Specimens of French Cipher (1689) Printed in John Wallis' Opera Mathematica

The celebrated codebreaker John Wallis frustrated Leibniz when he refused to disclose the art of deciphering. Instead, he referred Leibniz to specimens he had sent to Otto Mencke (Wikipedia) in a letter of 1/11 January 1697. The letter, including the specimens, were published in Wallis' Opera Mathematica, vol. 3, in 1699.

Wallis' Letter to Otto Mencke, 1/11 January 1697

(rough outlines)
.... (paragraphs omitted)
You make remarks about deciphering of secret writing and request me to give its rules.
However, it is a matter of great labor and devotion and (if I may say) great sagacity and cannot be understood by definite rules. Since there is a great variety in cipher (which daily increases), ability must be acquired by long experience. As it were, [a gladiator] makes a plan in the arena according to the circumstances.
I undertook this for the first time more than fifty years ago (when ciphers were simpler than these days). It was in an easy cipher (it was the first I had ever seen), which I mastered in two hours.
Afterwards, I undertook more difficult ones. It succeeded not badly.
This is not to say that I can decipher anything in any cipher. That some (due to very complicated ciphers or due to lack of ingenuity or due to conjectures made being less felicitous) escaped my ingenuity, I am ready to admit.
Such that you may see how hard and full of difficulty it is to solve a complicated cipher (as they are these days), I send you, among many, one as a specimen. Not that this is the most difficult I was involved in. Probably, I have deciphered something more difficult.
A letter from the Marquis of Bethune [François-Gaston de Bethune] (brother of the Queen of Poland [his wife being a sister of the Polish King Jan Sobieski] and an emissary of the King of France) was sent to Cardinal D'Estrées. Bethune sent its copy (enclosed in other letters) to the Marquis of Croissy (Secretary of the King of France). This came to my hands.
This I chose to be sent. I presented it such that anyone can know whether it is right or not.
About proper names designated by one number (and those like them), conjectures cannot be made but from circumstances. Even if I guessed wrong in one or other words, I will be easily forgiven. For it is evident that by the writer's fault (one number being put in place of another, even though he had the key before his eyes) errors were introduced. It happens often and is nothing to be wondered at.
Here, you have, first, the letter itself or its copy (written in cipher just as it came to my hands) exactly transcribed. Second, the interpretation I made (I marked which errors of the writer I correct or omissions I supply). Thereafter, that with interlinear explanation such that you see which number designates what (where I correct errors or supply omissions, I insert correction to the text in parentheses and similarly for omissions). Finally, the key as far as could be collected from the letter is added, by which you can examine each.
I wrote these such that your curiosity may be satisfied.
Wallis to Otto Mencke, 1/11 January 1697

First Letter from Marquis of Bethune to Cardinal D'Estrée, 6 September 1689 (NS)

Of the two specimens, the first is a letter from the Marquis of Bethune (a French emissary to Poland) to Cardinal D'Estrée. It deals with the papal conclave of 1689 (Wikipedia).

The first sheet of the letter as well as the cipher table in manuscript are reproduced in Figs. 1 and 2 of Beeley.

First Letter in Cipher

(Parenthetical annotations such as (l. ...) are Wallis' correction (i.e., "read").)

First Letter in Plaintext

First Letter in Cipher with Plaintext Interlined

Cipher used in the First Letter

This is a numerical code comprising numbers about 1-480, with low numbers up to about 64 being reserved for single letters. While the assignment of numbers are not alphabetical, words with the same initial letter occur in rows (i.e., the same ones digit). Wallis seems to have been aware of this because he conjectures that the name represented by 186 begins with D.

The "+ s" in the margin indicates "+" represents "s". In the manuscript, this is put in place of the blank sections 481-500.

At the footnotes, Wallis says a line placed below [a code number] indicates some imperfection or varied termination but these are occasionally neglected. For example, "335" is "Cardinal" but with an underline under "5", it may read "Cardinau" to form part of the plural form "Cardinaux". Similarly, "181 Pologne" may be used to form "Pologn-ois" and "177 France" to form "Franc-ois". "221 service" may read "servi" to form part of "servir."

A mark placed above [a code number] indicates the numbers themselves with the tens digit removed. While the mark looks like a line in the manuscript, it is represented by an apostrophe in the print. ("7bre" was a common way to spell "Septembre".)

Second Letter from D. de Teil to Louis XIV, 8 July 1689 (NS)

The second of the specimens is a letter by "D. de Teil" to the King of France. The writer is Caillet de Teil, a judge of the Parlement (Oeuvres complettes de M. l'abbé Coyer, p.290). The Marquis of Bethune, l'Abbé de Gravel, and de Teil were three ministers sent to Poland to promote the French interest.

The letter reports situations of Russia and advances suggestions on diplomatic policy.

Second Letter in Cipher with Plaintext Interlined

Second Letter in Plaintext

Cipher used in the Second Letter

Remarkably, Teil used a different cipher from Bethune's, though the general features are the same.

The footnotes mention some new features. Numbers 14, 16, 18, 20, 24, 28, 30, 34, 36, 38, 40, 44, 46, 48, 50, and 52 are not significant.

10, θ, and κ delete the preceding number. (E.g., see θ towards the end of the first paragraph.)

6', 8', and 9' repeat the preceding number.

ч ч delete everything in-between. (E.g., see the last line in cipher.)


Johannis Wallis, Opera Mathematica Vol.3 (1699) (Google), pp.659-672 (the appendix of letters starts on p.615.)

Philip Beeley, "Un de mes amis.": On Leibniz' Relation to the English Mathematician and Theologian John Wallis, Leibniz and the English-speaking world (2007), pp.63-82 (including three figures)

©2012 S.Tomokiyo
First posted on 3 November 2012. Last modified on 25 March 2015.
Articles on Historical Cryptography
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