different Person's Letter was written in a distinct Cypher, and that contriv'd with great Thought, yet they were all decypher'd by the Art and Ingenuity of a certain very famous Mathematician, who was hired by the Rebels.... Yet he had now at last this in him of a good Subject, that at this Time he discover'd nothing to the Rebels, which much concern'd the publick Safety; though he satisfied some of the King's Friends, that he could have discover'd a great deal. But all those whom it concern'd being sufficiently assur'd, that no Key of any Cypher had fallen into the Enemy's Hands; it was thought nothing but vain boasting, when the Rebels brag'd, that by the Help of their Friends they were able to find out the most hidden Secrets of the Royalists; till Mr. Matthew Wrenn (Son to the Right Reverend Bishop of Ely) who was intimately acquainted with this Mathematician, obtained of him some Copies of those Letters, as he had decypher'd them, and took Care to have them deliver'd severally to the Persons that wrote the Letters, who all acknowledged them for their own, and left no room to doubt of the Decypherer's Art.
(Life of Dr. John Barwick, p.251-252; see also sources noted by Underdown, p.296)

That the mathematician refers to Wallis appears from Barwick's letter to Chancellor Hyde of 13 February 1660 (CClSP, iv, p.561), referring to deciphering of letters taken at Naseby (see below) and the deposit in Oxford Library (see above). The same letter also reports Wallis' words (that Barwick heard from his friend, Matthew Wren (CClSP, iv, p.598)) that he knew no way of ciphering he could not discover.

The remarks on "publick Safety" may be supported by a letter from Rumbold to Hyde of 7 February 1660, reporting that the decipherer says he never did nor will give a true light into letters of importance or to those containing proper names. (CClSP, iv, p.550) Wallis himself attested later that sometimes he displeased those in power because he did not think fit to decipher some letters (cf. letter to Bishop Fell quoted below).

Middle Course

While Wallis is silent in his autobiography, written in his old age after he had gone through the Glorious Revolution as well as the Civil War, as to how he got through the Restoration, he says he endeavoured to act by moderate principles.

It hath been my Lot to live in a time, wherein have been many and great Changes and Alterations. It hath been my endeavour all along, to act by moderate Principles, between the Extremities on either hand, in a moderate compliance with the Powers in being, in those places, where it hath been my Lot to live, without the fierce and violent animosities usual in such Cases, against all, that did not act just as I did, knowing that there were many worthy Persons engaged on either side. And willing whatever side was upmost, to promote (as I was able) any good design for the true Interest of Religion, of Learning, and the publick good; and ready so to do good Offices, as there was Opportunity; And, if things could not be just, as I could wish, to make the best of what is: And hereby, (thro' God's gracious Providence) have been able to live easy, and useful though not Great
(Wallis' autobiography p.clxix)

Despite his aggressive spirit in many aspects of his life, including scientific controversy (Kemp p.9), Wallis' moderate principles in politics are at least supported by the fact that he signed the remonstrance against the King's execution (DNB). Thanks to his moderation as well as his ability, Wallis received favour after the Restoration. (By the way, Wilkins had married Cromwell's sister Robina in 1656 and was favoured under the Protectorate. Although the Restoration lost him his position, he was soon made Bishop of Chester.)

Accused for Deciphering the King's Letters Taken at Naseby

The most celebrated disclosure of the royalist secret papers during the Civil War would be that of the King's cabinet, taken when the royalists met a disastrous defeat at the Battle of Naseby (1645) (see another article). The letters published by the parliamentarians included deciphered letters. Wallis was accused for deciphering those.

The first of such accusations appeared in 1657 (still under the Protectorate regime) in A severe inquiry into the late Oneirocritica; or an exact account of the grammatical part of the Controversie between Mr. Thom. Hobbes and John Wallis, D.D., p.7. The author Henry Stubbe took side with Wallis' bitter opponent, saying "the doctor decyphered (besides others, to the ruin of many loyal persons) the King's cabinet taken at Naseby, and as a monument of his noble performance, deposited the original with the decyphering in the public library at Oxford." (Biographia Britannica p.4131).

At least, however, the collection of decipherings deposited by Wallis does not include papers taken at Naseby. While there was an accusation that Wallis took out the papers just before the Restoration and "altered what he pleased" (Biographia Britannica p.4131), no trace of such tampering has been found (Davys, Kemp).

Even the royalist Chancellor Edward Hyde (later the Earl of Clarendon) said the parliamentarians were not up to any deciphering.

Nor did I ever hear, that more of the King's Letters that were found at Naseby, than those which they found decyphered, or found the Cyphers in which they were writ, were decyphered. And I very well remember, that in the Volume they published, there was much left in Cypher, which could not be understood [Davys' note: only a few proper names and a very short passage in the 9th letter, as published by the Parliament, but marked 33 in the King's Words]; and which I believe they would have explained, if it had been in their Power:
(8 March 1660; Life of Dr. Barwick p.503)

While Hyde's understanding of the art of deciphering may not be reliable, Davys cites this and points out that it is natural to assume the keys to the ciphers used by the King were kept in the same cabinet as the papers and thus were captured at the same time (p.7).

(Furthermore, Wallis deposited transcripts, not the originals as stated by Stubbe. The originals were probably returned to the government.)

However, the story appears to have been widely believed and Life of Dr. John Barwick (the Latin edition was published in 1721) repeats the story.

Having said this, Wallis did decipher the King's letter. The deposited collection includes his deciphering of the King's letter of 3 February 1647 to his son (see another article) (Geneva, Astrology and the Seventeenth Century Mind, plate on p.25). Perhaps the particular controversy over deciphering of papers taken at Naseby reflects a special significance of Naseby for the memory of Charles the martyr.

In the mid 1680s, the story that Wallis deciphered the King's letters captured at Naseby appears to have been circulated again. Wallis wrote to John Fell, Bishop of Oxford, as follows in his letter of 8 April 1685. In February this year, Charles II had just died and been succeeded by his brother James II. Specific reference to James II seems to indicate some exchanges about the position of Wallis.

