H 0H ISTORIA MATHEMATICA - Elseviermedia.journals.elsevier.com/content/files/hm1-23043917.pdf ·...

FOUR DECADESOF EXCELLENCEIN THE HISTORYOF MATHEMATICS

HISTORIAMATHEMATICA

www.elsevier.com/locate/hm

Editors in Chief thomas ArchibaldSimon Fraser University, British Columbia, CanadaNiccolò GuicciardiniUniversità degli Studi diBergamo, Bargamo, Italy

40tH

Historia Mathematica publishes historical scholarship on mathematicsand its development in all cultures and time periods. In particular, thejournal encourages informed studies on mathematicians and their workin historical context, on the histories of institutions and organizationssupportive of the mathematical endeavor, on historiographical topics inthe history of mathematics, and on the interrelations betweenmathematical ideas, science, and the broader culture. This volumecelebrates the 40th anniversary of Historia by providing an anthology of some of the seminal papers published since 1973.

CONTENTS:IntroductionArchibald, T., Guicciardini, N.

the mathematical studies of G.W. Leibniz. on combinatorics.Knobloch, E. (1974)

Gauss as a geometer.Coxeter, H. S. M. (1977)

Investigations of an early Sumerian division problem, c. 2500 B.C. Høyrup, J. (1982)

On the transmission of geometry from Greek into Arabic.Knorr, W. (1983)

Changing canons of mathematical and physicalintelligibility in the later 17th century. Mahoney, M. S. (1984)

the calculus of the trigonometric functions.Katz, V. J. (1987)

Sharaf al-Dīn al-tūsī on the number of positive roots of cubic equations.Hogendijk, J. P. (1989)

Hilda Geiringer-von Mises, Charlier series, ideology, and the human side of the emancipation of appliedmathematics at the University of Berlin during the 1920s. Siegmund-Schultze, R. (1993)

How probabilities came to be objective and subjective.Daston, L. (1994)

Doubling the cube: a new interpretation of its significance for early Greek geometry.Saito, K. (1995)

Branch points of algebraic functions and the beginnings of modern knot theory. Epple, M. (1995)

L'intégration graphique des equations différentielles ordinaires. Tournès, D. (2003)

the first Chinese translation of the last nine books of Euclid's elements and its source.Xu, Y. (2005)

the Suàn shù shu , “Writings on reckoning": rewriting the history of early Chinese mathematics in the light of an excavated manuscript. Cullen, C. (2007)

On mathematical problems as historically determined artifacts: reflections inspired by sources from ancient China. Chemla, K. (2009)

the correspondence between Moritz Pasch and Felix Klein Schlimm, Dirk (2013)

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Introduction

Historia Mathematica was established by Kenneth O. May 40 years ago, succeeding an earlier newsletter sponsored by the International Commission on the History of Mathematics. May's initial vision of an international journal reflected his ideological views, and remains encoded on the cover. The ideal that the language of production of scholarship would not be an obstacle to its publication has in the meantime given way, for better or worse, to a world in which English is the new Latin. However, May’s ideal that this journal remain open to contributions of scholars from all over the world is intact thanks to its connection with the ICHM, the broad geographical distribution of the members of the Editorial Board, and the vision of our predecessors as editors in chief.1 We feel that we have the responsibility to continue this tradition by ensuring that history of mathematics is not reduced to the chronicle of past events and celebration of the great men of European mathematics that May so powerfully stigmatized. Historia is, and very much seeks to remain, the venue where scholars from all continents and scholarly backgrounds can meet to discuss and exchange their ideas, recent results, and views on the aims of our discipline. One of the most rewarding aspects of our work as editors is the broad correspondence we entertain with colleagues active on all continents. May, we hope, would be happy to see that Historia has met his intentions to promote a journal with an international public.

At the time of Historia’s founding, the field of journals accepting papers in history of mathematics included many journals that either are no longer with us or have modified their remit significantly. The Quellen und Studien of Neugebauer, Toeplitz and Stenzel was gone, as was Boncompagni's Bullettino. A fortieth birthday thus remains rare; Archive for History of the Exact Sciences can boast over 50 years, and continues strong. In the time interval between 1973 and today the practices of academic historiography have changed considerably. It's interesting to note that May himself was influenced by one of the most significant trends, quantitative history. Indeed, May showed from the very beginning a strong interest in defining Historia as a venue where historiographical issues were discussed. We believe that this is still a very important feature that is promoted in Historia in a number of ways: by including a section of Abstracts, a book review department that is authoritative in our field and that often includes thoughtful essay reviews, and rather long obituaries of past colleagues.

It occurred to us, and our project was enthusiastically backed by Valerie Teng-Broug and Evalyne Salome Wanjiru of Elsevier, that it would be proper to celebrate this anniversary with the present volume, an anthology of papers published in Historia Mathematica during these last 40 years. We had to choose a rather small selection to make a volume of 300 pages out of a list of nearly a thousand papers! Much good material could not be included within the covers of the present volume, but is of course accessible online at Historia’s webpage. The choice has not been easy, yet browsing the many papers that could have been included in this celebration anthology made us justly proud of the journal we are serving as editors. With no length constraint, a “greatest hits” volume would have been three times as long as this.

Looking back at the work of our colleagues, from the contributions of masters active in the 1970s such as Joe Dauben, Eberhard Knobloch, Ivor Grattan-Guinness, and Henk Bos, to

1 It is worth naming, with gratitude, all the past editors: Kenneth O. Ma , Joseph W. Dauben,

Eberhard Knobloch, David E. Rowe, Karen Hunger Parshall, Jan Hogendijk, Umberto Bottazzini, Craig Fraser, Benno van Dalen, June Barrow-Green.

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doctoral students who published their first results in our journal, provides both strong impressions of a heritage issuing from the recent past and intimations of developments that are near at hand. Yet, this feeling of continuity notwithstanding, a great deal has changed in our discipline. The “old days” of Kenneth May were characterized by the attempt to rejuvenate and refound a discipline which had come to a kind of maturity at the end of the nineteenth century, one in which activity had somewhat subsided during the two wars (with notable exceptions, of course). In reading May’s editorials one realizes that, in the 1970s, historians of mathematics were trying to define an institutional and cultural space where they could develop their research: the difficulty was to relate this research to that of mathematicians and mathematics teachers; namely, to claim for historical research a status and role within the mathematical community. In a sense this is no longer our main problem: mathematicians constitute our main readership and most of our authors have received their education in mathematics departments. Doctoral students, at least in Europe and North America, have now many possibilities of acquiring a professional training as historians of mathematics, though recent retirements have meant a certain reduction from the peak of such opportunities and on-going employment for even strong recent doctorates can be difficult to obtain. There is no doubt that a dialogue with the mathematical community is vital for our discipline: the connoisseurship of the practicing mathematician helps us greatly, often decisively, in a competent reading of past texts.2

The new frontier nowadays seems to be that of strengthening ties with historians of science, general historians, and philosophers of mathematics as well, scholars who are increasingly associated with and formed in history and philosophy departments. There is a pronounced interest and respect in those circles for the history of mathematics, which increasingly informs research. This is probably the main opportunity for growth for the history of mathematics in the immediate future.

One can mention here some recent facts: the success of the history of mathematics sessions organized by Karen Parshall at HSS meetings, the publication in Isis of two Focus sections related to the history of mathematics, the orientation towards a sophisticated methodological approach to history in works by colleagues such as Karine Chemla, Moritz Epple, Catherine Goldstine, Jeremy Gray, David Rowe, and Norbert Schappacher, to give only a few amongst many possible examples.3 Research in the history of mathematics nowadays, as some of the most recent papers published in Historia prove, is not declined along the divide between internalist vs externalist history: such a cleavage belongs to the past. Historians of mathematics are now happily at home with various methodological approaches to history and their tools, coming from fields as diverse as micro-history, intellectual history, the study of historical networks, anthropology and ethnography. The discipline is now sufficiently mature to establish a fruitful cooperation with colleagues working in history departments: while the technical difficulties of the subject remain a challenge for researchers in other fields, it is part of our task to provide the means for a cooperation that can only be mutually enriching.

The philosophy of science is no longer regarded as a normative theoretical enterprise detached from scientific practice. It is, instead, pluralistic: we have philosophers of biology, philosophers of physics, philosophers of medicine or engineering, and so on, in addition to the

2 See, Chang, Hasok “The Historian of Science: Painter, Guide, or Connoisseur.” Centaurus 50 (2008), pp. 37-42. 3 Amir R. Alexander (ed.) “Focus: Mathematical Stories.” Isis 97 (4) (2006), pp. 678-726, and “Focus: The History of Science and the History of Mathematics.” Isis 102 (3) (2011), pp. 475526.

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more traditional specialties, and many of these fields appear in combination as well. Further, philosophers are now seeking cooperation from fellow scientists: the problems they broach are those high on the agenda of practicing scientists. This bottom-up approach to the philosophy of science might not be so new after all, since some philosophers in the past, but especially some scientists, have focused their attention on the foundational issues raised by specific disciplines. The 21st century is witnessing a notable innovation in the field of the philosophy of mathematics as well: we are referring to a new wave of research, fostered by philosophers such as Paolo Mancosu, which attributes a much more important role to the history of mathematics than was ever assigned to it in the past.4 More and more philosophers of mathematics are now aiming to relate their own research to the particularities and idiosyncrasies of mathematical practice. This approach is not meant to substitute for or minimize the importance of more traditional research on foundational issues typically studied with the tools of the practitioner of mathematical logic. Mathematical logic is a flourishing discipline, and philosophers who rely upon the standpoint provided by mathematical logic find themselves in an exciting research field: suffice it to mention here the advances made in the logic of quantum mechanics, in the form of quantum computing and quantum cryptography. Yet, the theoretical lens of philosophers of mathematics is now getting wider and themes that fall outside the province of logicians and philosophers interested in foundational issues are emerging as driving forces in philosophical research. Here we can only list a few names. Philosophers like Mancosu, Michael Detlefsen and Marcus Giaquinto are tackling non-foundational epistemic issues relevant to mathematical practice such as the role of visualization and computer simulations in mathematical proofs, the role of aesthetic criteria in adjudicating theories, the notions of purity and simplicity as values that orient mathematical research, and the presence of a tacit dimension – in Pólya’s sense – in mathematical knowledge. These new viewpoints are conducive to a fruitful interaction between history and the philosophy of mathematics. The history of mathematics can provide the philosopher interested in mathematical practice with concrete examples of "mathematics in action," so to speak.

