Leibniz's machina deciphratoria under production by Klaus Badur and Wolfgang Rottstedt, using design suggestions by Richard Kotler to implement Nicholas Rescher's conceptual reconstruction of the device.

(top) Construction design for the Leibniz's Cipher Machine

(bottom) The Leibniz's Cipher Machine


An Account on the Occasion of the Initial Exhibition of the Reconstruction of Leibniz's Cipher Machine





Copyright 2012, University Library System, University of Pittsburgh

All rights reserved.

Library of Congress Control Number: 2012949833

Published by the Office of Scholarly Communication and Publishing,

University Library System, University of Pittsburgh, Pittsburgh, PA 15260. 2012.

Cataloging-in-Publication Data

Rescher, Nicholas.

Leibniz and cryptography : an account on the occasion of the initial exhibition of the reconstruction of Leibniz's cipher machine / Nicholas Rescher.

xii, 96 p. ; 23 cm.

Includes bibliographical references.

ISBN 978-0-9833584-2-8 (paper : alk. paper)--ISBN 978-0-9833584-1-1 (ebook)

1. Leibniz, Gottfried Wilhelm, Freiherr von, 1646-1716--Knowledge--Cryptography. 2. Cryptography--History--17th century. 3. Ciphers--History--17th century. I. Title.

B2598.R453 2012 2012949833

For Professor Herbert Breger

My sole predecessor in concern for Leibniz's work on matters of cryptology

Cryptolysis, ars solvendi aenigmata cryptographica, est summun specimen humanae penetrabililatis. ["Cryptology, the art of solving cryptographic enigmas, is the supreme specimen of human ingenuity.]

G. W. Leibniz

(to John Wallis, 1698)

[AIII,7 p. 759; AI,13, p. 300.]


Introduction: Reconstructing Leibniz's Cipher Machine.................................... xi

I. Leibniz and Cryptography......................................................................... 3

II. Leibniz's Machina Deciphratoria............................................................ 35

III. Pictographic Contextualization of Leibniz's Machina Deciphratoria............................................................49

IV. Leibniz's Own Work at Decipherment................................................... 61

Notes...................................................................................................................... 77


About the Author.................................................................................................. 95



Reconstructing Leibniz's Cipher Machine

During the 2010-11 academic year I launched into an investigation of Leibniz's dealings with matters of cryptology. In the course of this inquiry I read with surprise in the only recently (2001) published volume of Leibniz's Saemtliche Schriften containing several 1688 memoranda that Leibniz prepared that autumn for his audience with Leopold I, the Holy Roman Emperor. In them he described his machina deciphratoria, the cipher machine he had devised in the 1670s and had already briefly mentioned in a 1679 memorandum for John Frederick, the Duke of Hanover.

The information given in those 1688 memoranda regarding the workings of this machine--and in particular, its reliance on the Staffelwalze that was at the core of Leibniz's celebrated calculating machine--proved sufficient to enable its conceptual reconstruction. My engineer friend Richard Kotler helped to fill in some of the details of the gearing, and Klaus Badur of Hanover, who had earlier reconstructed a version of the calculating machine, undertook to arrange for the production of a physical model in collaboration with Wolfgang Rottstedt. The fruit of these efforts is the focus of the present exhibition of this rediscovered machine.

Although there had been earlier cipher devices such as slides or wheels, Leibniz's remarkable apparatus was the first actual cipher ma-chine. Vastly more reliable and easy to use, it had a sophistication not attained again until the Post-World War I era some 250 years after its day.

I am grateful to the University of Pittsburgh for allowing me to dedicate a portion of my research funding to the production of this model


and to Dr. Rush G. Miller, the University Librarian, for his cooperation in arranging for the present exhibition and his unfailingly supportive interest in the entire project. I am also grateful to Jeffrey A. Wisniewski, Kari E. Johnston, and John H. Barnett at the University Library System for their efficient support in regard to publication.

Details about Leibniz's machine and its historical significance are given in my essay on "Leibniz's Machina Deciphratoria: A Proto-Enigma Cipher Machine," in the journal Cryptologia. I am grateful to Dr. Craig Bauer, the editor of this publication, for permission to include this article in Chapter II.

Finally, I am grateful to three German Leibniz experts for their help with various aspects of this project: Dr. Herbert Breger, Dr. Sven Erdner, and Dr. Heinrich Schepers. To Dr. Breger belongs the distinction of being the first scholar to address Leibniz's interest in encipherment.

Nicholas Rescher

Distinguished University Professor of Philosophy

University of Pittsburgh


An Account on the Occasion of the Initial Exhibition of the Reconstruction of Leibniz's Cipher Machine3


Leibniz's Forays in Cryptography

1.     Cryptography's Place in Leibniz's Polymathic Project


It is unquestionably an exaggeration to say, with Voltaire, that men use speech only to conceal their thoughts from the view of others. But it is certainly the case that they sometimes do so.