I understand there have of late been complaints made of mee, that I deciphered the late King's letters, meaning those taken in the late King's cabinet at Naseby fight, and after printed. As to this without saying any thing, whether it bee now proper to repeat what was done above forty yeares ago; the thing is quite otherwise. Of those letters and papers, (whatever they were) I never saw any one of them, but in print: nor did those papers, as I have been told, need any deciphering at all, either by mee or any body else: being taken in words at length [*Wallis was mistaken here; see Hyde's letter quoted above], just as they were printed: save that some of them were, I know not by whom, translated out of French into English. 'Tis true, that afterwards some other letters of other persons, which had been occasionally intercepted were brought to my hands: some of which I did decipher, and some of them I did not think fit to do, to the displeasing of some, who were then great men. And I managed my selfe in that whole busines by such measures, as your Lordship, I think, would not bee displeased with. I did his Majesty, who then was, (K. Charles the first) and his friends many good offices, as I had opportunity, both before and after the King's death: and ventured farther to do them service, than perhaps some of those, who now complaine of mee, would have had the courage to do, had they been in my circumstances. And I did to his late Majesty, K. Charles the second, many good services, both before and since his restauration: which himselfe has been pleased divers times to profess to mee with great kindnes. And if either my Lord Chancellour Clarendon, or Mr. Secretary Nicolas, or his late Majesty, were now alive, they would give mee a very different character from what, it seems, some others have done. And I thinke his Majesty, that now is, [i.e., James II] knowes somewhat of it, and some other persons of honour yet alive.
Wallis to Dr. John Fell, 8 April 1685 (Peter Langtoft's chronicle p.clxx)

Codebreaking during the Reign of Charles II

That Wallis did deciphering during the reign of Charles II under Lord Arlington (Secretary of State during 1662-1674) is evidenced by Wallis' own letter of 15 August 1691 (see "John Wallis as a Cryptographer"), though the letter also tells that his work at that time was less than a tenth of that about 1691.

In May 1665, Mr. Wren, a secretary of Chancellor Hyde (now Earl of Clarendon), sent Wallis two letters to be deciphered, saying that the Lord Chancellor "knows that to your industry & sagacity the most accurate ciphers are as open as common alphabet, & this cipher being as he conjectures a very mean one will he supposes give you very little trouble to unmask it." (William James Roosen (1976, 2017), Age of Louis XIV: The Rise of Modern Diplomacy)

Wallis also deciphered letters to and from Clarendon in France where he went in exile in 1668. A letter from John Ellis to Wallis dated 7 October 1669 OS, appears to be one of many that forwarded materials to be deciphered to Wallis. Ellis worked for Sir Joseph Williamson, who managed intelligence activities of the restoration government. Williamson in turn reported to Lord Arlington, to whom it is considered the "My Lords" refers to. (Correspondence of John Wallis (1616-1703), iii, pp.252, xxix)

What I have received from you gives verry great satisfaction: but I pray for the future put the Number of the pacquet which you send mee in three or fouwer words; that so I may know & be sure I have all you send.
You will finde hereinclosed ten severall peeces in English whereof some are of a new but verry easie Cijpher. My Lords desire you would make what speed you with them. And for the frenche, I shall the Next post sende you a Considerable number.
(John Ellis to Wallis, 7 October 1669 OS)

(See another article for one specific example.)

Wallis' service never ended with the resignation of the Secretary of State in 1674. When the Earl of Argyll's letters in cipher were captured (see another article) in June 1683 upon discovery of the Rye House Plot (an attempt on the lives of King Charles II and his brother James), Wallis failed at deciphering (James Walker (1932), The Secret Service Under Charles II and James II)

Codebreaking after the Glorious Revolution

After the Glorious Revolution (1688), Wallis undertook deciphering for the new regime at the request from the Earl of Nottingham (Secretary of State during 1689-1693). At the time, King William III (William of Orange) led the alliance of England, the Dutch, the Emperor, and other princes to fight Louis XIV of France in the War of the Grand Alliance (1688-1697) and most of Wallis' decipherings at this period were in French. Specifically, at least letters between 14 June 1689 to 29 August 1703 were included (Kahn p.1005).

A collection of letters Wallis deciphered during 1689-1703, which Wallis intended to be printed, is now in Bodleian Library, MS Eng. misc. c.382. Letters deciphered in 1651-1701 are in British Library, Add MS 32499 (Jacqueline A. Stedall (2002), A Discourse Concerning Algebra: English Algebra to 1685 (Google), Beeley (2016) p.49).

First Codebreaking after the Glorious Revolution

Wallis' deciphering for the new regime was occasioned by an intercepted cipher letter in French from the camp of the army supporting James II (Wallis to Mr Harbord, 15 August 1691; see "John Wallis as a Cryptographer"). The army was laying siege (April to July 1689) on Londonderry, a stronghold for Protestants in Ireland. Wallis was asked to decipher it by the Secretary of State Nottingham and Richard Hampden, a Whig MP. The second he undertook was a similar letter and the third was a report from a French emissary in Poland.

Wallis' letter to Hampden dated 1 August 1689 reports his first achievements to the MP who requested it.

Yours of July 27, I received in a packet from the Earl of Nottingham last Sunday night after I was in bed. I wrote to his Lordship next day, an account of the difficulty I at first apprehended in what was desired, because of those papers being written in a hard cypher, and in a language of which I am not thoroughly master; but, sitting close to it in good earnest, I have (notwithstanding that disadvantage) met with better success (and with more speed) than I did expect. I have returned to his Lordship the papers that were sent me, with an intelligible account of what was there in cypher.
(Wallis to Hampden, 1 August 1689, Gentleman's Magazine, vol.58, p.851-2)

Wallis' letter to Hampden dated 3 August 1689 (see "John Wallis as a Cryptographer") was to provide some additional information. Perhaps with a sense of responsibility when faced with a fresh request from the new government, Wallis reported every minor technical detail of the deciphering (e.g., errors he found and guesses he made).

From his explanation, the French cipher assigned low numbers to letters alphabetically (e.g., 35(r), 36(s)), while larger numbers are used to code names and words. There seems to be some regularity in the arrangement of the code numbers, which allowed Wallis to make an estimate such as "a word which begins with c". (Another article shows a code table, which, though not alphabetical, would allow such prediction as "a name beginning with D".)

This was to be only the beginning of countless similar occasions to come. On the very day Wallis wrote the above detailed report, the next assignment arrived.

My Lord, from the time your servant brought me the letter yesterday morning, I spent the whole day upon it (scarce giving myself time to eat), and most part of the night; and was at it again early this morning, that I might not make your messenger wait too long.
(Wallis to Nottingham, 4 August 1689, Gentleman's Magazine, vol.58, p.852)

Deciphering of letters from French emissaries in Poland (1689)

Wallis' letter to Nottingham dated 18 August 1689 (see "John Wallis as a Cryptographer") shows his sense of urgency that he sent a preliminary report of his findings to the Secretary of State. Possibly, it refers to the above-mentioned third letter he undertook. It indicates maneuvers of France to induce the King of Poland to make war on Prussia and promote "a marriage of the Princess of Hanover with the Prince of Poland".

In face of the War of the Grand Alliance, Louis XIV tried to detach the King of Poland, John Sobieski, from the alliance with the Emperor without success. Louis' attempts mentioned by Wallis conform to such a background.