We invite our readers to rejoice on Historia’s 40th birthday, an event which we emphasize in addressing an invitation to our colleagues: historians of mathematics proper, and mathematicians, historians and philosophers as well. Please continue to grace this journal with the results of your original research and continue to support Historia by refereeing articles and providing reviews and abstracts. After all, the work of editors consists merely in the humble task of attempting to guarantee, in a timely way, an authoritative peer-reviewed publication of cutting-edge research in a broadly conceived field of history of mathematics.

Tom Archibald

Niccolò Guicciardini

4 Paolo Mancosu (ed.), The Philosophy of Mathematical Practice (Oxford: Oxford University Press, 2008).

HISltrlRIA MATHEMATICA 1 (1974), 409-430

THE MATHEMATICAL STUDIES OF G,Ws LEIBNIZ ON COMBINATORICS

BY EBERHARD KNOBLOCH, WEST BERLIN

SUMMARIES

Leibniz considered the "ars combinatoria" as a science of fundamental significance, much more extensive than the combinatorics of today. His only publications in the field were his youthful Dissertatio de Arte Combinatoria of 1666 and a short article on probability, but he left an extensive (hitherto unpublished and unstudied) Nachlass dealing with five related topics: the basic operations of combinatorics, symmetric functions in connection with theory of equations, partitions (additive theory of numbers), determinants, and theory of probability and related fields. This paper concentrates on the first and third topics as they appear in published sources and the Nachlass. It shows that Leibniz was in possession of many results not published by other mathematicians until many decades later. These include a recursion formula for partitions of n into k parts (first published by Euler in 1751), the Stirling numbers of the second kind (first published in 1730), and several special cases of the general formula for partitions that was published only in 1840 by Stern.

Leibniz fasste die "ars combinatoria" als eine Wissen- schaft von grundlegender Bedeutung auf, die weit mehr umfasste als die heutige Kombinatorik. Seine einzigen Veraffentlichungen auf diesem Gebiet waren seine Jugend- schrift Dissertatio de Arte Combinatoria aus dem Jahre 1666 und ein kurzer Aufsatz zur Wahrscheinlichkeitsrech- nung, aber er hinterliess einen umfangreichen, bisher unveraffentlichten und unbearbeiteten Nachlass, der sich mit fiinf entsprechenden Themenkreisen befasst : die kombina- torischen Grundoperationen, die symmetrischen Funktionen im Zusammenhang mit der Gleichungslehre, Partitionen (additive Zahlentheorie), Determinanten, Wahrscheinlich- keitstheorie und verwandte Gebiete. Der vorliegende Aufsatz beschr&!nkt sich auf den ersten und dritten Themen- kreis, soweit diese in gedruckten und vor allem nachge- lassenen Studien behandelt sind. Er zeigt, dass Leibniz viele Ergebnisse kannte, die von anderen Mathematikern erst viele Jahrzehnte spEter versffentlicht wurden. Zu

410 E. Knobloch HM1

ihnen gehijren eine Rekursionsformel fir Parti tionen einer natiirlichen Zahl n in k Summanden (zum ersten Ma1 von Euler 1751 vertiffentlicht), die Stirlingschen Zahlen 2. Art (zum ersten Ma1 1730 veraffentlicht) und mehrere Spezialf2lle der allgemeinen Formel fir Partitionen, die erst 1840 von Stern gefunden wurde.

Introduction

Compared to the large amount of extant (though mostly unpublished) material, our knowledge of Leibniz’s studies on combinatorics is very meagre. Leibniz himself spoke of the “ars combinatoria”, a philosophical term to which he attached different meanings in the course of his life, but which always embraced more than what we call combinatorics today and was not even restricted to mathematical problems. It is therefore advisable to concentrate the present discussion on the mathematical aspect of his “ars combinatoria”.

But even when used in this restricted sense the Latin term is not identical with “combinatorics” in the modern meaning of the word. As Leibniz understood it, the “combinatorial science” included not only algebra and the theory of numbers, but also affected all fields of mathematics known in his time. Louis Couturat [1901, 478-5001 has given examples of this.

In Leibniz’ early work Dissertatio de arte combinatoria [Leibniz 1666, hereafter abbreviated Ars Combinatoria], written while still under the influence of Lullism, this far-reaching meaning of combinatorics plays only a very minor role in regard to mathematics, while emphasis is given to the philosophical applications. It is therefore not astonishing that the book has its established place in studies on the history of logic, but so far has not been subjected to a thorough mathematical analysis, although a few details arrived at by Leibniz are repeatedly mentioned in the literature [Todhunter 1865, 31-33; Cantor 1901, 43-45; Tropfke 1924, vol. 6, 68; Hofmann 1948, 4-5; Hofmann 1949, Z-31.

Even the most recent studies on the Ars Combinatoria by Dy. Kutlumuratov [1964, 19-331 and Michel Serres [1968a; 1968b, 409-421; 19691 hardly go beyond the earlier descriptions insofar as the mathematical aspects are concerned. Thus, although Serres [1968a, 119; 1968b, 4091 is right when he observes that “Le De Arte est le parent pauvre du commentaire Leibnizien,” one can hardly agree with his concluding statement that “L’apport du De Arte B la science mathematique se resume 3 cell” [1968a, 124; 1968b, 4151. Contrary to his plans Leibniz never published any further mathematical contributions to the Ars Combinatoria except for a short essay on the theory of probability [Leibniz 16901, but hundreds of mostly uncollated manuscripts among the more than 7300 pages of mathematical material he left behind

HMl Leibniz on combinatorics 411

bear witness of his numerous studies in this field. Disregarding those studies exclusively concerned with the

theory of numbers or with algebraic problems, the relevant notes may be roughly grouped under five headings: 1. Combinatorial theory in a narrower sense (basic combinatorial operations). 2. Symmetric functions (together with the theory of equations). 3. Partitions (a part of additive theory of numbers). 4. Deter- minants (elimination of unknowns in systems of linear equations and equations of higher degree). 5. Theory of probability and related fields (theory of games, calculation of rents and interest).

For practical reasons a selection has to be made. The last two groups are most easily treated separately. The number of extant manuscripts on them is quite different. Leibni z kept more than 900 drafts on determinants (the algebraic index notation) , of which only a very few have appeared in print. These will be discussed in a separate publication [Knobloch 19751. On the fifth topic Leibniz produced only a few sometimes quite erroneous studies. The more important ones have recently been discussed by Biermann [1965, 79-851.

Because of Leibniz’s conception of the “ars combinatoria,” studies on one of the first three topics often embrace parts of the other two, which makes it quite difficult to draw boundaries between them. For example, Leibniz tried in numerous studies to determine the partitions of integers into a given number of summands, i.e. special combinations in his terminology. He recognized the interest of the problem for probabilities in dice games and for constructing magic squares and cubes. He had the same application in mind, however, when he sought the number of certain symmetric functions of a certain degree or an algorithmic resolution of general equations of the fifth and higher degree. The symmetric functions again are the subject of numerous special studies dealing with polynomial powers, power sums, the reduction of symmetric functions, the number and variability of their terms, or calculations with them. Leibniz handles all these problems with combinatorial methods in the modern sense.

It appears from the foregoing description of the drs Combinatoria that manuscripts that at first sight seem irrelevant may nevertheless contain valuable information. In the present paper we restrict ourselves to a discussion of Leibniz’s major results on the first and third topics, except for a short intro- ductory analysis of his early work as a whole, since those from the second group are discussed elsewhere [Knobloch 19723.

1. The Dissertatio de drte Combinatoria

1.1. Systematic arrangement and introduction Leibniz prefaces his dissertation with a table of contents,

a proof of the existence of God which we shall not discuss here,

412 E. Knobloch HMl

and a compilation of mathematical definitions of which the more important ones are given here. [1]

Permutations he calls “variationes ordinis” combinations are called “complexiones”. When referring to combinations of a special class, he writes “com2natio” or “combinatio”, “con3natio” or “conternatio” etc., a way of writing which was used earlier by Marin Mersenne [1635, 135 ff.; 1636, Livre des Chants, 134; see Knobloch 19741. Leibniz soon gave up these expressions, though, and after his stay in Paris (1672-1676) frequently used “combinationes” as a general term, even when not referring to combinations of the second class, thus following the practice of many of his precursors. The class of which a combination is made is called “exponens”. When all possible combinations (without repetition) are meant, he uses “complexiones simpliciter”. He introduces no special term for variations, but circumscribes them as “variationes tam complexionis seu materiae quam situs seu formae” similarly to Kaspar Schott in his Magia Universalis [1658 vol. 3, 6901 or Sebastian Izquierdo in his Pharus Scientiarum [ 1659, 3191.

Leibniz subdivides his dissertation into 12 problems, but these by no means serve as a usable organization of its contents. This also appears from the very different lengths of individual chapters. The longest chapter (problem 2) alone takes up 28 pages of the Academy edition, while the shortest fills a mere 2 l/2 lines. On the other hand, the final paragraph following the formulation of the 12th problem refers to all of the last 6 problems, whereas the second chapter includes part of the first. Leibniz indirectly gives an explanation in his introduction for this lack of balance. He writes that he is primarily concerned with three important points, namely “problemata, theoremata, usus”. Where he thinks it worthwhile, as in the discussion of the first and fourth problems, he has added “theoremata” to the problems, but for all problems he has discussed “usus” . It is above all their selection and presentation which lead to the lengths of the chapters differing to such an extent that the second one is longer than all the other 11 chapters taken together. In that chapter Leibniz applies the “ars combinatoria” to the “Ars complicatoria Scientiarum seu Logica inventiva” [Leibniz 1666, LSB VI, 1, 2011.