The symbolic encoding of information and its concealment and revelation was of paramount interest to Leibniz throughout his entire career from beginning to end, and was a topic that stimulated his mind in many directions. And cryptography, so Leibniz tells John Bernoulli, is a part of this project that is well deserving the attention of a mathematician.1 The art de dechifrer . . . est une matiere encor demy-mathematique,2 and finding the key to a cryptogram is akin to finding the solution of equations in algebra.3

Thus in a brief 1674 sketch of the Art of Innovation (ars inveniendi) states that this includes the ars explicandi crytophemata and, like the latter, admits of pursuit via appropriate general rules.4 And in a long letter to E. W. von Tschirnhaus of May 16785 Leibniz describes cryptography (the ars deciphratoria) as an integral part of the scientia generalis that has close connections with algebra and constitutes a key component of the ars combinatoria.6 Despite the power of the analytic method it proves insufficient in cryptography, where a more extensive (longior) procedure of synthesis will prove necessary.7 Moreover, encoding transforms the



foundation of a body of information from one format to another--much as with the representation of geometric figures from diagramatic to algebraic representation in Cartesian geometry. His note of 1678 on the ars inveniendi remarks that the ars deciphrani represents a sector of the field where analysis alone will not suffice for discovery, and observes that while analysis is generally more difficult, synthesis is more laborious.8 As to the type of synthetic reasoning involved, Leibniz likens the type of reasoning invoked in decipherment to finding good moves at playing chess.9

That everything can be said by the use of numbers is a key thesis of Leibniz's universal characteristic.10 And in a way, the object of the ars cryptographica is the inverse of the Leibnizian characteristic: the latter seeks to make language more perspicuous and transparent, the latter

Gottfried Wilhelm Leibniz (1646-1716)

Engraved by B.Holl / Published in London by W.S.Orr & Co.



more difficult of access. Coding and decoding of information in symbolic systems are, after all, inverse procedures and the steps that can make these processes simpler can be reversed to render them more complex and obscure. And Leibniz insisted that in this way advances in cryptography can serve to convey instructive insights into the ways of scientific inquiry. For as Leibniz saw it, cryptanalysis is something of a paradigm for scientific method, the ars faciendi hypotheses.11 Thus he observes that the investigation of causes is easier when different phenomena exhibit a commonality, even as facilius est cryptographemata solvere, si plures literas occultando sensu secundum eandem clavem scriptas.12

In the Nouveaux Essais Leibniz writes: "L'Art de decouvrir les causes des phenomenes, ou les hypotheses veritables, est comme l'Art de dechiffrer."13 For in scientific explanation "a hypothesis is like the key to a cryptograph, and the simpler it is, and the greater the number of events that can be explained by it, the more probable it is."14 Leibniz accordingly endorsed fully the idea--already found in Bacon's Novum Organon and in the 1586 Traicte des Chiffres of the French algebraist and diplomatist Blaise de Vigniere15--that science aims to decode the secrets of nature. In just this way he claimed in relation to the convervation of force that "j'ay toutes les raisons de croire que d'ay dechifre une partie de ce mystere de la nature."16

Leibniz saw what he called "the method of hypotheses" as a key tool of scientific inquiry and the deciphering of a cryptogram was his favorite illustration of the workings of this method of hypothesis-utilization:

A hypothesis of this kind is like the key to a cryptograph, and the simpler it is, and the greater the number of events that can be explained by it, the more probable it is. But just as it is possible to write a letter intentionally so that it can be understood by means of several different keys, of which only one is the true one, so the same effect can have several causes. Hence no firm demonstration can be made from the success of hypotheses.17

To be sure, this method in its application to issues regarding nature is never certain and demonstrative.

For perfectly universal propositions can never be established on this basis (viz., induction based on the experience of particular cases) because you are never certain in induction that all individuals have been consid


ered. You must always stop at the proposition that all the cases which I have experienced are so. But since, then, no true universality is possible, it will always remain possible that countless other cases which you have not examined are different.18

In empirical application to the contingencies of nature the method is always conjectural and yields no more than a probability. As Leibniz sees it, a meaningful decoding is its own verification19 since in cryptography we deal with a finite body of text and so can attain demonstrative certainty in favorable conditions. However, the observable data of nature's "texts" are limitless so that our "decryptions" thereof afford no more than the moral certainty of high probability.20 But of course an insufficiency of texts leads to an underdetermination of possibilities and defines description which, after all, requires a sufficiency of data: aliquando enim tam pauca verba alphabeto incognito scripta habentur, ut prorsus impossible sit humano ingenio clavem reperiri.21 Accordingly, while Leibniz envisioned a deep methodological kinship between the use of hypotheses in scientific explanation and in cryptology,22 he granted that clever guesswork can sometimes surpass the more laborious path of method. And he held that in "the art of deciphering . . . an ingenious conjecture often greatly shortens the road."23

Just as with his interest in combinatorics, so Leibniz's interest in a universal character also had a direct bearing upon cryptography.24 For even as in translation the text of one language is encoded in the vocabulary of another,25 so an artificial language like the universal characteristic functions in such the same way as encrypted communication. And a universal language can clearly provide an excellent nomenclator for coding. But mere coding is not as yet excipherment and is here that the ars cryptographica comes into its own.