By the way, if "the Princess of Hanover" refers to Sophia-Charlotte (Wikipedia), there must be some mixup, since she had married the prince Frederick of the Elector of Brandenburg in 1684. According to Histoire de Jean Sobieski, Roi de Pologne (in Oeuvres complettes de M. l'abbé Coyer, Tom. 7 (1783)) (Google), France did support marriage of the eldest son James of the King of Poland against a Brandenburg (Prussia) candidate. In 1688, a son Ludwig of the Elector of Brandenburg and James of Poland vied for the hand of Ludwika (Wikipedia), who had a vast inheritance of the Radziwil family (p.261). The French took side with James of Poland in order to detach the King of Poland from the Emperor. Even in 1689, after Ludwika married into the Neuburg family, the King of Poland sought confiscation of Ludwika's inheritance by asserting that she was once engaged with James but his sue was not accepted. France would have offered to James a princess of the French blood but the Poles wanted a prince's daughter (p.298).

On the other hand, Emperor Leopold proposed a match with a daughter of the Elector Palatinate (sister of the Empress), which resulted in a marriage in March 1691. The Marquis of Bethune (Beely identifies François-Gaston de Bethune, marquis de Chabris (1638-1692), while The Cambridge Modern History indexes the person as Maximilien-Aspin; the year of death of the former matches known histories), an envoy of France, struggled to prevent the match, which irritated the court in Vienna so much that a marriage contract included expulsion of the envoy from Poland (Authentic memoirs of John Sobieski, King of Poland, Google, p.257). Rivalry between France and the Emperor in Poland escalated so much that the ambassador from Vienna accused the Marquis for scheming to put a puppet of France on the Polish throne during the life of Sobieski. The ambassador evaded an answer when Sobieski asked for evidence (Histoire, p.301-302). When a courier sent to Vienna was assaulted by someone, the Emperor was infuriated and urged Sobieski to expel Bethune from the Polish soil. In the end, Louis XIV transferred Bethune as ambassador to Sweden before he was banished (Histoire p.304, Authentic memoirs p.259).

Interception of French letters from Poland appears to have been an important source of intelligence. Wallis deciphered one dated 6 September 1689 (see another article) and the following shows forwarding similar materials.

My Lord Nottingham commands me to send you the inclosed Letters from the French ministers in Poland lately intercepted: there are 2 of the 4 of September, 2 of the 5th and 12 of October. My Lord desires that you will decipher them as soon as you can, being of great importance to his Majestie and his allies; he desires that you will begin with the postscript of Mons. DeVeil's Letter of the 5th of Octcb. giving an account of a conference with the Great General of Lithuania. The Letters are of a fresh date, and therefore it is the more important to have them soon deciphyred, in regard some light may be gained by them as to what is at this time on foot.
Richard Warre (under-secretary to Nottingham) to Wallis, 1 November 1689 (Monthly Magazine, vol. XIII, p.446)

This time, Wallis had difficulty with new ciphers.

My Lord, I send you the other deciphered letters. All the rest are in new cyphers, and different from one another, and I cannot yet see which way to attack any of them, and therefore cannot tell when I shall be able to send you another; they change their cyphers so often, and study every time to make them harder than before. I beseech your Lordship to excuse me to his Majesty, if herein I cannot be so quick as I would. But it shall be faithfully endeavoured by, my Lord, your Honour's very humble servant.
(Wallis to Nottingham, 9 November 1689, Gentleman's Magazine, vol.58, p.852)

First Complaint

In November 1689, Wallis found assignments from the secretary of state were too much. He wrote to Nottingham in a letter dated 23 November 1689, that he had "been indisposed for a week, or more, and part of the time very ill," observing, that it was "hard service to keep the fansy so long upon the stretch, with so much intenseness as is requisite to decyphering." (Gentleman's Magazine, vol.59, p.3)

Next month, he went so far as to say the secretary should not complain of the time taken for the job, unless he would not let Wallis quit the service.

But I have been indisposed as to my health, and my eye-sight begins to fail me to such a degree as it hath never done till now; insomuch, that I must be forced to quit this service altogether, or be excused if I cannot forward it with that expedition which I could wish. I have lost the sight of one eye in the service already this winter (and that which was much the best of the two), and the other is but dim, so that I am forced to favour it, lest I lose both. I trust your Lordslup Will favour me so far, as to let his Majesty know my unhappy circumstances.
(Wallis to Nottingham, 22 December 1689, Gentleman's Magazine, vol.59, p.3; Date corrected according to EMLO, accessed on 14 January 2013)

Tackling Louvois' Letter to Lauzun in Ireland (1690)

Regarding the situation in Ireland, Nottingham's sense of urgency is apparent in telling Wallis to return a solution by the bearer of the intercepted letters to be deciphered. The letters were from Louvois to the Count of Lauzun, who was sent to Ireland by Louis XIV to support James II deposed by the Glorious Revolution.

I send you herewith some Letters from Mons. Louvois to Mons. Lauzune, which I desire you will decypher as soon as possibly you can; they seeming to containe matters of great importance for their Majesties service, and that you will begin with the letter which is of the latest date, and proceed to that which is of the next latest date and so on in that order, letting the letter of the oldest date to be that which you last decypher. As soon as you have done the letter of the freshest date, I desire you will send it back immediately by this messenger, whom I have ordered to stay till you dispatch him; and, as soon as you have decyphered any of the other letters, I desire you will send each of them as fast as they are done by a messenger, whom I will pay. Pray make what haste you can herein, since it seems to be of very great importance to their Majestys, ...
Nottingham to Wallis, 13 July 1690 (Monthly Magazine, vol. XIII, p.446)

Nottingham's haste may be understandable considering that the Jacobite forces under James II and Lauzun had just been defeated on 1 July 1690 in the Battle of the Boyne, when Lauzun's letters in cipher were found in his possessions. To this letter were attached letters from Louvois dated 1, 25, 26, 27 May, and 10 June and another "subscribed by a French name in a French hand, which I [Wallis] cannot read" dated 22 June.

In four days, Wallis managed to break the cipher from the unknown sender (Seignelay). On returning the solution, he could not help telling Nottingham that deciphering, without any key, a letter, written by a great French minister, in a cipher, which he had never seen before, to return it by the same messenger was more than he could do.

Wallis could not decipher Louvois' letters, which were not written in the same cipher. Wallis said he had "never yet mastered any of his ciphers; and I suspect somewhat of peculiar in his way of cyphering, which I have never yet had the good hap to light upon." (Monthly Magazine, vol. XIII, p.447) Harry Thompson prints one of the undeciphered letters, dated 27 May 1690. The cipher is a numerical code consisting of numbers up to numbers in the 400s.

Wallis referred to the increasing difficulty of French ciphers.