He must have felt the danger of staying with this subject too long, for towards the end of the second part he interrupts a train of thought by stating that he would say more about the methods of proofs used “if we did not fear that, in our endeavour to explain everything, we were torn away from the progress of our reasoning” [Leibniz 1666, LSB VI, 1, 1991. [21 In 1690 the Frankfurt bookdealer Crbker published an unauthorized reprint of the dissertation, to which Leibniz reacted in the February issue of the Acta Eruditorum [Leibniz 1691, LSB VI, 2, 549-5501. There he was self-critical enough to point out the insufficient


structure of his work (“oeconomia operis, in qua multa possent mutari in melius” i.e. the structure of the work, much of which can be improved).

From a modern mathematical point of view the drs Combinatoria lends itself to the following subdivisions:

1. Problems 1 - 3. Combinations 2. Problems 4 - 12. Permutations

2.1. Problems 4 - 6. The three basic types 2.2. Problems 7 - 12. Permutations with restrictions

and the usefulness of permutations Leibniz says that he has given proofs for the solutions of

only a few problems, and that he owes the second part of the first, the second, and the fourth problems to other authors, whereas he himself is the author of the others. Three things he thus credits to earlier authors: the determination of the number of combinations without repetition (CwoR) of two objects, of all possible CwoR, and of permutations without repetition (PwoR) . These are indeed solved by Christoph Clavius [1585, 33- 36] in his commentary on the Sphaera by John of Sacrobosco. Leibniz further remarks that he does not know who first solved these problems, but that Clavius in his commentary explains very satisfactorily everything that has been known for a long time. To this wrong conclusion he is led by the results he has found in Clavius. That the solutions were not found by the latter Leibniz learned from Clavius and from the second major source for his drs Combinatoria, the Hours of Mathematical and Philo- sophical Refreshment by Daniel Schwenter [1636, 68-71; continued by Harsdorffer 1651 and 16531.

1.2. Problems 1 - 3 Problem 1 gives an answer to the question of how the CWOR

of any given class can be found. Leibniz is led to a wrong conclusion by Clavius. He writes that combinations of two objects are well known and given by n(n-1)/2, “generaliorem modum nos deteximus , ” i . e . “the more general method has been detected by us!” Leibniz’s method is to take the values from the arithmetic triangle, the formation of which he characterizes as C(n, k) = C(n-1, k) + C(n-1, k-l). It should be pointed out that he starts with zero the column of the number of objects chosen, although in this and in a later passage [Leibniz 1666, LSB VI, 1, 1971 he explicitly excludes combinations of no elements since he gives the number of all possible combinations as 2n-1. Even this is an independent development beyond Clavius, who had not even considered combinations of one object. Leibniz then pre- sents eight propositions, the contents and proofs of which he asks the reader to take from the arithmetic triangle. Almost all of them had been stated before Leibniz and are not repeated here.

Problem 2 determines the total number of all CwaR of n objects. Leibniz’s reason for taking 2n-1 (a result cited by Sanchez 1953, 138) is hard to understand a1.d even harder to explain once it

414 E. Knobloch HMl

has been understood. The reason has to be seen in the “discerptio” (“Zerf~llen” or partitioning) employed in the Practica Italica. [3] Here we have the first hint of Leibniz’s. studies on additive number theory, which has been known as “partitio numerorum” since Euler.

Among applications Leibniz discusses two questions: how many variations (ordered subsets or permutations) without repetition (urrol?) are there of a certain size or of all subsets? His solution, which he gives in words only, is first to form C(n, k).k! for each k and then to add all the values determined in this way. The method is correct, if not as elegant as that of his precursors. Mersenne [1635, 1331 already applied the formula n(n-1) . . . (n-k+l) and added the individual results whereas Izquierdo [1659, 327f.l and Johannes Caramuel de Lobkowitz [1670, vol. 2, 942-9431 used a recursion rule for the direct determination of the number of the Vwor of all classes. Most interesting in this context from a mathematical point of view are numbers 76 and 77 of section X [German translation in Schmidt 1960, 53-54, annotations on page SOOf.] where Leibniz gives a general solution of the following problem: how to find all possible combinations or all combinations of a given size for a given “caput”. Since he defines a “caput” as a definite subset of definitely given elements which have to be contained in the desired combinations, the problem reads in modern terms: how many combinations of a certain size or of all possible sizes contain a certain number of given elements?

At the beginning of chapter 7, where Leibniz discusses the same problem with respect to permutations, he deliberately refers to this passage [Leibniz 1666, LSB VI, 1, 2191. It took him until January 1676 to answer the question for variations with repetition (VWR). [4] Earlier Pierre HBrigone [1634, vol. 2, 1221 and following Andreas Tacquet [1665, 3791 had considered only the special case for a single fixed element.

By choosing c objects from a set of n objects including the “caput” of m elements, the solutions to the two problems, which Leibniz expressed only verbally, were Znmm-l and C(n-m, c-m) respectively. For m # 0 it should read 2n-m. His examples show that Leibniz created the terms “Ollio” (read “nullio”) and “lnio” (read “unio”) as early as his Ars Combinatoria. He even forms the word “SuperOllio” for C(i, k) = 0, where k > m.

In problem 3 Leibniz determines the number of combinations of sets (“classes”) for a given number of sets and the elements within the sets. This has to be understood as a general formulation for the number of VwR. Noteworthy is the related work on “partitio numerorum” (no. 23, 25). Leibniz speaks of “discerptiones” or “ZerfPllungen” (i.e. partitions), thereby introducing the special term which is still used today. He defines partitions as subsets of combinations. Only those “complexiones” are called “discerptiones” which, when taken together, equal the


total. This then means nothing else but combinations (“Kombina- tionen”) or, when the order of summands is considered as well, permutations with a given sum (“Variationen zu bestimmten Summen”) which have been used for a long time. Leibniz writes that it is possible to determine partitions of a certain size as well as of all sizes. In agreement with this he uses the same names as for common combinations, namely “discerptiones dato exponente” and “simpliciter”. There are n/a (n even) or (n-l)/2 (n odd) bipartitions of a number n when the order of summands is disre- garded, and n-l partitions otherwise. Later remarks indicate that Leibniz already was experimenting with partitions of more than two summands.

1.3. Problems 4 - 12 Beginning with problem 4, Leibniz studies permutations.

He determines the number of &OR by means of the recursion rule n? = n(n-l)! and illustrates it by presenting numerous so-called Protean rhymes, i.e. rhymes which permit an unusually large number of rearrangements of their words without violation of the metrical laws. Problem 5 is dedicated to the circular permutations (cyclic transpositions) the number of which he correctly determines as n!/n = (n-l)!. His solution in problem 6 for PwR is wrong, though.

Especially interesting are his calculations of the number of VwR for special types of repetitions, which I have discussed in the context of Mersenne’scombinatorial studies [Knobloch 19741.

The final six problems hardly seem to deal with mathematics. Nevertheless Leibniz here, too, solves a number of interesting combinatorial questions which have previously been overlooked. Problem 7 considers permutations that contain a “caput”, i.e. a subset that is mapped onto itself by the permutation, or, in a special case, remains invariant. He distinguishes six possible types of “caput” by its having one or several elements, by the presence or absence of homogenous elements which can be placed in a given position in the same way as those already placed), and whether or not it is monadic (possessing no homogenous elements). Thus, for example, in the most general case where the m elements of the “caput” of a total of n(U msn) can be permuted, the result is (n-m)?m!c(il+al, il). . .c(ik+ak, ik) # llm<n, given that in the jth case there are i. (j=1,2...,k) homogenous elements within the “caput”, which ase homogenous with aj outside it.

Leibniz suggests the application of these rules for the solution of the remaining problems. In problem 8 he determines the number of permutations which have several “capita” in common. Problem 9 then precisely states the conditions under which this is not possible: firstly, when several “capita” partly or completely take the same place in a permutation, and, secondly, when the same (monadic) element occurs within several capita. The final three problems deal with the cases in which permutations

416 E. Knobloch HMl

or “capita” are useful or useless, a question which still received some passing attention in Bernoulli’s Ars Conjectandi [1713, 78; Haussner 1899, vol. 1, 811. This is not a purely mathematical decision, but has to be made in cooperation with the sciences from which each special case has been taken (e.g., origin of elements calls on natural science, and transposition of rhymes requires metrics).

2. Studies in Combinatorial Theory in a Narrower Sense

It was only during his stay in Paris (1672-1676), and especially after he had made the acquaintance of Christian Huygens, that Leibniz became thoroughly involved in the details of mathematical problems. Among other things, he took up his combinatorial studies again. By the end of 1672 he had worked his way through some mathematical literature [Hofmann 1949, 6ff.; 1966, 425 ff.], but some of it, such as Pascal’s [1665] treatise on the arithmetic triangle, he had read so superficially that he made several wrong statements about it during his first visit to London in 1673. When he claimed that he himself and not Pascal had discovered the law of the additive formation of combinatorial numbers [Hofmann 1949, 161 he unwittingly showed how little he knew at that time about the work of the earlier mathematicians on combinatorial problems.

Although Leibniz had his reservations about his own Dissertatio de Arte Combinatoria only a few years after its publication, the effect.it had on his later studies, even those made after 1690, is clearly discernible. Of course he made himself familiar during his Paris stay with the related mathematical literature (Pascal) and even with the newest publications of his day (such as the Elgmens des Mathgmatiques by Jean Prestet [1675]) and was motivated by them to the study of more advanced problems. One of these was the question of how many numbers can be formed from a certain number of numerals with a given number of digits [Prestet 1675, 350; Bernoulli 1713, 130f .; Haussner 1899, Vol .I, 127; LH 35, XIV, 1 f. 3061.

Leibniz was primarily interested in the laws of formation of the combinatorial numbers and the arithmetic triangle. He also took up the study of CwRandvwR, partly with the same questions in mind as in the Ars Combinatoria (“caput”-theory) [LH 3.5, III B, 14f. l-2; Rivaud 1914-1924, Nr. 12811. During these studies he discovered (if one interprets his results) that there are C(n-1, k-l) partitions of a number n into k summands, if the order of summands is considered [LH 4, V, 9 f. l-7 De L'Horizon de la Ei.octrine Humaine].