Leibniz's' interest in these issues had been enlivened by reading John Wilkins 1668 Essays towards a Real Character and a Philosophical Language and he was also familiar with this scholar's earlier Messenger: Showing how a man may with Privacy and Speed communicate his thoughts to a Friend at a Distance (London 1641).26 Wilkins was the first co-secretary of the Royal Society (along with Leibniz's friend Henry Oldenburg), and his work evoked much contemporary interest in cryptology. And already his early work in combinations the issues of codes and ciphers fell well within Leibniz's virtually boundless range of interest and information, 6


and cryptography had an integral and significant place within the project of scientia generalis that was ever a glint in Leibniz's eye. He viewed it as a natural field for the deployment of rules in rational procedure--exactly in the formalized manner to which he was always deeply partial.27

As a bibliophile, Leibniz was well aware of the literature on the subject. In 1689, during his Italian journey, he prepared an elaborate "must have" inventory for a bibliotheca universalis selecta, some 35 closely packed printed pages in length. This list included a dozen items on steganography cryptology, and verbal concealment.28

In May of 1683, Leibniz's long-time helper, collaborator and correspondent J. D. Brandshagen--and eventually one of his most useful links to England where Brandshagen spent much time29--wrote to Leibniz about a now-lost letter that Leibniz had written to him in April.30 Brandshagen complains that the codes mentioned in earlier correspondence--based on the monoalphabetic cyphers issuing from JACOBUS and LABYRINTHUS31 as key words--will not work in the present instance, but that SALOMONIS will do the trick in that the text can be deciphered on this basis. Already earlier on, Leibniz had recommended such a cipher based on QUIRNHEIM to his correspondent Johann Wilhelm Mers von Quirnheim,32 alternating the direction of substitution.

And there can be no question that Leibniz had good theoretical insight into matters of cryptography. One clear sign of this is his brief paper on Praecepta artis decriptoriae of the middle 1680s.33 Although it deals primarily only with the issue of finding the language of an encrypted text, it betokens a familiarity with the relevant literature. And in a letter of March 1693 to Count Platen, the Hanoverian prime minister, Leibniz forwarded to him a book entitled Steganographie by J. S. Haes, the librarian at the court of Hesse-Cassel.34 He remarks that this work rightly notes the salient characteristics of a good cipher (1) that it be difficult to decipher, (2) that it be easy to write out, (3) that its use be hard to detect, with its messages easily mistaken for ordinary letters, and (4) that encipherment be simple.

His interest in cryptographical matters crops up at many places in Leibniz's correspondence. Thus in June of 1689 Leibniz reported to his great friend J. D. Crafft that he has received the "character-book" from Munich "und habe den clavem felicissime ausgefunden."35 At some point in 1690 Crafft borrowed this book from Leibniz and their subsequent correspondence referred to it as "the encrypted book" (das cifrirte Buch).36 7


In March of 1691 Crafft promised to return it soon, and he explains the key to Leibniz.37 At one stage, a postal intermediary between Leibniz and Crafft was Philip Wilhelm von Hoernigk (d. 1714), who became Wirklicher Geheimer Rat and archivist in Passau. In his correspondence with Leibniz during the 1680's von Hoernigk at first sometimes included a few encrypted passages.38 In January of 1691 Leibniz sent him a letter in which he enclosed another to Crafft which reminded him to return the encrypted book. In his reply von Hoernigk remarked that the book was doubtless still in Crafft's possession.39 The book dealt with alchemical matters, but its contents disappointed Crafft through their insufficiency of detail: "Es sind keine chymische process drinn, sodern alles auf Ertze gerichtet."40 As early as Leibniz's service in Mainz, he and Crafft agreed to use a cipher based on the key word LABYRINTHUS for confidentiality in their communication. 41

Leibniz's correspondence of the late 1690s indicates that the Bernoullis too had some interest in the ars deciphrandi.42 Leibniz saw it as only natural that mathematicians should be interested in cryptography; he viewed cryptography as analogous to algebra, and finding the key to a cipher an analogous to finding the solution of a set of equations.43 Moreover, Leibniz's interest extended from cryptography to cryptographers. For example, in response to a question, one of Leibniz's Parisian correspondents explained to him in a letter of March 1695, that Antoine Rossignol (1600-1682), Seigneur de Juvisy, conseiller du roi, and celebre par les dechifrements was Maitre des Comptes at the French court.44 His interest in Viete and--as we shall see--above all Wallis further attests to this.