My Lord, I sent your Lordship, by last Thursday's post, an account of one of those letters which were sent me to decypher; since when I have not been idle, but employed a great deal of pains and (your Lordship would say, if you had been to see it) a great deal of patience, upon the rest, but without success. They do so oft change their cyphers, and even their methods of cyphering, and make them now so intricate (finding that so many have been discovered), that even to myself it seems more strange that I can decypher any, than that I miss of some; which I thought necessary thus to signify, that your Lordship may not impute the failure to want ef diligence, or want of due devotion to his Majesty's service.
(Wallis to Nottingham, 22 July 1690, Gentleman's Magazine, vol.58, p.852; also cited in Monthly Magazine, p.253 without date)

(In 2020, Norbert Biermann published the plaintext and ciphertext of the five letters from Louvois and the one solved by Wallis (from Seignelay). He found Louvois' cipher in the archives. The cipher has entries up to 451, in which not only syllables and words/names, but also letters are mixed in random order.)

Deciphering of Simple Ciphers (July 1690, January 1692)

About the same time (26 July), Wallis reported to Sir Henry Capel (Wikipedia), Privy Councillor, his solution of a simple cipher, which he "received this morning, and finding the cipher not hard, I have this day dispatched it" (Monthly Magazine, vol. 13, pp. 560-561, vol. 14 pp. 251-252).

The packet forwarded to him contained two letters, one of which enclosed the letter in cipher. The letter only contains about a dozen names or phrases such as "Sir Will. Sharp", "Scotland", and "three commissioners" in "two distinct ciphers". Wallis observed that "some particular spellings and otherwise" indicated the writer to be a Scotchman.

Wallis provided his guess that "Wilson" meant the King and an expression "Wilson's being factor for an Irish bankrupt marchand" might mean the King's acting in the room of the deposed James II. (Such doublespeak was typically used by Jacobites.)

Later, Nottingham forwarded a similar letter to Wallis.

I desire you will decypher the enclosed letter, if you can, and send it to me. There is not much of it in cypher, which may make it the more difficult to find out the meaning; but if that could be discovered, I believe it may contain something of importance for their Majtys service. I am your most humble servant, Nottingham.
(Nottingham to Wallis, 14 January 1692, Monthly Magazine, vol. 14, p.521)

(This piece is quoted here to be compared to the tone of 13 July 1690 above.)

Wallis responded with a solution as early as 19 January 1692. As it turned out, the few words in cipher were in three ciphers used "promiscuously." Each word in cipher was marked to indicate which cipher applied. Moreover, many nulls were used.

Secrecy of Deciphering Activities

Wallis' letter to Nottingham dated 20 February 1690 [Thursday] (see "John Wallis as a Cryptographer") reflects his alarm about talk of his achievements. Should his activities be known, Wallis feared France might change the ciphers, which had actually been done more than once, for he had at that time "allready nine or ten of their ciphers by mee". Wallis thought the deciphering activities must be kept secret.

Codebreaking for Foreign Courts

Wallis provided his service to the Elector of Brandenburg, an important ally of England in her struggle with France. Many intercepted letters from the French King's envoys in Poland were sent from the Elector to Smettan, his envoy in London, which Wallis deciphered and sent back. They were written in a great variety of ciphers (including different methods of ciphering) (W. Wallis, Sermons, p.xlvii).

It was probably in this context that in 1690 the envoy from Brandenburg in England made corrections in Wallis's deciphering. The secretary of the envoy wrote, "The Envoyé of Brandenburg having made some small corrections in the Letter deciphered by you, which may be of use to you if any others of the same cypher should again be sent to you I have sent those letters to you by my Lord's Order, who desires, that when you have perused those corrections you will send the letters back to me again, for his Lordship." Four days later, Wallis pointed out that most of the corrections were those made by himself and the others were inaccuracies in the original letter. (William James Roosen (1976, 2017), Age of Louis XIV: The Rise of Modern Diplomacy)

Anyway, the Elector sent the deciphered letters to the King of Poland. The enraged King immediately sent for the French envoys and, having made them hear their own letters read, ordered them to depart his dominions instantly (W. Wallis, Sermons, p.xliv).

Wallis boasted about this episode in his letter to Johnston, ambassador at Berlin, dated 9 June 1691 (W. Wallis, Sermons, p.xliv-xlvi) as well as a similar letter to Harbord, dated 15 August 1691 ("John Wallis as a Cryptographer").

Having said this, while the Marquis of Bethune, a French envoy, indeed struggled to prevent an alliance between Poland and the Emperor and his schemes led to a situation he could no longer stay in Poland, circumstances of his removal described in history books are different from the above episode (see above).


Wallis received fifty pounds from the Earl of Nottingham for his deciphering of the first three letters (Wallis to Mr Harbord, 15 August 1691; see "John Wallis as a Cryptographer"; On the same day, Wallis also wrote directly to Nottingham, see Sermons p.xxxv). After that, materials to be deciphered that "came faster than we could well dispatch them" (ibid.; see also Wallis to Nottingham, 12 November 1689) kept Wallis in a continual employment. In about a year he deciphered some hundred sheets but no further remuneration was forthcoming. Only after he complained to Hampden could he be rewarded with another fifty pounds. However, after that, again, he was kept without payment and he complained bitterly about the treatment in the above-mentioned letter to Mr. Harbord.

Wallis did not hesitate to use an unreserved language, which drew another one hundred pounds.

I have at length mastered this cypher, and have found it, as I did expect, perplexed enough (the most intricate, I think, of any that I have decyphered); and I take it for a good-hap that I have been able to decypher it at all. But, my Lord, it is hard service and I am quite weary. If your Honour were sensible how much pain and study it cost me, you would pay me; and there is a proverb, of not riding a free horse too hard.
(Wallis to Nottingham, Gentleman's Magazine, vol.58, p.852; Sermons, p.xxix)
My Lord, however I am neglected, I am not willing to neglect their Majesties services, and have therefore resumed the letter which I had laid by, and which I here send decyphered. Perhaps it may be thought worth little, after I have bestowed a great deal of pains upon it, and those valued accordingly. But it is not the first time that the like pains hath been taken to as little purpose by, &c. J. W.
(Wallis to Nottingham, 25 August 1691, Gentleman's Magazine, vol.58, p.852; Sermons,
Sir, I thank you for yours of the 25th, and am sorry you think yourself neglected; I am sure you have not been by me, who have taken very many occasions of representing your good services to their Majesties, and I know the Queen has you her thoughts for some preferment that may be worthy of you, and at the same time has ordered me to present you one hundred pounds, which I shall pay to such as you shall authorize, under your hand, to receive it.
(Nottingham to Wallis, [27 August 1691], Gentleman's Magazine, vol.58, p.852)

Later Years

In 1692, the amount of extant correspondence concerning cipher is much less than in the preceding years. In August 1692, Queen Mary II offered through Nottingham the deanery of Hereford but Wallis declined it, hinting that favors for his son and his son-in-law Blencowe would be more welcome by saying "I have a son-in-law, Mr Sergeant Blencowe, of the Inner Temple, a Member of Parliament, an able Lawyer, and not inferior to many on the Bench, of a good life and great integrity, cordial to the Government and serviceable to it" (History of Parliament Online,, Galileo Project).