From both early and later manuscripts can be seen the importance which Leibniz attached to an appropriate classification and designation of the combinatorial operations. Even in drafts written after 1690 [LH 35, XII, 2 f. 122 or LH 4, V, 9 f. l-71


he repeats the classification of the general conception of variations he had employed in his dissertation. New, however, are his special terms. In agreement with modern usage he speaks of combinations without necessarily meaning sets of two elements, which he calls “simple” or “doubled” depending on whether they are CwoR or CwR. Special weight is given to the fact that combinations can also be formed of no elements [LH 35, I, Sf. 53; I, 29 f. 1-18, especially f. 7v-8r]. Permutations (without repetition) he calls “transpositions”; variations (with repetition) of two, three,... elements are called “bigae”, “trigae”, etc.

Characteristic of Leibniz is the close interrelationship between purely mathematical and philosophical thinking. This is very obvious in his De 1'Horiwn de la Doctrine Humaine [ LH 4, V, 9 f. l-71, developed from a thought in the Ars Combinatoria. Combinatorial laws lead him to the assumption of the cyclic nature of history to the extent that everything said or done will repeat itself someday.

3. Studies on Partitions

3.1. His tori cal background It is often maintained [Enestrom 1912/13, 352; Mllller

1907, 74; Riordan 1958, 1071 that the history of the problem of finding the number of partitions of a natural number into integer summands begins with Euler’s writings on the “partitio numerorum”, to which he was stimulated by the Berlin mathematician Philippe Naud6 the Younger, whose writings are compiled by Scriba [1970, 141-1421. As we have seen, and as the study of the unpublished manuscripts clearly shows, this belief is unjus- tified. G. Enestrtim was the first to regret that the pre- Eulerian history of the problem seemed to be quite unknown [ 1912/13, 3521.

Karl Friedrich Hindenburg, the founder of the so-called Combinatorial school in Germany, pointed out as early as 1778 and again in 1779 [Hindenburg 1778, 15; 1779, 871 that Christian Huygens and Jakob Bernoulli had already solved special problems in this field while studying the laws governing dice games [Huygens 1657, 529; Bernoulli 1713, 21-25; Haussner 1899, vol. 1, 22-281, but his remarks were soon forgotten, as was his reference to Leibniz’s first statements on partitions in his Ars Combina- toria [Hindenburg 1779, 211.

Another special problem on partitions appears around 1200 A.D. in Leonardo di Piss’s Liber Abaci [1202; 1857, 297f .], which reads : Which proper subset T of the set Z of integers is constructed in such a way that each integer can be parted into a sum of its elements? In case of the existence of such a subset it is asked in addition that T be minimal and that no element appear more than once.

418 E. Knobloch

The charming history of this so-called Bachet’s weighing problem has been told in another publication [Knobloch 1973a, 142-1511. Here we mention that the “perfect partitions” might be interpreted as solutions and that Leibniz occupied himself also with this problem [LH 35, XII, 1 f. 171.

Finally, Marin Mersenne,in his contributions to combinatorics contained in the two volumes on musical investigations written in 1635/36, lists the various partitions of certain numbers into equal or different summands, without regard to order, in order to describe the individual types of classes of combinations and permutations with repetitions [Mersenne 1635, 139; 1636, Livre des Chants, 1301. By this procedure he in fact determines the special numbers of the Bose-Einstein statistic.

3.2. Leibniz and his successors Leibniz was almost exclusively interested in those parti-

tions that were formerly called combinations with repetition and fixed sums, and which he called “true partitions” [LH 35, XII, 1 f. 102-1031. Permutations with repetitions and fixed sums, “absolute partitions” as he called them, he made use of only for the determination of combinations where the order of summands does not matter. It was his aim to find inductively, i.e. with the aid of tables, independent formulas for the number of partitions with a definite number k of summands, in other words, to calculate the number of solutions of the Diophantine equation [Rieger 1959, 3561:

(1) n1 + n2 + . . . + nk = n I<_ nl( n2( .--< nk

The still existing manuscripts reflect his numerous errors and misleading attempts, but one should not forget how long the mathematical world had to wait after Leibniz for the general solution of this problem. The mathematician and astronomer Ruggiero Giuseppe Boscovich, who worked in Italy, formulated the problem as follows [1748, 931: “Dato un numero, trovare quante sieno tutte le sue divisioni possibili in parti intere, senza determinarle a una a una” (Given a number, to find how many partitions of it are possible into integers, without determining the individuals one after the other). Boscovich, however, was content to describe his efforts in a general manner and to inform his reader that he had abandoned the problem because it contri- buted nothing to his primary subject, the polynomial theorem. At about the same time Euler was occupied with problems of partitions that led to a first publication in the same year [1748, EO I, 8, 313 -3381. However, neither in this nor in later publications did Euler give an explicit formula for (l), but only a recursion law. [S]

Hindenburg [1779, 88f.l frankly admitted that there was still a problem to be solved, but was as little interested in its solution as had been Boscovich. He was well aware of the difficulties when he wrote 16 years later [1695, 412f.l:


Note: that the solution of this problem is one of the most difficult ones that exist in the field of combinatorics, which becomes quite clear from Leibniz's remarks on this problem in a letter to Johann Bernoulli (more than 30 years after publication of his Ars Combinatoria). [61 Johann Christoph Weinggrtner, a disciple of Hindenburg, even

declared in his textbook [1800, vol. 1, 2961 on combinatorial analysis that such a formula would be very difficult to get. It was not until 1840 that Moritz Stern derived the long-sought result with the help of the Eulerian recursion law:

-2 q 1 Here and below Pn stands for the number of partitions of a

number n in the sense of (l), e stands for the number of partitions of n into m summands, P, h is the number of partitions of n, the smallest summand of whiih is h, and pfj h is the number

of partitions of n into m summands of which the smallest summand is h.

3.3. Terminology Leibniz's terminology for partitions, just as for symmetric

functions, is not consistent [Knobloch 1972, 280f.; 1973b, 92-961. In his drs Combinatoria he speaks of "discerptiones, Zerfallungen" as mentioned above, and defines them as special cases of "complexiones" (combinations). The Latin term "discerptio' he uses most, and it appears in numerous manuscripts up to his death. [7] When he wants to refer to specific partitions into 1, 2, 3, 4... summands, he writes "uniscerptiones, biscerptiones, triscerptiones, quadriscerptiones..." and sometimes also "lscerp- tiones, 2scerptiones..." evidently following his former usage for combinations of certain sizes in the Ars Combinatoria [LH 35, III A, 28 f. 29v-30r; XII, 1 f. 103v, LBr 705 f. 77v]. I have found only two places where Leibniz applies the general term "discerptio" to the special partition into two summands [LH 35, XII, 1 f. lSr; III A, 28 f. 30r].

His terminology for partitions into two summands is quite in agreement with his general habit of forming words with the prefix "bi" in order to describe the formation of pairs. There are numerous examples of this, as "biconcursus, bielectio, bifactor, bifidus, biga, bilitera" etc.

In his later writings Leibniz uses "divulsio" in the same sense as "discerptio", but less frequently. For special partitions he creates the forms "bidivulsiones, tridivulsiones" etc., but analogous to the forms of "discerptio" also uses the

420 E. Knobloch i-Ml

abbreviated forms of “univulsio, bivulsio” etc. [LH 35, IV, 5 f. 17 and f. 19; III A, 15 f. l-9; Leibniz, Mathematische Schriftea Vol. 7, 1651.

Beyond this one finds occasional uses of 1. “partitiones” or “bipartitiones” [LH 35, II, 1 f. 7-37, especially f. 26’ (Rivaud 1914-1924, Nr. 1233 E); XII, 1 f. 341-342, especially f. 34Zv (Rivaud 1914-1924, Nr. 730)]. 2. “sectioned”; in the same manuscript [LH 35, XII, 1 f. 232-2331 Leibniz also and evidently only there, introduces a new symbol for P$: ;qEzq. 3. “dispersio” [LBr 705 f. 126’ and 12Sr Diatribe Algebraica de Multinomiis].

3.4. Leibniz's enumeration of partitions in comparison to thoseof his successors

In numerous tables Leibniz followed the same method of ordering: he arranged the summands of a partition by decreasing size and partitions by the size of successive sunnnands. When he wanted to enumerate all possible partitions of n he proceeded by number of summands. He hardly ever deviated from these two basic methods.

Both his methods differ from those of his successors Boscovich, Euler and Hindenburg. Here we refer only to methods of presenting the partitions of a class. All three authors reversed Leibniz’s principle by arranging the summands by increasing size. While Euler himself did not emphasize his presentation and sometimes also arranged the summands by decreasing size like Leibniz when enumerating all possible partitions [Euler 1751, EO I, 2, 193; 1748, EO I, 8, 318 and 3331, Hindenburg gave strict orders as to how “combinations to a special sum” were to be treated [Hindenburg 1779, 73f.; and somewhat inexactly, 1778, 6-91. His procedure differed from that of Boscovich insofar as he arranged the partitions by the first, while Boscovich used the last summand [Boscovich 1738, 901, and then the partitions of the next smaller class of the rest were placed either after or before this summand by both authors. In this way Boscovich enumerated the partitions in the same sequence as Leibniz. In those two cases [LH 35, XIV, 1 f. 169-170; III A, 28 f. 29-301 however, where Leibniz arranged the summands of a partition by decreasing size, but the partitions themselves by increasing size of the last summand, he enumerated the partitions in the same order as Hindenburg did. Like Hindenburg he also tried to attach appropriate letters to summands.

Leibniz constructed tripartitions from bipartitions by split- ting each second summand into two summands again. It was Hindenburg’s achievement to develop a general form of this method applicable to all partitions, but the idea stems from Leibniz. He was quite proud of it, because Euler [1753, EO I, 2, 2541 had still confessed that “Qui autem actu omnes partitiones dinumerare voluerit non solum in immensum laborem se immergit, sed omni

kit41 Leibniz on combinatorics 421

etiam attentione adhibita vix cavebit, ne turpiter decipiatur” (But whoever wants to enumerate really all partitions will not only begin an immense labour, but will also hardly avoid being deceived, even if he is as attentive as possible). Leibniz’s numerous efforts in this direction demonstrate how right Euler was.