One of the few of Leibniz's discussions of cryptography that is more than perfunctory is a short paper of the mid-1680s labeled Praecepta artis deciphratoriae,45 whose deliberations relate principally to determining the language of the text being deciphered. It does, however, indicate familiarity with the then-current publications the field. And one principle of which Leibniz was acutely aware and which he repeatedly stressed is that the smaller volume of encrypted material that is available, the more difficult the code is to break. Indeed, with a simple nonalphabetic (Caesarian) transposition cipher it is no more than an exercise in combinatorics to determine the amount of text required for a good chance of decipherment. (And here it also it becomes possible to graph the length of text against the probability of successful decryption.)8


1.     Leibniz and Secret Communication


Already from the outset of Leibniz's correspondence with Baron Boineburg they used a (simple monoalphabetic) cypher to conceal names and salient expressions.46 And even Leibniz's very first letter to duke John Frederick of Hanover in March of 1673 bore witness to his awareness of the utility of encypherment in official correspondence.47 Moreover, when Leibniz corresponded with the Hanoverian chancery secretary (Kanzleisekretaer) Friedrich Wilhelm Leidenfrost, they regularly enciphered various names.48 And he also sometimes employed a nomenclator code in sensitive scripts--especially in dealing with commercial and diplomatic matters--as well as issues relating to reunion of the churches.49

The Leibniz House in Hanover, Germany

Courtesy of the Library of Congress 2002713727 / ca. 18909


In Leibniz's plans for a comprehensive library, books on cryptology and related issues (steganography, codes, cyphers, etc.) always find a place.50 And Leibniz appears to have shared this literature. One of his few explicit discussions of rules for cryptography, the "Praecepta artis deciphiratoriae" of ca. 168551 is substantially an extract from the Mysterium artis sleganographiae of L. H. Hiller (Ulm, 1682), where only monoalphabetic cyphers were considered.

Two episodes show clearly that Leibniz had little interest in secret communication as such. The one relates to steganography, the other to anagrams.

Steganography is the procedure of hiding secret messages in open texts by such devices, say, as lettering only every fourth word of the text count as part of the concealed message or using punctuation to signal the words that count (e.g. second after a period, third after a comma). With Leibniz this topic is inseparably connected with Johann Sebastian Haes (also Haas), librarian at the ducal library in Hesse-Kassel, a versatile scholar and an assiduous Leibniz correspondent during the 1690s. Haes wrote a book on steganography52 a notice of which he sent to Leibniz in January 1692 in the hope that he would pass it on to Duke Ernest August.53

In a rather perfunctory manner, Leibniz conceded that steganography may indeed have some use.54 But Haes does not let the matter rest. He exalts the merits of stenography,55 and pleads with Leibniz to recommend his book to Count Platen,56 the Hanoverian prime minister, as providing for more efficient cryptography than the established procedures. (Throughout early 1693 Haes became almost frantic about this issue.57) Leibniz clearly takes little interest in the matter, although he describes Haes to Platen as "son intention est belle et utile, sur tout aux grands seigneurs."58 Haes ultimately became rather distraught about there being no reaction from Platen.59

As regards anagrams, Newton had famously projected one to stake his claim to his discovery of fluxions in the face of keeping its processes secret. And others too resorted to this practice.60 Leibniz's correspondent, the eminent Dutch mathematician Christian Huygens (1629-95) publicized his solution to Bernoulli's suspended-chain (catena) problem by an anagram, exactly in the manner of Newton in relation to fluxions--and the fashion of the day. Huygens described this in detail to Leibniz who had also solved this problem,61 but Leibniz disapproved of this secretive 10


proceeding.62 But Huygens reiterated his view, insisting "je vous remontray la necessite du Chifre pour pouvoir connoitre ce qu'un chacun auroit trouve au sujet du Probleme de Mr. Bernoulli,"63 and subsequently adding that Leibniz ought to give "vos inventions sous la couverture du chifre, comme je vous l'avois conseille plus d'un fois."64 But Leibniz marginally asks himself "pourqoi prendre cette peine inutilement" when publication is the natural pathway to priority.65 Leibniz was no friend of mystery-mongering. As he saw it, the fruits of research should be available to the universal benefit of the republic of learning.