After Nottingham was dismissed from secretaryship in November 1693 because of the increasing Whiggish trend at the time, Wallis provided his service to his successor, Shrewsbury. The following accounts an occasion when Wallis almost gave up the task as desperate.

I must beg your Lordship's pardon, that I have not hitherto been able to decypher the papers you sent me for that purpose. I have bestowed true pains upon them (of which it is no use to complain, since this will not make the matter easier) and have made it almost my whole study and employment, by day and by night, ever since, without being able to accomplish it; and, I find, I must bestow a great deal more, if ever I do. Your Lordship understands cyphers so well, as not to think it a business of one sitting (I should have said of one day or week) to decypher a new cypher, so intricate as this, without a key. I have been thinking, more than once, of giving it up, as infeasible: but I shall, with your Lordship's leave, spend a week or two more upon it before I give it up as desperate: and if, after all, I must be obliged to leave it, your honour will at least pity me, for having bestowed so much labour in vain.
(Wallis to Shrewsbury, 5 January 1695; Gentleman's Magazine vol.59; also wrongly cited in Monthly Magazine, xiv p.252)

I trouble your Honour with this letter, to let you know I am not idle, though I cannot yet give such an account of the papers sent as I could wish. I have already employed about seven weeks upon them, and have studied hard thereon eight or ten hours a day (or more than so) very oft, which (in a business of this nature) is hard service (for one of my years), unless I would break my brains at it. It was a saying of King Cha. I that it was some skill at least to know when a game is lost. And it is so at decyphering, to know when it is not to be done. The cypher is strangely confused, perplexed, and intricate, even to such a degree, that it must needs be very troublesome to use even by those that have the key; and how is it then to me that have not?
(Wallis to Shrewsbury, 8 February 1695, Gentleman's Magazine, vol.59, p.113; Sermons, p.xxxii)

This time, Wallis managed to decipher the letter after ten weeks' hard study.

I now send the other two pieces, which I had not before deciphered. The decyphering of all which hath been the result of ten weeks hard study. I hope your Lordship will think this, with what I have done (which every body could not do) of this kind for two years last past, may deserve a better recompence than some good words; for really, my Lord, it is hard service, as well as difficult, requiring much labour, as well as some skill; and such, as, were it not to serve his Majesty, I should, not easily be hired to undergo.
(Wallis to Shrewsbury, 23 February 1695, Gentleman's Magazine, vol.59, p.113; Sermons, p.xxxii; Monthly Magazine, vol. 14, pp.252-253)

A glance at a list of extant correspondence concerning cipher indicates Wallis' deciphering activity was not required during 1696-1701. In 1699, Wallis saw fit to publish specimens of his deciphering in Opera Mathematica (see another article).

Wallis' correspondents

Wallis' correspondents about cipher include the following (EMLO).

Nottingham, secretary of state from March 1689 to November 1693, May 1702 to April 1704;

Shrewsbury, secretary of state from February 1689 to June 1690 and March 1694 to December 1698;

Richard Hampden, a Whig MP, (August 1689);

Richard Warre, under-secretary to Nottingham, (August 1689-May 1690);

Sir Henry Capel (July 1690), mentioning Ireland;

John Methuen (24 December 1690), mentioning Poland;

Henry Sidney, secretary of state from December 1690 to March 1692 (March-April, August 1691, April 1692), mentioning Swedish;

George Robert (June 1691), mentioning Dublin, Robert Pawling (December 1691-July 1692), mentioning Elector of Brandenburg, King of France, King of Poland, Schmettau (Brandenburg envoy), present, etc.;

William Deeds (January 1693);

Sir John Trenchard, secretary of state from March 1693 to April 1695 (August-October 1693), mentioning Turkey;

James Vernon "at the Secretaries office" (March, August 1695);

Sir William Trumbull, secretary of state from May 1695 to December 1697 (April, November 1695) involving "Royal Grant for Services in Decyphering";

William Aglionby, under-secretary to Nottingham, the latter being re-appointed secretary of state May 1702 (July 1702);

Archbishop of Canterbury (September, December 1702), mentioning cipher, plot, trick, assassinate, contriver, Aglionby, Prince of Darmstadt, Nottingham, etc.;

John Ellis (September 1702), mentioning hard cipher, William Blencowe.

Private Ciphers

Rosicrucian cipher (1657)

Wallis' letter dated 22 July 1657 includes his deciphering of a cipher sent by his acquaintance (said to be Richard Lawrence) as a challenge. The cipher turned out to be a Rosicrucian cipher (Correspondence of John Wallis, i).

It is wondered whether this challenge was occasioned by Henry Stubbe's accusasion in 1657 of Wallis' alleged deciphering of the King's letters captured in Naseby (see above).

Caesar cipher (1692)

In a letter dated 9 June 1692 addressed to the ambassador in Berlin, Wallis recommends a simple Caesar cipher. This is because the ambassador's request had been for an "easy Cipher", which yet may be tolerably safe because the ambassador was to use it himself and could not well spare much time for it.

(By the way, it is noted that Wallis says in this letter that he knew James II, then in exile in France, used ciphers not better than these.)

Inquiry from Leibniz

Wallis' codebreaking was known abroad and Wallis was repeatedly asked by Leibniz, who was employed in the court of Hanover, to publish his achievements. But Wallis never disclosed the art of deciphering. When asked by Bishop Burnet, he only made generic remarks "My Lord, nothing is too hard for sagacity and industry." (Davys p.51)

This section follows exchanges between Wallis and Leibniz (in Latin).

First Request (1697)

Leibniz, a generation younger than Wallis, was influenced by works of Wallis as a mathematician. He must have been gratified when Wallis wrote a favorable review of his Hypothesis Physica Nova in the 1670s (Matthew Stewart, The Courtier and the Heretic, p.109 of the Japanese translation). Leibniz had worked on codebreaking in the 1680s but could not go beyond the basics (Beely). Soon after he started direct correspondence with Wallis on scientific matters in the mid 1690s, he asked Wallis to write something about the art of deciphering (Leibniz to Wallis, 19/29 March 1697).