3.5. Determination of numbers of partitions It was Leibniz’s primary interest to determine the values

of Pn and 4, from which he also hoped to arrive at the number of symmetric functions of a certain degree. lie tried throughout his life (evidently without asking for help from any of the numerous mathematicians he corresponded with, although Johann Bernoulli might have been able to help him [S]). In the final years of Leibniz’s life the Wolfenblittel school principal Theobald Overbeck seems to have given him voluminous tables of partition numbers which led him to the discovery of several laws of recursion. Overbeck was a diligent worker, but on the whole no match for Leibniz.

The above-mentioned solut)on to Pz given in the Ars Combina- toria Leibniz summed up as P, = [n/2] not later than 1677 [LH 35, III A, 28 f. 29-301. This is an important fact when one realizes that Weinglrtner [1800, Vol. 1, 2973 was not able to go beyond separate cases for n even or odd.

Above all Leibniz tried to determine the value of P3 to which he devoted three studies that will be discussed fieie in greater detail. Still in his Paris time, he formulated a rule erroneously called “universal solution” which, however, can be transformed into a formula for pi. Two later attempts failed because of false reasoning or calculating. The methods were right, and therefore it is possible to bring Leibniz’s various efforts to a successful and improved conclusion.

3.6. The three studies on Pz The first attempt was made about 1673 [LH 35, XII, 1 f.

15 (Rivaud 1914-1924, Nr. 520B) and III B, 14 f. 4 (Rivaud 1914- 1924, Nr. SZOA)]. Leibniz tried to derive a method for n = 8 as an example that leads to a complete listing of all tripartitions . To this end he takes up a thought from his Ars combinatoria: Vripartitions are derived from bipartitions by adding another summand.” Consequently he first enumerates all possible ordered bipartitions of 8:

7+116+215+314+4(3+512+611+7

6.1.1 hld-m,3-rkrds-

5.2.1 4.2.2 A+CJ- -

4.3.1 3.3.2

From these n - 1 = 7 bipartitions he then obtains the requested tripartitions by renewed partition of the first summand into all

422 E . Knob1 och HM1

possible bipartitions (without regard to order). He notes that during this process repetitions occur with increasing frequency. Only the m-th tripartition of the m-th bipartition yields a new tripartition. The 1st to (m-l)th partition of the m-th bipartition are among the 1st to the (m-l)th tripartitions of previous bipartitions. If therefore the m-th column contains (m-l) partitions, it contains nothing but repetitions. Therefore all those bipartitions that contain no more than (m-l) partitions can be excluded. If one bipartition can be excluded, all the following ones are to be omitted as well, since the first summand continually decreases and thus permits progressively fewer partitions.

From these observations follows the Leibnizian rule which in modern terms reads:

To find all k-partitions P$ of a natural number n, find all (k-l)-partitions of the numbers (n-l) to 1; in each case one has to subtract from P:-1 (Xm<n-1) the number of preceding numbers (that is, n-m+l). Hence,

(3) Pk n = "=zl

pk-l - v+l n-v

The summation is to be continued as long as the summands are non-negative, e.g. PB 3 = P; + (P;-l) + (P2,-2) = 3 + (3-l) + (2-2)= 5. In reality (3) is not a universal rule, as Leibniz thought, but is only valid for the one case k = 3 from which he prematurely generalized . As an example of the limited ap

5 licability of the

rule, (3) yields the wrong result Pz = P: + Pg -1 = 4 + (3-l) = 6. For k = 3 one gets

(4)

By putting the number of bipartitions in place of Pz-v-l, we find

It thus appears that Leibniz inductively, without complete proof, found a solution that Moritz Stern derived 170 years later. The result can be further simplified to

p3 = [(n2 +3)/112] n

a compact expression Leibniz came very close to in his second attempt about 1689 [LH 35, XII, 1 f. 102-1031. There again he starts from special examples, in that case n = 6 and 7. He first makes use of all possible partitions, which means that he again observes the arrangement of sunnnands as in his first


attempt. For n = 6 we have the following partitions into three summands :

411 321 231 141 312 222 132

213 123 114

or 1+2+3+4=10. In general

n-2 1 k = (";')

k=l

is the number of all possible tripartitions of a number n. The total consists of all those partitions that contain summands once, twice, or even three times. According to the possible orders they appear six times, three times or twice. The frequency of their appearance in the most difficult first case is determined by gradual exclusion of the last two possibilities.

a. Three equal summands This third case (2 + 2 + 2) cannot appear more than once

and then only when n is divisible by 3:

(7) a = 1, if n : O(mod 3)

a = 0, if n # O(mod 3)

b. Three equal summands By dividing the remaining amount A = ((n-l) (n-2) - (2 or 0))/2

by 3 one gets all d/3 partitions with summands that are either all different or with two equal summands. The partitions of this second group now appear once and only once. Leibniz prematurely determines their number by means of his example n = 7 as

(8) n - 0 vel.1 , i.e. n 2 [I 2

The two bars are meant to indicate that we here have a condition of divisibility with respect to 2. Actually, his example n = 6 shouldhave set him right, since [6/2] is 3, whereas only the partition 4 i 1 + 1 is possible. His (8) is correct only when n is an odd number and indivisible by 3, that is when n E 1, 5 (mod 6) is true, since in both of these cases every pair between 1 + 1 and [n/2] + [n/2] must occur. When n E 3 (mod 61, however, we must subtract the partition that is numbered as that one with three equal summands. When n is an even number, the last pair [n/2] + [n/2] cannot be used for a partition into three summands, and in the case n s 0 (mod 6) the partition consisting of three equal summands must be subtracted as well. Thus the possibilities can be accounted for in three cases:

424 E. Knobloch

n - O(mod 6)

HMl

B=4

(9) n E 2,3,4(nwd 6) n-B [ 1 - 2 B=2

n f 1,5(mod 6) n [I ?

B=O

C. Three different summands With these values Leibniz!s method of solution can be

continued analogously, and we obtain

(10) f (4 - [” ; 81) = f ((n-1) (n-2) ; (2 or 0) - p ; “1)

as the number of partitions with three different summands. Divi- ding by 2 is necessary because the partitions with three different summands appear twice after the subtraction of the partitions with two equal summands. Thus

(11) P;‘, = a + [y] + $( f$ - [CL]) = a + $( [(n-lk(n-2)] + [y!.])

Leibniz’s own final formula

ad + 2 + 10 0 vel 1 - 3 0 vel 1 12

is incorrect because of the error mentioned, although recognizably similar to the modern solution (6). One bar above 0 vel 1 indicates a condition of divisibility with respect to 3.

His third and most extensive investigation; of which I am offering only a bare outline here, was made in 1699 and probably led to his comment to Johann Bernoulli concerning partitions [LH 35, III A, 15 f. 104; the following four sketches f. 5-6, f. 7, f. 8, f. 9 must be added]. Leibniz calculated P3 with the aid of certain partitions into two summands which he &led “useful”. The concept “useful ” results from his order of the summands . The partitions into three summands arise from a first item z as well as all partitions into two summands of the accom- panying complement y, of which only those whose first part v is not larger than z are useful.

He performs the solution in four steps. In the first two he makes a separate investigation of the cases in which all (total) or only some (partial) partitions into two summands of a complement are useful. In the second two he calculates the number of partitions of the two types of complements. The sum of both of these numbers produces the desired value P;.

a. All partitions into two summands are useful. Let us assume that v is the first summand in the first partition into two surmnands of any complement y. In his table Leibniz finds


the proposition that z+v is always equal to n-l. If we disregard an oversight, he subsequently proves that there is a relation- ship z,>n/2<--> OS z-v.

b. Only some of the partitions into two summands of y are useful. Once again Leibniz goes to his tables and finds the statement that the number of partitions into two summands which are not useful is y-l-v and that the number of useful partitions of y for a given v is, therefore, [y/2] - y + 1 + v.

C. Sum of all totally useful partitions into two summands. Only useful partitions into two summands supply all the numbers between 2 and [(x1+1)/2]. The desired sum of [2/2] + [3/2] + . . . + [ (n+1)/4] results in

(121 s = L

2( f f f+l) f+l -- -

2 c 1 2 ,fr % [ I)

d. Sum of all partially useful partitions into two summands. On the basis of (a) all complements y> f have only partially useful partitions into two summands. Since y = n - v, the number of partially partitions into two summands, on the basis of (b), is

[ 1 y +2v+l-n

Since the largest summand of the last partition into three summands is v = [(n+2)/31, as Leibniz again finds in his tables, the Q = n - [(n+l)/2[ - [ (n+2)/31 values from (13) for the various values of v from y = n - v = 1 + [(n+1)/2] to v = [(n+2)/3] must be added together. Although the difference between two consecutive values for v, v-l is not constant, Leibniz adds these Q values with the summation formula of an arithmetic series, a step permissible only for even Q. Insertion of the first value v = n - 1 - [(n+l)/2] and of the last value v = [ (n+2)/3] in (13) produces

(14) R= [$(l + [?I)]- 2[9 + [$(n - [?])I + 2[5q= [%I+ B

Leibnizls final result is

(15)

Because of when n E difficulty 10,ll (mod

(16)

P3 =s+ n f"P=S++([~] +l3)

the error involved, Leibniz’s result is only correct 0 , 2, 3 , 4 , 5 , 7 (mod U). There is, however, no in extending it to the cases in which n z 1,6,8,9, 12). (P being an odd number), and we obtain

Pi = s + f([Q$qR + [Y](F] + y))

whereby y = -If + - [T] )] + 2[?]

426 E. Knobloch HMI

Keeping in mind the necessary limitation to an appropriate n, we can state that for the very first time in the history of mathematics Leibniz found an explicit formula for ~2.