Leibniz's reaction to the issue of stenography and anagrams indicate that secret communication as such really had little interest for him. Cryptography, on the other hand, because of its clearly mathematical involvements, is something else again. Its theoretical interests, its relations to algebra, and its involvements in the ars combinatoria gave this topic an entirely different standing in the mind of Leibniz. And on occasion he put it to practical use as well.66

1.     Leibniz's Wallis Project: 1697-1701


In the era of the War of the Spanish Succession all major European capitals had their Black Chambers where the needs of decipherment were amply provided for. All of these involved people of extraordinary talent. In England there was John Wallis (after Newton England's ablest and most creative mathematician), in Vienna there was Giuseppe Spedazzi67 (who was also an able composer), and in Paris there was the great cryptographer Antoine Rossignol and his disciples.

As early as 1673 Leibniz had remarked that the "de doctrina divinandi seu de hypothesibus . . . pars est doctrina de chiffris construendis solvendisque, quam vellem a Wallisio accurate tradi."68 However, the latter 1680s witnessed renewed stimulus to Leibniz's interest in cryptography. In his (anonymous) review of Wallis's Treatise of Algebra (1685) in the June 1686 issue of the Acta Eruditorum of Leipzig, Leibniz noted the analogy between solving equations and deciphering cryptograms, and expresses a wish that Wallis should provide some example of his work in this area.69 After Leibniz started corresponding with the man himself in early 169770, he reiterated this wish to Wallis71 who responded that he has already sent come samples of his work to the Acta Eruditorum,72 and 11


went on to provide Leibniz with a copy of this material. When he saw Wallis' decipherment Leibniz was truly astounded, and in his subsequent correspondence with Wallis, Leibniz persisted with this quest for further details about this summum specimen humanae penetrabilitatis.73

Wallis' communication presented the decipherment of two encrypted French diplomatic communications. The ciphers were different but functioned similarly, the symbols in each being either single objects or groups of two or three, with some standing for letters of the alphabet and others encoding syllables or words. The encypherment was accordingly fairly complex through combining several distinct elements.

John Wallis (1616-1703)--"the father of British cryptography"74--had since 1649 been Savilian Professor of Geometry at Oxford where he con

John Wallis (1616-1703)

National Portrait Gallery, London / after Sir Godfrey Kneller, Bt, oil on canvas12


tinued until 1703. He was a scholar-mathematician of almost Leibnizian versatility and Leibniz's editor C. I. Gerhardt aptly termed him "the Nestor of English mathematicians."75 He was an immensely gifted cryptographer whose services were deemed invaluable by every British administration from Oliver Cromwell to Queen Anne. He provided invaluable service to the crown (i.e., William III) in deciphering communications captured from French and Jacobiate forces in Ireland.76 His splendid portrait by Sir Godfrey Kneller commissioned by Samuel Pepys now in the Examination Schools in Oxford speaks volumes. In the background here lies volume three of his Opera mathematica which contained the decipherment of those two 1689 diplomatic dispatches. The material deciphered by Wallis revealed the hostile intentions of "a treaty (or intreaty rather) of the French King [Louis XIV] with the King of Poland presently to make war on Prussia."77 This achievement was rewarded by the elector (later king) Friederick III of Prussia by a handsome sum as well as the gold medal and chain which rests on that book in the Oxford portrait. A knowledge of the arcana of the cryptographic art clearly brought significant rewards. (And where remuneration was concerned Wallis was no less eagerly importunate than Leibniz.)

Wallis had been in Leibniz's thoughts for a long time. Already in his Mainz period, Leibniz had heard of Wallis and his cryptographic achievements,78 and in his 1673 De methodi quadraturarum usu in seriebus Leibniz drew the analogy between the search for a rule in a series or tabulation with the search for the key to a cipher.

Perusal of Wallis' 1996 communication had a powerful impact upon Leibniz. He was impressed, indeed virtually awed--presque etonne--by Wallis' cryptographic achievements, deeming them amazing (merveilleuse)79 and vaunting his skill as "virtually unequalled."80 An extensive correspondence soon unfolded between them.81 Leibniz had not just respect but admiration for Wallis's work in code-breaking, and valued it as rivaling and indeed exceeding the best that that cryptographic adepts of contemporary France were able to produce.82

What intrigued Leibniz especially was that while Wallis published some of his decipherments, he never disclosed his method for obtaining them.83 And Leibniz was convinced that Wallis was only revealing the top of the iceberg in his published accounts, and that a great deal of additional information would actually be required as basis for decipherment.8413