In this letter, after discussing mathematical topics, Leibniz broached the matter by quoting an anonymous review of Wallis' Algebra (1685), which appeared in Acta Eruditorum ("Acts of Scholars"; a scientific journal which Leibniz and Otto Mencke established in 1682 in Leipzig, referred to as Acta Lipsiensia or Acta Lipsica ("Acts of Leipzig") in their correspondence): "The person who reviews your Algebra in the 1686 issue of Acta Lipsiensia, p.283 wishes that you write something about the art of discovering secretly written matters." The reviewer, noting Wallis' excellence in solving cryptograms, observed it had relation with mathematics and pointed out that what had hitherto been written was quite imperfect. Referring to Viète, who deciphered ciphers of Philip II of Spain (see another article), the reviewer emphasized the importance of leaving specimens to the posterity.

Actually, the anonymous reviewer was none other than Leibniz. Apparently, Leibniz wanted Wallis, skilled both in mathematics and deciphering, to discuss the art in a scientific way (Beeley). In the mid 1680s, when the review appeared, suspicion of Wallis' deciphering of the letters of Charles I (some forty years before) was much talked of (Wallis to Bishop Fell, supra) (perhaps because of some talk about the position of Wallis upon accession of James II). Probably this reached Leibniz' ears. Ten years after the anonymous review, Leibniz now had a chance to ask the master in person.

After the self-staged quotation of the review, Leibniz expressed fear that the art of deciphering might perish with Wallis and wished that Wallis teach young people so that the past glory could be enjoyed without requiring labors from scratch all over again.

Wallis' reply (Wallis to Leibniz, 6/16 April 1697) pointed out the infinite variety in cipher and the difficulty, which was already great and was increasing day by day, and stated that the art of deciphering could not be reduced to definite rules. At least, he made a general remark that he would start from a conjecture and maintain or reject it by ascertaining whether they were likely to succeed or not until some plausible meaning was established. Then, he told Leibniz that he had sent a transcript of a cipher letter with its deciphering to Otto Mencke, the editor of Acta Eruditorum.

Specimens Sent to Otto Mencke (1697)

The letter to Mencke (Wallis to Mencke, 1/11 January 1697; see another article) had also been a reply to the latter's wish for rules about deciphering of secret writing. As in writing to Leibniz, Wallis had said it could not be reduced to definite rules, but enclosed specimens of deciphering.

The specimen selected by Wallis was a cipher letter dated 6 September 1689 NS from the Marquis of Bethune, French emissary in Poland, to Cardinal D'Estrée. Wallis included the cipher letter itself, deciphered text, ciphertext interlined with deciphering, and the cipher table: the presentation method he used forty years before in the collection deposited at the Bodleian Library.

When reporting this to Leibniz (Mencke to Leibniz 22 May [1 June] 1697), Mencke pointed out political complications in publishing this, not to speak of difficulty of finding a publisher (Beeley). In the end, Wallis would publish this himself in Opera Mathematica (see here).

While disclosure of the deciphered text with the original cipher letter might reveal the kind of cipher used, it discloses nothing about how it was achieved. Leibniz told Wallis that a specimen would have been useful (Leibniz to Wallis, 28 May / 7 June 1697). Leibniz also complained to Mencke that Wallis should have sent the method of deciphering as well as the solution (Leibniz to Mencke, August - beginning of September 1697).

Wallis' reply (Wallis to Leibniz, 30 July 1697) to Leibniz' letter in May merely noted that he had written to the editor of Acta Lipsiensia in January but did not know whether it was received.

Asking for Reasoning Process (1697-1699)

While Leibniz admitted that the art of deciphering ("Ars cryptolytica") could not be reduced to definite rules, noting that otherwise it would not be a big deal, he asked for a demonstration with specimens as to how Wallis proceeded with his analysis (Beeley, Davys p.31) (Leibniz to Wallis, 28 September /8 October 1697). Even without general rules, if the process of trials and errors for some specific case of deciphering is disclosed, the idea, if not the specific procedure, might be useful for similar activities.

In the next spring, Leibniz admired the specimen sent to Mencke, which he saw, and again asked for disclosure of the process (Leibniz to Wallis, 24 March 1698). (In the meantime, the Elector of Hanover died in January 1698 and was succeeded by his son, later King George I of England.)

In these exchanges, the two scholars did not just waste paper in futile exchanges about deciphering, which were actually only mentioned in last parts of their letters for discussing other topics such as mathematics. At the end of the year, again, Leibniz repeated his request at the end of his letter (Leibniz to Wallis, 29 December 1698). He wished that Wallis would not let the admirable art of deciphering perish and, admitting that it could not be reduced to sufficiently general rules, the deficiency might be filled by examples, with the trace of reasoning being always attached. Leibniz wished that Wallis give the benefit to the posterity. He even said if scribes were necessary for the task, there might be ways. It is wondered whether the reference to the scribes implied his request was backed by the new Elector of Hanover or it was within his discretion as a librarian and a court councillor. (At this stage, it seems the latter was the case.)

Wallis' reply (Wallis to Leibniz, 16 January 1699) again did not satisfy Leibniz. Wallis stated everyone was not equal in deciphering, which requires a special talent, and a capable man might be reluctant to undertake it when he found the labour it involves. He compared the work of deciphering to hunting, saying hunting varies in accordance with the variety of the game. (The exchanges up to this point were printed in Opera Mathematica, published in 1699.)

The matter seems to have been half dropped, when the next letter of Leibniz (Leibniz to Wallis, 30 March 1699) only mentioned it in the postscript as a reminder, saying he did not want the summit of human subtleties to perish. Several letters were subsequently exchanged without bringing up the matter (Wallis to Leibniz, 20 April 1699; Leibniz to Wallis 4 August 1699; Wallis to Leibniz, 29 August 1699).

Resumption in Earnest (1699-1700)

In November 1699, Leibniz revived the matter (Leibniz to Wallis, 24 November 1699). Unlike his letters thus far, in which he only mentioned it as a subsidiary topic, he now brought it up foremost. Leibniz stated while all of Wallis' recent book (probably referring to Opera Mathematica published this year; see here) was worthy, nothing was more pleasing than the wonderful specimens of his art of cryptanalysis.

Leibniz proceeded with a remark that this did not fulfill their appetite and proposed that young people with ability and zeal be sent to England to be trained by him. He even stated that he was under command to ask under what conditions Wallis would accept the task. This statement made clear that the request was from the court of Hanover.

Wallis took several months to reply to the revived proposal of Leibniz (Wallis to Leibniz, 29 March 1700). Wallis said he was at a loss what he should say about the matter of cryptography and offered an explanation different from what had hitherto been professed. He said since cipher was commonly used in matters of great moment, dissemination of the art of deciphering could be prejudicious. He cited the authority of the King to declare he would not disclose it to anybody (if it be possible at all) and further said he should not indiscriminately propagate it to others without consulting the King. Wallis now made clear his refusal of disclosing the art. Still, he concluded with an offer that he would exert himself if Leibniz had anything to have solved.

