LEIBNIZ AND CRYPTOGRAPHY

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 *

LEIBNIZ AND CRYPTOGRAPHY

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

by

NICHOLAS
RESCHER

UNIVERSITY LIBRARY SYSTEM, UNIVERSITY OF
PITTSBURGH

PITTSBURGH, PA

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.]

Contents

*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 *

*References...............................................................................................................91
*

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

**xi**** **

Introduction

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 *Sämtliche**
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

**xii**** **

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*

LEIBNIZ
AND CRYPTOGRAPHY

An Account on the Occasion of the
Initial Exhibition of the Reconstruction of Leibniz’s Cipher Machine**3 **

I

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

**4 **

**LEIBNIZ’S FORAYS IN CRYPTOGRAPHY **

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.*

**LEIBNIZ AND CRYPTOGRAPHY **

**5 **

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 *Traicté** des Chiffres
*of the French algebraist and diplomatist Blaise
de Vignière15—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
dechifré 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

**LEIBNIZ’S FORAYS IN CRYPTOGRAPHY **

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 **

**LEIBNIZ AND CRYPTOGRAPHY **

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 **

**LEIBNIZ’S FORAYS IN CRYPTOGRAPHY **

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 Hörnigk (d. 1714), who became Wirklicher
Geheimer Rat and archivist in Passau. In his
correspondence with Leibniz during the 1680’s von Hörnigk
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 Hörnigk
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 *célébre**
par les dechifrements *was Maitre
des Comptes at the French court.44 His
interest in Viète 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 **

**LEIBNIZ AND CRYPTOGRAPHY **

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 (*Kanzleisekretär*) 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. 1890***9 **

**LEIBNIZ’S FORAYS IN CRYPTOGRAPHY **

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 **

**LEIBNIZ AND CRYPTOGRAPHY **

proceeding.62 But Huygens reiterated his view,
insisting “je vous remontray
la necessité du Chifre pour
pouvoir connoitre ce qu’un chacun
auroit trouvé au sujet du Problème de Mr.
Bernoulli,”63 and
subsequently adding that Leibniz ought to give “vos
inventions sous la couverture du chifre,
comme je vous l’avois conseillé 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 **

**LEIBNIZ’S FORAYS IN CRYPTOGRAPHY **

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 canvas***12 **

**LEIBNIZ AND CRYPTOGRAPHY **

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** étonné*—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.84**13 **

**LEIBNIZ’S FORAYS IN CRYPTOGRAPHY **

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 wäre zu wündschen dass H. Wallasius nicht nur solutionem Epistolae cryptographicae, sondern auch modus solviendi geben hätte. Ich glaube aber dass
er aus diesen
einizigen brief clavem also
wie er sie
hier gegeben nicht finden können.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 **

**LEIBNIZ AND CRYPTOGRAPHY **

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 mas**15 **

**LEIBNIZ’S FORAYS IN CRYPTOGRAPHY **

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 porté à
l’assiduité*) 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 **

**LEIBNIZ AND CRYPTOGRAPHY **

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 à
ce qui est desiré*.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 François Viète.

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 **

**LEIBNIZ’S FORAYS IN CRYPTOGRAPHY **

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 favorisé.” But just to play safe, he continued: “Je vous supplie au reste du vous souvenir du garçon Finnois parent de M.
Brenner et de ce garçon 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 causé de l’étonnement.” Having repeatedly asked him to publish his
methods only to have him counter that “il n’y a point de règles 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 curiosités 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
façon, 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 **

**LEIBNIZ AND CRYPTOGRAPHY **

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 nommé 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 tutoring**19 **

**LEIBNIZ’S FORAYS IN CRYPTOGRAPHY **

�.
1690-1696.
Leibniz tells various correspondents that Wallis should be urged to write
more about the �.
March 1697.
Leibniz first recommends Wallis himself to take on an apprentice in the ars decriphrendi. �.
November 1698.
Leibniz begins urging Ferdinand of Tuscany to fund a Wallis apprentice. (A I
16, p. 576.) �.
December 1698.
Leibniz asks M. G. Block for details regarding a young Swedish calculating
prodigy. (A III 7, p. 969.) �.
February 1699.
Leibniz urges von Fuchs, Privy Councillor in Berlin,
to secure Brandenburg funding for a Wallis apprentice. (A I 16, pp. 577-78.) �.
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. �.
March
1699–August 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.) �.
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.) �.
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.) �.
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). |