This reaction engendered what one might call "Leibniz's Wallis Project."85 For throughout the years from 1697 to 1701 Leibniz again and again told his correspondents--above all those who might themselves have contact with this genius--that Wallis must be persuaded to ensure the perpetuation of his cryptographic knowledge. So in October 1690 Leibniz urged Henri Justel in London that Wallis should be persuaded to publish something on the art de dechifrer.86 And in a letter to Halley in June 1692 Leibniz urges that Wallis should not allow his cryptographic insights to die with him.87 Similar plaints went to various of Leibniz's English contacts.88 And in a long letter to Thomas Burnett of February 1697, Leibniz urges that Wallis should be induced to write about the codebreaker's art de dechiffrer "in which he achieved amazing success already in his youth."89 Moreover, he sent the same message to any Englishman in touch with Wallis, telling Alexander Cunningham "Je souhaiterois que M. Wallis nous voulut donner les lumieres qu'il a sur l'art de dechifrer,"90 also telling Thomas Smith that "Vellem vir egregius aliquid nobis daret de Arte solvendi aenigmata cryptographica, in qua vix quenquam parem sese habere ostendit."91

In a letter of 11 January 1697 Wallis sent to Leibniz his deciperment of an encoded French diplomatic dispatch together with his key.92 But Leibniz was disappointed. For as he wrote to Otto Menke, the editor of the Leipzig Acta:

Es waere zu wuendschen dass H. Wallasius nicht nur solutionem Epistolae cryptographicae, sondern auch modus solviendi geben haette. Ich glaube aber dass er aus diesen einizigen brief clavem also wie er sie hier gegeben nicht finden koennen.93

And Leibniz reiterated this wish to Wallis himself, flatteringly describing his cryptographic work as fastigium quoddam subtilitatis simul industriaeque humanae.94

In the period between early 1697 and early 1701 there were ten exchanges of letters between Wallis and Leibniz. From the very start of this correspondence and recurring in all but two of Leibniz's contributions to the prolonged exchange (namely his letters of 4 August 1699, [No. XIV in Gerhardt's numbering] and of Spring 1700 [No. XVIII]), there is a stubbornly repeated request to the effect: "Seeing that you are now past 80 years old, do please take on an apprentice in cryptography so that your 14


methods will not be lost to posterity"95 Moreover, Leibniz explicitly tells Wallis of his eager curiosity about his amazing (mirifica) skill.96

Immediately after the correspondence had been begun by Wallis in late 1696, Leibniz in his very first letter opens this campaign for a clever young man to become Wallis's97 cryptographic apprentice. Commenting on Wallis' 1686 paper he continues: His ego nunc meas preces adderem, nisi gravis aetas tua obstaret . . . Si qui tamen adessent Tibi juvenes ingeniosi et discendi cupidi, possent coram paucis verbis a Te multa discere, quae interesset non perire.

Contact with Wallis and his work profoundly changed Leibniz's views of cryptography. Initially Leibniz was hopeful that rules of practice (regulae) could take one far in developing the ars deciphratoria.98 Initially--at least up to 1674--Leibniz had hopes that decipherment could be widely achieved by methodical rules of procedure.99 For Wallis carefully explained that cryptography is not subject to definite rules (certis regulis) but is a matter of ad hoc contrivances whose complexity is ever in the increase.100 Codebreaking, so Wallis insists, is rather a rambling hunt (vaga venatio) than a method.101 As Wallis saw is, there can be no general rules in cryptography because "every new Cypher almost being contrived in a new way, which doth not admit it any constant Method for the finding out of it."102

Reluctantly, Leibniz conceded that cryptanalysis cannot be practiced by following rules specific instructions (praeceptis),103 an acknowledgement which evokes from Wallis a stress on the very special dispositions and skills that the craft requires.104 In the end, Leibniz could not but admit Wallis' protestations that the cryptographic art consists in special devices that admit no general rules.105 While it was a fundamental conviction with Leibniz that ars had to be founded in scientia and praxis based on the teachings of theoria, Wallis led him to the reluctant realization that cryptography might be an exception to the rule.

Initially in his 1686 review of Wallis' Algebra Leibniz spoke of cryptography itself as a scientia rather than as an ars. But corresponding with Wallis seems to have made him increasingly unsure of this. And he eventually conceded that Cryptographematum solutionem certa methodo absolvi non posse.106

Leibniz was, however, rightly convinced that the cryptographic art could certainly be taught by example.107 (It was, in fact, though just such apprenticeship that the craft was actually transmitted by its mas15


ter-practitioners to their own sons or relations at the courts of Europe throughout the 17th century.)