Tenors of this last exchange is phrased as follows in Wallis to Tilson, 20 March 1701, which reminded the authority of payment due to him (see the next section).

I may add this allso; That I have been solicited by Mijn Heer Leibnitz, more than once, in behalf of the Elector of Hanover: who is willing to send hither some young men, whom he desires I would instruct therein; leaving it to me to make my own proposals on what terms I would undertake it. To which I have returned answere, That I shal be ready, my selfe to serve his Electoral Highness if there be occasion: but the skill of doing it, being a curiosity which may be of use to my own Prince, I do not think it proper to send it abroad, without his Maties leave.
(Wallis to Tilson, 20 March 1701)

Cryptologic Observations

The first reason Wallis gave for not disclosing the art of deciphering was that the process of deciphering need to be adapted to the materials at hand and thus could not be reduced to definite rules. This would be something that anyone who had ever tried his hand would realize. When Leibniz requested demonstration with a specimen of how his reasoning proceeded, Wallis still did not comply. When Leibniz came up with a specific proposal of dispatching young people to England to be tutored by Wallis, saying he was under the command of his master, Wallis resorted to the authority of the King and made clear his refusal.

While there were some who thought dissemination of the art of deciphering was beneficial as a means of preventing intrigues, as seen in John Falconer's Cryptomenysis Patefacta published in 1685 (cf. another article), there continued to be many who considered disclosure of the art of deciphering is prejudicious to the national security.

When one looks at the two specimens of French cipher published in Opera Mathematica (see here), it may be seen what Wallis feared. They indicate a striking weak point of the French cipher: low numbers were reserved to code single letters. The first thing Wallis did must have been to focus on such low numbers. For a skilled person as Wallis, it would have been little more difficult than simple substitution ciphers (given sufficient amounts of materials). (To make the matter easier, letters were arranged alphabetically, albeit with gaps in code numbers.) Any demonstration would reveal this strategy of first tackling the low numbers. Probably, Wallis feared that if he disclosed such a technique, this fundamental structure would be changed to a more complicated scheme of randomly assigning numbers to single letters and other words and names.

Such a scheme did exist in the seventeenth century. Examples include the ciphers used by J. Peterson (1653), Henry Manning (1655), and William Lockhart (1656-1658) (see Codes and Ciphers of Thurloe's Agents), a cipher for Spanish (PRO SP 106/6 no. 34; see another article), and ciphers used by the Spanish in the late seventeenth century (see Cg.52, Cg.53, Cg.54, Cg.55, and Cg.56 in Spanish Ciphers in the Seventeenth Century). More may be found in the British and French archives. It would be common during the American Revolutionary War (see WE006 and WE007 used by John Jay (see here) and WE008 used by Benjamin Franklin (see here)).

It is rather surprising that the French still used such a vulnerable cipher, after the great Rossignol. Possibly, these two particular ciphers were not the best of the French ciphers.

Notes on Latin vocabulary used by Wallis and Leibniz

cryptographema, -atis n. cryptogram (similar to cryptogramma, -atis commonly found in Latin dictionaries.

solutio cryptographematum solving a cryptogram, deciphering, codebreaking

de cryptographematis explicandis on solution of a cryptogram

cryptographicus, -a, -um, adj. of or relating to cryptography (an adjectival form of cryptographia, -ae commonly found in Latin dictionaries. Cf. geographia/geographicus).

res cryptographica the matter of cryptography

cryptolyticus, -a, -um, adj. deciphering or cryptanalysis in modern English.

ars cryptolytica art of deciphering

de cryptolyticis on deciphering (used as a noun)

Wallis' Successor

Wallis' effort to convey his arts to his son, John, did not bear fruit. While some say he lacked the father's genius (John Wallis as a Cryptographyer, p.83), Wallis wrote, in a letter dated 28 February 1694, "I have been shewing my son, John Wallis ... from time to time, how I proceed; and teaching him (as far as it is to be taught) how to do the like, and have made use of his assistance (when we were together) in decyphering many letters, who is of capacity enough to understand it, but complains of the fatigue, as not being worth his while to undergo, having other business to attend; and nothing but a long practice (besides a natural sagacity) can render a man expert at it." The son, who was a barister-at-law, was "at that time much better off ... than to acccept" deciphering, as Wallis' great-great-grandson put it (W. Wallis, Sermons, p.liii-liv)

Nonetheless, Leibniz' concern that the art of deciphering might perish with Wallis did not materialize. Wallis counted on his grandson, William Blencowe, a son (one among three sons and four daughters (Wallis' autobiography)) of his daughter. Wallis could obtain from Chancellor Somers a promise of a salary for training his grandson (Kahn p.169, Calendar of Treasury Papers, W. Wallis, Sermons, p.lii-liii). This was in 1699, the very year Leibniz resumed his request in earnest. Although William Blencowe (1683-1712) was then yet sixteen, there was no time to be wasted for Wallis, who was already eighty-three.

However, payment was not carried out for some time and Wallis had to remind the authority of his due in 1701. As seen in the above-mentioned Wallis' letter to Tilson, 20 March 1701, Wallis emphasized that he had been solicited by Leibniz more than once in behalf of the Elector of Hanover to send some young men to be instructed by Wallis. In his memorial (Calendar of Treasury Papers) to the King at about the same time, Wallis emphasized that Blencowe "had deciphered one of the best English ciphers and a very good French one."

In his letter to Tilson, Wallis proposed a change in the condition allowed by the first agreement, i.e., payment of fees for instructing his grandson. Whereas fees for instructing would naturally be limited to lifetime of Wallis, who was already above eighty years of age, Wallis tried to let his grandson Blencowe be allowed to receive an annual pension. As can be seen in the letters cited in the subsection "Remuneration" above, Wallis' position as a decipherer was never an official one. Officially, it was only that a mathematician in Oxford happened to be skilled in deciphering and thus received requests for deciphering from time to time. He was not even guaranteed any payment in return. Wallis wanted that his grandson's position as a decipherer would be one that drew a fixed salary.

Wallis' petition seems to have been granted and Wallis and Blencowe received a pension jointly, retroactive to 1699. When Wallis died in 1703, William Blencowe became the first official Decipherer of England.