3.7 Further results concerning the theory of partitions The difficulties encountered when calculating pf3 evidently

deterred Leibniz from attempting to determine P$ for larger k, but he found many other results, in particular the laws of recurrence. As early as 1673 he formulated e = p,"-l = 1 [LH 35, XII, 1 f. 151. Probably more than ten years later he derived a recurrence formula that is valid only under certain conditions and expresses 6 with the aid of e-l and es2 [LH 35, XII, 1 f. 232-2331. Nevertheless, in the years 1712- 1715 by induction he found but did not prove the following interesting regularities [LBr 705 f. 77-781:

Pk k = P + PkI1

(Euler’s Rule of Rec~rren~ek[Eul~r11751, EO I, 3, 1911)

n-l "I& p;:',+" = ki1 p;+,

v=o

'n =l+P n-l,1

+P n-2,2

+ .*. + 'n-[n-2],[n/2]

P n n = '2n

His study of symmetric functions led him, in addition, to two partition problems, which, however, he neither formulated as such nor investigated intensively. He compiled a table which produces the values

lk P," (k = 1,2 . . . . n) v=l

in the special case n P

n = .E, PV n

without realizing that

$ P; = P;+k v=1

As a result of his investigation of powers he calculated the number of all the partitions of a number n, which only contain the summands 1,2,3 and 1,2,3,4 respectively, without recognizing the fact that the results coincide with Pz+3 and P$+~ respectively.

A further use of partitions, namely the attempt to divide products (i.e. forms or the terms of symmetric functions) into a certain number of factors [LH 35, IV, 5 f. 171, led Leibniz (before 1700!) to Stirling numbers of second order [Sterling, 1730, 81, an,d the solution of special problems of permutations.

l-341 Leibniz on combinatorics 427

NOTES

These definitions are often mentioned in the literature, for examples Serres [196Sa, 121-123; 196Sb, 412-4131. A German translation is given in Schmidt [1960, 581. This expression is used again in the drafts LH 35, XII, 1 f. 17 and LH 35, I, 29 f. l-18, especially f. 6r. Leibniz possibly knew it from Lantz [1616, 521. LH 35, III B, 14 f. l-2 De numero jactuum in tesseris, written in January 1676 [Rivaud 1914-1924, Nr. 12811; Biermann has discussed this study in two papers [1954; 19561. Euler [1770; EOI, 3,141] published an explicit formula only for the number of ways in which a number n+X can be given by n dice, each of them consisting of n sides carrying the first m integers. Hindenburg alludes to Leibniz's letter of July 28 (August 7) 1699 to Johann Bernoulli [Leibniz, Mathematische Schriften, vol. 3, 6011. For example LH 35, III A, 4 f. 26; III A, 28 f. 29-30; III B, 14 f. 4 [Rivaud 1914-1924, Nr. 520 A]; XII, 1 f. 15 [Rivaud 1914-1924, Nr. 520 B]; XII, 1 f. 14; XII, 1 f. 102-103; XII, 1 f. 232-233; XIII, 1 f. 163; XIV, 1 f. 169-170; XIV, 1 f. 176; LBr 705, f. 77-78; LBr 705, f. 120. See note 6. When 'Leibniz asked Johann Bernoulli whether he had ever tried to determine the number of partitions, he got no answer.

1.

2. 3.

4.

5.

6.

7.

8.

REFERENCES

Bernoulli, Jakob 1713 Ars Conjectandi Hrsg. von den Gebriidern Thurneisen und Nikolaus I. Bernoulli, Base1 Brussels 1968).

(Reprint

Biermann, Kurt Reinhard 1954 Uber die Untersuchung einer speziellen Frage der Kombinatorik durch G.W. Leibniz Forschungen und r'orschritte 28, 357-361.

I 1956 Spezielle Untersuchungen zur Kombinatorik durch G.W. Leibniz Forschungen und Fortschritte 30, 169-172.

1965 Uberblick iiber die Studien von G.W. Leibniz zur Wahrscheinlichkeitsrechnung Forschungen und Fortschritte 39, 79-S.

Boscovich, Ruggiero Giuseppe 1747 Metodo di alzare un' infinitinomio a qualunque potenza Giornale de' Letterati di Roma, 393-404.

1748 Parte prima delle Riflessioni sul metodo di alzare un infinitinomio a qualunque potenza Giornale de' Letterati di Roma, 12-27.

1748 Parte seconda delle Riflessioni sul metodo di alzare un infinitinomio a qualunque potenza Giornale de' Letterati di Roma, 84-99.

428 E. Knobloch HMl

Cantor, Moritz 1901 Vorlesungen iiber Geschichte der Mathematik. Vol. 3 (1668-1758). Leipzig (Reprint Stuttgart, New York, 1965) .

Caramuel de Lobkowitz, Johannes 1670 Mathesis Biceps vetus et Nova. 2 vols. Campania (Cursus Mathematicus, Vol. 1 and 2).

Clavius, Christophorus 1585 In Sphaeram Joannis de Sacro Bosco Commentarius. Rome.

Couturat, Louis 1901 La Logique de Leibniz. Paris. (Reprint Hildesheim, 1969).

Enestr%m, Gustaf 1912-1913 uber die Utere Geschichte der Zerfsllung ganzer Zahlen in Summanden kleinerer Zahlen Bibl. Math. (3)13, 352.

Euler, Leonhard 1748 Intrnductio in Analysin Infinitorum. Lausanne . (L. Euleri Opera Omnia, Ser. I, vol. 8, Leipzig- Berlin 1922, and vol. 9, Genf 1945 [designated by EO I, 8 and EO I, 91).

1751 Observationes analyticae variae de combina- tionibus Comm Acad Sci Petrop 13 (for 1741/43), 64-93 [EO I, 2, Leipzig-Berlin 1915, 163-193; Enestrsm-Index 158, designated by E 1581.

1753 De partitione numerorum Novi Comm Acad Sci Petrop 3 (for 1750/51), 125-169 [EO I, 2, 254-294; E 1911.

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ACKNOWLEDGEMENT

I am very indebted to Mr. D. Busche and Mr. C. Kraft for their help in translating this paper.

HISTORIA MATHEMATICA 4 (1977), 379-396

GAUSS AS A GEOMETER

BY H, S, M, COXETER, UNIVERSITY OF TORONTO, TORONTO M5S 1Al

;his paper was presented on 4 June 1977 at the Royal Society of :anada’s Gauss Symposium at the Ontario Science Centre in Toronto.

SUMMARIES

In an attempt to reveal the breadth of Gauss's interest in geometry, this account is divided into six chapters. The first mentions the fundamental theorem of algebra, which can be proved only with the aid of geometric ideas, and in return, an application of algebra to geometry: the connection between the Fermat primes and the construction of regular polygons. Chapter 2 shows his essentially 'modern' approach to quaternions. Chapter 3 is a sample of his work in trigonometry. Chapter 4 deals with his approach to the geometry of numbers. Chapter 5 sketches his differential geometry of surfaces: his use of two parameters, the elements of distance and area, his theorema egregium, and the total curvature of a geodesic polygon. Finally, Chapter 6 shows that he continually returned to the subject of non-Euclidean geometry, which was so precious and personal that he would not publish anything of it during his lifetime, and yet did not wish to let it perish with him.

In einem Versuch die Breite des Interessen von Gauss an der Geometrie zu enthiillen, ist dieser Bericht in sechs Kapitel eingeteilt. Das erste erwzhnt den Fundamentalsatz der Algebra, der nur mit Hilfe geometrischen Ideen bewiesen werden kann, und anderseits eine Anwendung der Algebra auf die Geometrie: die Verbindung zwischen Fermats Primzahlen und der Konstruktion regelmXssiger Vielecke. Kapitel 2 zeigt seine im wesentlichen 'moderne' Darstellunq der Quaternionen. Kapitel 3 enth;ilt eine Kostprobe seiner Arbeit iiber Trigonometrie. Kapitel 4 beschaftigt sich mit seiner Darstellunq der Geometrie der Zahlen. Kapitel 5 skizziert seine Differentialgeometrie der Fl&hen; seinen Gebrauch zweier Parameter, die Elemente von Abstand und FlSche, sein theorema

Copyright 0 1977 by Academic Press, Inc. All rights of reproduction in any form resertzed.

380 H. S. M. Coxeter HM4

egregium und die totale Kriimmung eines geodatischen Polygons. Kapitel 6 zeigt schliesslich wie er St&dig auf das Thema der nicht-euklidischen Geometrie zuriickkehrt, die ihm so wertvoll und persijnlich war, dass er nichts darlfber zu seinen Lebzeiten verbffentlichen wollte, und trotzdem wtfnschte er nicht, dass sie mit ihm unterginge.

1. CYCLOTOMY

The idea of representing the complex number x + yi by the point (x,y) in the so-called ‘Argand plane’ should be attributed to Caspar Wessel (1745-1818), whose memoir on the subject was presented to the Copenhagen Academy of Sciences in March 1797 [Ball, 1960, 4711. He interpreted addition as translation and multiplication as rotation. This idea enabled Gauss [1799a] to develop the first of his three proofs of the fundamental theorem of algebra: that every rational integral algebraic function of one variable can be resolved into real factors of the first or second degree.

One such function had already interested Gauss in 1796,

when he was only 19: the cyclotomic polynomial (x” - 1)/(x - 1). He proved that a regular n-gon can be constructed with ruler and compasses whenever all the odd prime factors of n are Fermat

primes 2 2k

+ 1, each occurring in the first power only [Gauss, 17961 . The big surprise was the case k = 2. Although he did not trouble to simplify the actual construction for the 17-gon [see, e.g. Coxeter 1969, 271, he expressed its feasibility by writing out the expression

2ll cos i7= - 16 l + $17 + $(34 - 2 J17)

+ $4117 + 3 /17 - J(34 - 2 417) - 2 /(34 + 2 417)).

(To be quite precise, instead of 2a he wrote P).