As Leibniz saw it, a decipherer must be (1) clever (ingenious and equipped with natural sagacity--especially in mathematics) and (2) patiently hardworking (sedentarius and porte a l'assiduite) with patientia laboris.108 But in due course he also added (3) being generally knowledgeable and erudite.109 For example, in seeking the key to a cryptogram that is based on a key word substantive information regarding the context may well prove useful.110 For in decipherment as in hermeneutics, knowledge of contextual information may prove critical as a guide to probability.111

To keep Leibniz at bay Wallis sent him a copy of his Acta Eruditorum paper.112 But in the face of Leibniz's dogged persistence, he ultimately yielded some ground to Leibniz's insistence that it might be a good idea for him to take on an apprentice. But he stressed that--given that encryption is usually only used "in matters of great moment"--he could not proceed without royal approval (inconsultis nostro principe) seeing that "it could much inaccomodate our friends no less than our enemies if the art of revealing secret writing were widely known."113 At last Leibniz became satisfied that he has at last made real progress.114 After all, how could an intelligent monarch fail to foster so important an instrument of human knowledge? Leibniz's last surviving letter to Wallis closes with the plea that he should tanto ingenii humani specimine ars inveniendi provehetur.115 But how was this venture to be funded?

Leibniz's hope of funding an apprentice cryptographer for Wallis found extensive and persistent expression in his correspondence with Ferdinand the hereditary prince of Tuscany.

During his Italian sojourn in 1698-90 Leibniz was put in touch with Prince Ferdinand de Medici of Tuscany (who later came to the throne as Ferdinand III), and impressed with the solution of a mathematical problem-challenge.116 Leibniz deemed this mathematically interested prince as the ideal sponsor for a Wallis disciple. In a letter of November 1698 Leibniz mentioned a certain prodigy in mnemonics and then continued: "I know of someone--[viz. Wallis]--with amazing skill at deciphering, so skilled that I myself am awed at what I have seen him do."117 He urged the prince to fund a young apprentice for each of these prodigies and offers to supply candidates, urging haste on account of Wallis' great age impressing upon Ferdinand the importance of cryptography.16


In his response, Ferdinand suggests that he himself knows a young man capable of developing both skills--mnemonics and cryptology.118 Replying in January of 1699 Leibniz urged the claims of his own candidate for the mnemonics post and indicates that the cost would come to at least 400 Roman scudi per year with even a single year able to produce good results.119 The prince responded in February of 1699 by wishing Leibniz good luck with the project.120 Leibniz still did not let the matter rest but returned to it again in relation to his demarche on Wallis who, regrettably, n'a pas encore pu se resoudre a ce qui est desire.121 Finally in his response of June 1700 the prince dryly encouraged Leibniz to pursue his own effort in this direction.122

Despairing of further progress in this Italian direction Leibniz turned elsewhere, suggesting in February 1699 to Paul von Fuchs, the versatile minister of state in Brandenburg that the elector there should fund a disciple for Wallis,123 and observing that France had been well served by employing cet admirable dechifrateur, the notable algebraist Francois Viete.

To identify a suitable candidate for his projected Wallis apprentice, Leibniz wrote in March of 1699 to the polydidact Johann Andreas Schmidt (ca. 1660-1726)--whom he had supported for appointment as professor of theology at the University of Helmstedt--asking him to recommend a suitable young scholar and describing the needed qualification of combining nature sagacity with practice.124 In his reply Schmidt suggested an otherwise unidentified young man in Jena.125 And in December 1698 Leibniz pressed M. G. Block for particulars regarding a young Swedish calculating prodigy, unquestionably with a view to his Wallis project. 126

Nor did Leibniz neglect possibilities closer to home. In March of 1699 Leibniz prepared a memorandum for the periodic joint-session of the privy councilors of Hanover and Celle127 in which he urged his ongoing plan for finding a young apprentice for Wallis. He wrote:

The most celebrated decipherer now living in Europe, is found in England. He is a superb mathematician and stands in correspondence with me. Since he is now a man of eighty years it is of concern that the great things he had achieved in this art will be lost with him. I have often remonstrated with him for the public good that he finally be prepared to instruct in some 17


suitable young man who is gifted with a similar inclination to calculation and effort.128

In his proposal Leibniz claimed (with questionable accuracy) that in the end Wallis agreed with his suggestion.

Finally, Leibniz seems to have found his man. In April of 1699 he wrote a long and elaborately detailed letter to the celebrated philologist Johann Gabreil von Sparwenfeld (d. 1627), Master of Ceremonies at the court of Charles XI of Sweden, detailing at considerable length his project of a Wallis-disciple.129 He raises the problem of funding the project and mentions a Mons. Block for the job, describing him as "un honneste homme, et qui merite d'estre favorise." But just to play safe, he continued: "Je vous supplie au reste du vous souvenir du garcon Finnois parent de M. Brenner et de ce garcon qui peut faire des grands chifres dans sa teste." Leibniz evidently nursed hopes that the court of Sweden might take up the good cause with financial support. He characterized the art de dechifrer as "un des plus grands echantillons de l'esprit humain," and he describes his friend Wallis as "asseurement des premiers en Europe pour cela" whose achievements "m'ont cause de l'etonnement." Having repeatedly asked him to publish his methods only to have him counter that "il n'y a point de regles generales dans cet art," has urged that he should take an apprentice to learn by example what cannot be transmitted by discourse. Leibniz then elaborated his plea that some great prince should fund such an apprentice in the interests of "le bien public, et particulierement sur l'avancement des sciences." In his reply Sparwenfeld informed Leibniz that he is unable to suggest someone suitable for apprenticeship in cryptography, a subject "dont vous parlez si juste et si bien."130