List of Letters

List of letters between Wallis and Leibniz/Mencke

Leibniz to Wallis, 6 December 1695 (OM p.652)

Wallis to Leibniz, 1 December 1696 (OM p.5, LB p.5)

Wallis to Mencke, 1 January 1697 (OM p.659)

Leibniz to Wallis, 19 March 1697 (OM p.672, LB p.11)

Wallis to Leibniz, 6 April 1697 (OM p.674, LB p.15)

Leibniz to Wallis, 28 May 1697 (OM p.678, LB p.23)

Wallis to Leibniz, 30 July 1697 (OM p.681, LB p.29)

Leibniz to Wallis, 28 September 1697 (OM p.685, LB p.40)

Wallis to Leibniz, 21 October 1697 (OM p.687, LB p.43)

Leibniz to Wallis, 24 March 1698 (OM p.687, LB p.44)

Wallis to Leibniz, 22 July 1698 (OM p.688, LB p.45)

Leibniz to Wallis, 29 December 1698 (OM p.691, LB p.52)

Wallis to Leibniz, 16 January 1699 (OM p.693, LB p.56)

Wallis to Leibniz, 30 January 1699 (listed in EMLO)

Wallis to Leibniz, February or March 1699 (listed in EMLO)

Wallis to Mencke, 29 March 1699 (mentioned in EMLO)

Leibniz to Mencke, 1699 (mentioned in EMLO)

Leibniz to Wallis, 30 March 1699 (LB p.62)

Wallis to Leibniz, 20 April 1699 (LB p.65)

Leibniz to Wallis, 4 August 1699 (LB p.68)

Wallis to Leibniz, 29 August 1699 (LB p.71)

Leibniz to Wallis, 24 November 1699 (LB p.73)

Wallis to Leibniz, 29 March 1700 (LB p.75)

Leibniz to Wallis, 3 September 1700

Wallis to Leibniz, 5 November 1700 (LB p.81)

Leibniz to Wallis, November 1700 (listed in EMLO)


Biographies of John Wallis

Wallis' autobiography ... Wallis' letter dated 29 January 1697 to Thomas Smith (a fellow at Oxford University), written in response to repeated requests. It is printed in an appendix (p.cxl) to the preface of the editor of Peter Langtoft's chronicle (1725) (Google).

C.J. Scriba, "The Autobiography of John Wallis, F.R.S.," Notes and Records of the Royal Society, 25 (1970), 17-46 (JSTOR)... Printing conventionally known text of the autobiography and Wallis' first draft in parallel.

Biographia Britannica (Google)

Oxford DNB


J. A. Kemp (ed.), John Wallis's Grammar of the English Language (1972) ... Introduction reviews various aspects of Wallis' life.

William Wallis, Sermons now first printed from the original manuscripts of John Wallis ... to which are prefixed memoirs of the author (1791) ... Including memoirs by a great-great-grandson of Wallis, including quotations of letters, which appears to be excerpts from the editor's contributions to Gentleman's Magazine.

"The Reverend John Wallis, F.R.S. (1616-1703)" Notes and Records of the Royal Society, 15 (1960), 57-67... Printed in an issue describing people involved in the establishment of the Royal Society.

G. Udny Yule, "John Wallis, D.D., F.R.S. (1616-1703)," Notes and Records of the Royal Society, 2 (1939), 74 (The present author has not seen this.)

J. Scriba, Complete Dictionary of Scientific Biography at

O'Connor, John J., Robertson, Edmund F., "John Wallis", MacTutor History of Mathematics archive, University of St Andrews

Letters and Other Papers

Johannis Wallis, Opera Mathematica Vol.3 (1699) (Google)…Letters are from p.615, in which pp. 659-672 print specimens of letters deciphered by Wallis (excerpted in another article); referred to as OM in the list above

Gothofredi Guillelmi Leibnitii, Opera Omnia (Google) (1768)

Gottfried Wilhem Leibniz, Leibnizens mathematische schriften: Briefwechsel (Google, Internet Archive) (1859) (also including letters after 1699; referred to as LB in the list above)

Philip Beeley and Christoph J. Scriba (ed)., Correspondence of John Wallis (1616-1703) (2003- ) (three out of eight volumes have been published as of 2012)

David Eugene Smith, 'John Wallis as a Cryptographer' (1917) (pdf) ...Printing some letters in the 1690s.

Gentleman's Magazine, (vol.58 (1788), vol.59 (1789), p.3, 113) ... Printing some letters.

Monthly Magazine and British Register, vol. XIII, pp.446-447, 560-561; vol. XIV, pp. 252-253, 521-522 ... Printing some letters (Scans can be found at Google for vol. XIII (albeit defective) and vol.XIV)

John Davys, An Essay on the Art of Decyphering (1737) ... Printing Wallis' introduction to the collection of deciphering that he deposited to the Bodleian Library.


Early Modern Letters Online (EMLO), Beta Version (accessed in January 2013) (online)

C.J. Scriba, "A Tentative Index of the Correspondence of John Wallis, F.R.S.," Notes and Records of the Royal Society, 22 (1967), 58-93. (This excludes inter alia the letters Wallis exchanged with his government concerning his deciphering and the deciphered letters except where these have already been published.)

National Register of Archives at The National Archives ... List of locations of original papers related to Wallis.

John Wallis Bibliography


Philip Beeley, "Un de mes amis.": On Leibniz' Relation to the English Mathematician and Theologian John Wallis, in Leibniz and the English-speaking world (2007), pp.63-82 (Reproducing cipher letters deciphered by Wallis)

Philip Beeley (2016), "Breaking the Code. John Wallis and the Politics of Concealment" in G.W. Leibniz und der Gelehrtenhabitus, pp.49-82 (DOI:

Norbert Biermann, "'I Suspect Somewhat of Peculiar in His Way of Ciphering'" (Universität der Künste Berlin, DOI:

David Kahn, The Codebreakers (1967) ... Reproducing (p.168) Louis XIV's letter dated 9 June 1693, deciphered by Wallis.

Geneva, Astrology and the Seventeenth Century Mind p.25 ... Reproducing Charles I's letter dated 3 February 1647

Harry Thompson, The Man in the Iron Mask [the present author has only seen the Japanese translation]... Printing in the appendix a letter in a French cipher that Wallis could not solve.

Life of Dr. John Barwick (Google)

Calendar of the Clarendon State Papers (CClSP), vol. IV (Internet Archive)

Underdown (1960), Royalist Conspiracy in England

Richard S. Westfall, "John Wallis", The Galileo Project (Various data and documents, etc.)

'Wallis, John (1650-1717)' (History of Parliament Online)...Introducing John Wallis, Jr.

'Blencowe, John (1650-1717)' (History of Parliament Online)...Introducing John Blencowe.

'A Recently Discovered Sir John Blencowe Document', 3 August 2009 (page from Blencowe Families' Association)...A page of Blencowe's descendants.

©2012 S.Tomokiyo
First posted on 14 November 2012. Last modified on 28 February 2021.
Articles on Historical Cryptography
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