2. ROTATIONS AND QDATERNIONS

Gauss [1818, 3581 expressed the general rotation (about a line through the origin in 3 dimensions) by the orthogonal matrix

$a' + b2 - c2 - d2) - ad - bc ac + bd

ad - bc

ac + bd

- b2 + C’ - d2) - ab - cd

ab - cd +(a2 - b2 - c2 + d2)

tlM 4 Gauss as a geometer 381

.*nd remarked that the product of two such rotations is represented by the product of two symbols

(a,b,c,d) (a,B,y,fj) = (A,B,C,D),

#here A,B,C,D are certain bilinear expressions in the other letters. He thus anticipated the representation of a linear transformation by a matrix and also the rule for multiplying quaternions. All that is lacking for perfect agreement is a trivial matter of sign: Hamilton's a + bi + cj + dk is not (a,b,c,d) but (a,-b,-c,-d).

3. PLANAR AND SPHERICAL PENTAGONS

Gauss [1823b] discovered a pretty theorem in what we would now call affine geometry. Decomposing a convex pentagon 12345 into triangles in various ways (see Figure l), he used trigonometry to express the area w in terms of the areas

a = 123, b = 234, c = 345, d = 451, e = 512

of the triangles at the five corners: w is the greater root of the quadratic equation

2 w - (a+b+c+d+e)w + (ab+bc+cd+de+ea) = 0.

I wonder how one could interpret the smaller root! It may have been about this time that Gauss's interest in

spherical trigonometry led him to investigate Napier's Pentagramma Mirificum: a star-pentagon with five right angles [Gauss 1823a; Knott 1915, 1701.

Each side of the inner pentagon, being an arc of the great circle polar to the opposite vertex, is equal to the external angle at that vertex, as in Figure 2. Denoting the squared tangents of these five sides (or angles) by ct,B,y,G,~, he uses Napier's equation

cos 1 = cot 3 cot 4

(in which 1,2,3,4,5 can be cyclically permuted) to set up the following table:

Quadrate der Tangenten Secanten Gleichungen

a Y6 l+a=y6

B 6E l+f3 6E

Y ca 1+-y Ea

6 a$ 1+6 a8

E BY l+E By

HM4 Gauss as a geometer 383

FIGURE 2


He reduces the geometry to algebra by using the identity

(l+y)(l+R-6~) - (l+@)(l+y-~a) = ~{(l+a-Y6) - (I+&-a6))

to show that, among the five equations in his third column, any three imply the remaining two. In other words, if the matrix

a 1 0

16 810

01E:ylO

01 10

01 1

has all its two-by-two determinants equal to 1, the entry below y must be a, with 6 to the right of it and 6 below this 6: the diagonal entries are cyclic, with period 5 [Coxeter 1974, 221. Carried by his enthusiasm, Gauss continues by writing down his 'schbne Gleichung'

3+a+f3+y+A+E=aBy6E.

(Napier's second set of five equations, beginning with

cos 1 = sin 2 sin 5,

arise from sin21 = aly6, etc.)

4. CRYSTAL LATTICES AND QUADRATIC FORMS

In his review of a dissertation by L. A. Seeber, Gauss [1831a] expressed a positive definite quadratic form

2 2 ax + 2bxy + cy

or 2 2 2 ax + by + cz + 2a'y.z + 2b'zx + 2c'xy

as the square of the distance from the origin to the point (in 2 or 3 dimensions) whose oblique coordinates (or more precisely, affine coordinates) are (x,y) or (x,y,z) [Coxeter 1951, 393; 1969, 51, 225, 335-3371.

He saw that integral values of the coordinates yield systems of parallel lines or planes forming a 'lattice' of the kind used in crystallography, and that a unimodular transformation of the coordinates will yield new systems of parallels intersecting one another in the same discrete set of points: a lattice in the strict sense. (In modern terminology, this is a discrete set of points whose set of position vectors is closed under subtraction, that is, along with any two of the vectors the set includes also their difference.) In other words, Gauss used a lattice to represent a class of equivalent positive definite forms.

Every lattice has a certain minimal distance d between pairs of its points, and a certain area or volume J for its unit

184 4 Gauss as a geometer 385

r-e11 (the basic parallelogram or parallelepiped whose coordinates involve only zeroes and ones). In terms of the corresponding

quadratic form, d2 is the minimal positive value for integers

y and Y (or x,Y,z), 2 while J is the determinant

a c' b'

or c' b a'

b' a' c

4 binary or ternary form is said to be extreme when the ratio

J/d" (n = 2 or 3) is as small as possible. Gauss showed that any binary or ternary extreme form is equivalent to

2 2 X + XY + Y

or 2 2 2

X +y +z + yz + zx ,

respectively. He thus came within a hairbreadth of determining the closest lattice packing of circles or spheres.

5. CURVATURE OF SURFACES

Every first course on the differential geometry of surfaces is essentially a description of Gauss's treatise Circa superficies curvas [Gauss 18271. He begins by observing that the condition for a point (x,y,z) to lie on a given surface amounts to expressing the rectangular coordinates x,y,z as functions of two parameters, p and q. This parametrization yields a tangent vector

(dx,dy,dz) = (adp + a'dq, bdp + b'dq, cdp + c'dq) ,

where a =i$ , a' =z+ , etc.; a normal vector

(A,B,C) = (bc'-cb',ca'-ac',ab'-ba') = z% , a (~4 a (X,YI , a0' a(P,Q); 3

and a first differential form

ds2 = E dp2 + 2F dpdq + G dq2,

where E= a2 + b2 + c2 , F = aa' +bb'+cc' ,G=a '2 + b'2 + /2 *

(This expression for ds, the element of distance on the surface, is the essential first step in the development of Riemannian geometry and the general theory of relativity.) The element of area is

dS = Jh dp dq,


where

A = A2 + B2 + C2 = ,

and the curvature K (which we now call Gaussian curvature) is given by

+ FIz ac- aE aA_ 2ax aF+4aF aF_ 2aF a>) ap a4 a4 ap a4 a4 ap a4 0 ap

+G(!% aG ap ap

-2A I a2E - -

I aq2

aF G

+

2 1

This theorema egregium looks somewhat less formidable in terms of Christoffel symbols [Coxeter 1969, 3671. As it does not involve A,B,C, it shows that K can be computed by measure- ments made on the surface itself, without reference to the 3- dimensional space in which it was originally assumed to lie. In other words, K is a ‘bending invariant’, unchanged by the kind of distortion that takes place when a flat sheet of paper (for which K = 0) is rolled up to make a cylinder or cone. Its most obvious property is that it is positive or negative according as the surface is egg-shaped or saddle-shaped: K > 0 at the top of your head, but K < 0 on your neck.

In terms of ‘geodesic polar coordinates’, with radius p and angle q [Coxeter 1969, 370-3721, we have simply

2 2 2 ds = dp t G dq .

Since now E = 1 and F = 0, theorema Tgregium yields

1 a‘JG K=-JG - -

aP2

This enabled Gauss to find the total curvature of a geodesic triangle ABC in the form

Ji KdS=A+B+C-IT.

More generally, the total curvature of a geodesic n-gon is equal to the ‘angular excess’ of the n-gon as compared with an ordinary flat n-gon. In other words, it is 27~ minus the sum of the external angles of the n-gon (see Figure 3). In the case of a compact simply-connected surface such as a sphere or ellipsoid or egg, we can take the geodesic polygon to ‘circum- navigate’ the whole surface, decomposing it into two regions,

Gauss as a geometer 387

FIGURE 3

each in turn serving as the interior of the polygon. The external angles of the two regions differ only in sign and thus cancel out, leaving LC

11 K dS = 4~r

for the whole surface. Thus Gauss [1827, 2461 came very close to the Gauss-Bonnet theorem and to the Euler-Poincare character istic [Struik 1961, 155-160; Coxeter 1969, 3741.

Having found that the total curvature of a geodesic triangle ABC on the general surface is

II ABC K ds = A + B + C - 71,

he may have noticed that the area of a geodesic triangle AEJC on a surface of constant negative curvature is

(a - A - B - C)/jKI.

This expression leads naturally to our final chapter.


6. NON-EUCLIDEAN GEOMETRY

The earliest surviving record of Gauss's thoughts on the foundations of geometry is the letter [Gauss 1799bJ to his friend Bolyai Farkas, in which he said "If one could prove the existence of a triangle of arbitrarily great area, all Euclidean geometry would be validated". Five years later, Gauss [1804] considered the possibility of an equilateral and equiangular polygon k d c f g . . . (Figure 4) which goes on indefinitely without ever crossing the line k $I perpendicular to k d. This is what we would now call a horocyclic polygon.

4

k d

FIGURE 4

Gauss [1813] complained that "In the theory of parallel lines we have not yet progressed beyond Euclid. This is the shameful part of mathematics, which sooner or later must take on a quite different shape". Gauss [1816] wrote to C. I. Gerling in Marburg, pointing out the error in Legendre's 'proof' of Euclid's parallel postulate. It is clear from this and other letters that he had already begun to accept the possibility of a new geometry in which there is an absolute unit of length which he called the constant: something analogous to the 'radian' which is the absolute unit of angle. He must have discussed this with F. L. Wachter, who wrote: "If your anti-Euclidean geometry is true, why does the constant remain undetermined?"

On the other hand, his confidence was wavering, for Gauss [1817] wrote to H. W. M. Olbers: "I come more and more to the conclusion that the necessity of our [Euclidean] geometry cannot

!-IM 4 Gauss as a geometer 389

be proved. Perhaps in another life we shall attain insights into the nature of space which are now beyond our reach".

One year later, Gerling sent Gauss [1818, 1801 a memorandum by his colleague F. K. Schweikart [1780-18591, who declared that there are two kinds of geometry: Euclidean and 'astral'. In the latter, the angle sum of a triangle differs from II, and is consequently less than TI, decreasing when the area of the triangle increases. Gauss [1819] replied to Gerling, praising Schweikart and pointing out that the angular defect of a triangle not only increases with its area but is proportional to the area.

The first occurrence of the term Nicht-Euklidische Geometrie seems to be when Gauss [1824] wrote to Schweikart's nephew, F. A. Taurinus: "I can find no contradiction arising from this non-Euclidean geometry . . . I have for some time expressed a wish that Euclidean geometry were not true [in the real world] for then we would have lein absolutes Mass a priori."