In fact Leibniz had already had substantial epistolary dealings with M. G. Block, some even touching on cryptology. In July of 1698 Block had written a long letter to Leibniz with much autobiographical detail in which he states that the late Baron R. C. von Bodenhausen has entrusted to his executors some papers with "observations, proces et curiosites de le nature, de la Medicine, de la chymie, etc." of which "la plus grande partie dont il estoit jaloux est ecrite avec un chifre d'une telle facon, qu'il semble presqu'impossible de la dechifrer."131 Bodenhausen had entrusted the cipher to Block whose own opinion of this material was low. However, Leibniz made efforts to get hold of it as well as further Bodenhausen 18


papers.132 Earlier on, in his own correspondence with Bodenhausen, Leibniz had repeatedly recommended using his favorite LABYRINTHUS cipher.133

But as regards Block also Leibniz did not put all his eggs in one basket. In July 1700 Alphonse des Vignoles (1649-1744), destined to be Leibniz's successor as director of the mathematical section of the Berlin Academy, wrote to him in response to a query about potential cryptologists that he has met "un Avocat de Berlin nomme M. Bauermeister qui est fils d'un Conseiller de Bernbourg" who possesses some knowledge of deciphering.134 Moreover, he also knows of another promising young man called Cibrovius who is reported as having une disposition admirable pour dechifrer.135 Gradually Leibniz accumulated some possibilities.

It appears from this proliferation of contacts that Leibniz simply did not care who--be it Hanover-Celle, Tuscany, Brandenburg, Sweden-- should supply or fund the Wallis apprentice as long as this was done before Wallis' remarkable skills became lost upon his death. Only when it was clear that this unhappy event was imminent did Leibniz give up on his project. (See Display 1.) Seemingly, the secret cipher that Leibniz wanted most urgently to decrypt was that of Wallis's cryptological modus operandi. Wallis himself, however, was not receptive, insisting that the diffusion of cryptographic knowledge would do more harm than good: "Nostris utique Amicis non minus quam Inimicis magno fore posset incommodo, si Ars, occulte scripta recludendi, passim innosceret."136

1.     The Aftermath


Leibniz's Wallis project was not entirely in vain. David Kahn summarized the situation as follows:

[Worried] that Wallis and the art might die together, [Leibniz] pressed his request that he instruct some younger people in it. Wallis finally had to say bluntly that he would be glad to serve the elector [of Hanover] in this way if need be but he could not share his skill abroad without the king's leave. The shrewd old cryptanalyst, who was frequently asking for more money for his solutions, then used Leibniz's arguments to his own advantage in successfully urging the secretaries of state to pay for his tutoring19


Display 1

The Chronology of Leibniz's Wallis Project

1.                   1690-1696. Leibniz tells various correspondents that Wallis should be urged to write more about the art de dechiffrer.

2.                   March 1697. Leibniz first recommends Wallis himself to take on an apprentice in the ars decriphrendi.

3.                   November 1698. Leibniz begins urging Ferdinand of Tuscany to fund a Wallis apprentice. (A I 16, p. 576.)

4.                   December 1698. Leibniz asks M. G. Block for details regarding a young Swedish calculating prodigy. (A III 7, p. 969.)

5.                   February 1699. Leibniz urges von Fuchs, Privy Councillor in Berlin, to secure Brandenburg funding for a Wallis apprentice. (A I 16, pp. 577-78.)

6.                   March 1699. Leibniz urges the Celle-Hanover Hauskonferenz to fund a Wallis apprentice. (A I/6, p. 121) He later reiterates this plan to Count Platen, the prime minister.

7.                   March 1699August 1700. Leibniz asks J. A. Schmidt of Helmstedt/Marienthal to recommend a suitable prospect as cryptographic apprentice, and elicits the nephew of former Professor Hoffmann of Jena. (A I 16, pp. 639, 656, 662.)

8.                   April 1699. Leibniz explores funding for Wallis's disciple with von Sparvenfeld in Stockholm, and suggests to him that he has a promising prospect in view, viz. M. G. Block. (A I 6, p. 727.)

9.                   March 1700. Wallis seemingly yields to Leibniz's repeated urgings to take on an apprentice, provided that William III is agreeable. (GMath. IV, p. 76.)

10.                July 1700. Alphonse des Vignoles writes from Berlin that he can suggest two plausible candidates for the Wallis apprenticeship. (G. C. Bauermeister and C. L. Cibrovius).