Gauss [1828] gave a proof, independent of the parallel postulate, that the angle sum of a triangle is less than or equal to Il. Apparently he forgot that this had already been done by Schweikart. He went on, in the same spirit, to prove many of the propositions in Euclid, Book I, forgetting that Euclid himself had done the same thing!

The really exciting developments began when Gauss [1831b] wrote (but did not publish) the wonderful essay on Parallel lines concerning which he wrote to H. K. Schumacher (in May, 1831): "After meditating for some forty years without writing anything out ,.. I have at last begun to put down a few of my thoughts, so that they should not perish with me."

B

A

C

FIGURE 5


Gauss defines parallel lines by letting a ray rotate clock- wise about A, beginning with the position AB (Figure S), SO that the angle at A gradually increases. Among such rays, there is no last one that meets the ray BN, but there is a first one that fails to meet BN. This ray AI’+! is said to be parallel to BN. In other words, AM is the Dedekind cut between the rays that meet BN and those that do not. He extends the notion from rays to lines by proving that A can be replaced by any other point on the line AM, and B by any other point on the line BN. He proves that this relation of parallelism is symmetric and transitive, and that parallel lines do not meet when extended backwards. The details are straightforward but tricky, making use of Pasch’s ideas on order, long before Pasch was born [Coxeter 1969, 176-190, 265-2681.

Two months later, Gauss wrote to Schumacher again, pointing out an error in the latter’s ‘proof’ that the angle sum of a triangle is r. In non-Euclidean geometry, he said, there are no similar figures that are not congruent. It is possible for all three angles of a triangle to be zero, in which case one might draw it as in Figure 6.

FIGURE 6

(The middle one of these three versions is almost the way Poincare would have drawn it in his inversive model, fifty years later! )

Referring to Figure 7, Gauss remarked that, as C recedes from A, the difference

3 DBC - 3 DAC

does not tend to zero (as it would in Euclidean geometry). In this remark he came close to Lobachevsky’s proposed test for the nature of astronomical space, using parallax [Bonola 1906, 941.

Another significant remark: in non-Euclidean geometry a

HM4 Gauss as a geometer 391

FIGURE 7

circle of radius r has circumference

71 k(e r/k _ e-'/k 1,

where k is a constant, greater than any distance we can measure. In 1829, Gauss's old friend Bolyai Farkas had written a

textbook on geometry and allowed his bright young son, Bolyai Janos, to add an 'Appendix' on The Absolute Science of Space. The first copy that was sent to Gauss went astray, but the elder Bolyai sent a second copy in January 1832. In February, Gauss [1832] wrote about it to Gerling in great enthusiasm, saying that Janos had rediscovered all his ideas and results and developed them with great elegance, though so concisely as to offer considerable difficulty to anyone not familiar with the subject. In March he replied to Bolyai Farkas. That letter is important for two quite different reasons: first, because it had (unintentionally) a devastating effect on the over-sensitive Janos; secondly, because it includes a mathematical proof embodying the same kind of perfect simplicity as Pythagoras's proof that/2 is irrational or Euclid's proof that there is no greatest prime. Here is a slightly abridged version of the letter:


"If I begin by saying that I am unable to praise this work, you will surely be surprised. But I cannot say otherwise. To praise it would amount to praising myself. Indeed, the path taken by your son, and the results to which it led, coincide almost exactly with the meditations that have occupied my mind for the past 30 or 35 years. Although I intended not to let this work be published during my lifetime, I had resolved to write it all down so that it should not perish with me. It is accordingly a pleasant surprise for me that I am spared this trouble, and I am overjoyed that it happens to be the son of my old friend who has superseded me in such a remarkable way.

"The surface which your son calls F, being a sphere of infinite radius, could well be called a Parasph&-e ['horosphere'], the curve L a Paracykel ['horocycle']; and the name Hypercykel ['equidistant curve'] could be used for the set of all points in a plane at constant distance from a line. But the names are unimportant: the chief thing is the substance, not the form.

"I have a somewhat different way to proceed. As a specimen, here is my pure-geometrical proof that the angle-sum of a triangle differs from 180' by an amount proportional to the area of the triangle. "I. Three lines , parallel in pairs in opposite directions (Figure 8) form a trebly asymptotic triangle T. "II. T has a finite area, say t. [Otherwise it could be approximated by an ordinary triangle of arbitrarily great area, but that would make the geometry Euclidean.] "III. A line de and two rays parallel to it from an outside point a form a doubly asymptotic triangle (Figure 9) whose area is a function f (0) of the external angle at a, increasing steadily from f(0) = 0 to f(r) = t. "IV. Figure 10 shows that f($) + f(T-4) = t. "V. Figure 11 shows that f(@) + f(q) + f(i'r-4-Q) = t. "VI. Since f(Q) + f(+) = t - f(r-4-q) = f($+$), f(4)/@ is constant, namely

f(4) t -=-- Q r

"VII. Figure 12 indicates how a trebly asymptotic triangle T may be decomposed into a finite triangle ABC of area Z (say) and three doubly asymptotic triangles whose areas are

flM4 Gauss as a geometer 393

FIGURE 8

FIGURE 9


FIGURE 10

FIGURE 11

I-&l4 Gauss as a geometer 395

FIGURE 12

~1 = f(A) = ; A , B = f(B) = ; B, y = f(C) = 4 C.

Thus

t=afB+y+Z=;(A+B+C)+Z,

and finally

z+T - A - B - C).

"There does not seem to be any such simple expression for the volume of a non-Euclidean tetrahedron . ..I'

It is difficult to understand how this wonderful letter could have offended Farkas's son. The sad fact remains that Bolyai Janos was so disappointed that he withdrew from scientific activity and always regarded the 'Prince of Geometers' with an unjustifiable aversion [Bonola 1906, 1011. Certainly Gauss had no desire to hurt him. E. T. Bell [1937, 2921 mentions "the great mathematician's acute sensitiveness to all forms of suffering . . . The novels of Sir Walter Scott were read eagerly as they came out, but the unhappy ending of Kenilworth made


Gauss wretched for days and he regretted having read the story."

REFERENCES

Ball, W W R 1960 A Short Account of the History of Mathematics New York (Dover)

Bell, E T 1937 Men of Mathematics London (Gollancz) Bonola, Roberto 1906 ~a Geometria Non-euclidia Bologna

English translation by H S Carslaw 1955 New York (Dover) Coxeter, H S M 1951 Extreme forms Canad. J. Math. 3, 391-441 - 1969 Introduction to Geometry 2nd ed New York (Wiley) - 1974 Regular Complex Polytopes Cambridge (University

Press) Gauss, C F 1796 Werke 1, 462 - 1799a Werke 3, l-102 - 1799b Werke 8, 159 - 1804 Werke 8, 161 - 1813 Werke 8, 166 - 1816 Werke 8, 168, 175 - 1817 Werke 8, 177 - 1818 Werke 8, 180, 358-360 - 1819 Werke 8, 181-182 - 1823a Werke 3, 481-482, 484-485 - 1823b Werke 4, 406-407 - 1824 Werke 8, 186-187 - 1827 Werke 4, 217-258, 341-347 - 1828 Werke 8, 190-193 p 1831a Werke 2, 192-196 - 1831b Werke 8, 202-218 - 1832 Werke 8, 220-225 Knott, C G 1915 Napier Tercentenary Memorial Volume London

(Longmans, Green) Struik, D J 1961 Classical Differential Geometry Cambridge,

Mass. (Addison-Wesley)

Historia Mathematics 9 (1982) 19-36

INVESTIGATIONS OF AN EARLY SUMERIAN DIVISION

PROBLEM, c, 2500 B,C,

BY JENS HBYRUP ROSKILDE UNIVERSITETCENTER,

4000 ROSKILDE, DENMARK

SUMMARY

Two Sumerian school tablets from c. 2500 B.C.--con- taining respectively a correct and an erroneous solution of the same problem of dividing a very large amount of grain into rations of 7 sila each--are analyzed. It is suggested that the error committed cannot reasonably be explained by two earlier conjectures on the method of solution (normal "long division," and multiplication by the reciprocal), but only by conversion to an inter- mediary unit and calculation in two steps analogous to the principle of "long division." Also discussed are some possible implications of this result for contemporary Sumerian arithmetical abilities and general techniques.

Nous analysons deux tablettes d'ecole sumeriennes gravees environ 2500 ans avant notre &z-e. L'une est une solution correcte et l'autre, une solution erronee d'un m&me problsme, h savoir la division d'une tres grande quantite de grain en portions de 7 sila. On montre que les deux methodes de solution proposees jusqu'a maintenant (l'une analogue a la methode moderne, l'autre &ant la multiplication par le nombre inverse) ne suffisent pas ?J expliquer l'erreur commise, et qu'il faut supposer une conversion en une unite inter- mediaire suivie d'un processus en deux &tapes de "division de nombre complexe". Nous discutons des implications de cette analyse sur notre connaissance du savior et des pratiques arithmetiques contemporains des sumeriens.

In an important paper on the prehistory of Babylonian mathe matics, Marvin A. Powell [1976] discussed two tablets from Fara (ancient &ruppak) from c. 2500 B.C., which he identified as two school exercises (one cQrrect and one erroneous) dealing

0315-0860/82/010019-18$02.00/O Copyright 0 1982 by Academic Press, Inc.

AN rights of reproduction in any form reserved.

19

20 Jens Hplyrup HM 9

671 R 671 F

FIG. 1. The two school tablets from c. 2500 B.C. dealing with the distribution of one "granary" of barley in rations of 7 sila. The reproductions are basically those of Jestin [1937], but corrections are made where (and only where) Jestin's version was unambiguously contradicted by the photographs (cf. Figs. 2 and 3). Note that No. 671 begins at the reverse of the tablet (if Jestin's identification of obverse and reverse is to be relied upon, as it probably is), and that the result is written in three lines.

HM 9 Sumerian Division Problem 21

with the same problem. The two tablets were originally published by Raymond Jestin [1937, Nos. 50, 6711. They are shown in Fig. 1 in a corrected version of Jestin's somewhat imprecise reproduction, and in photographs in Figs. 2 and 3. In transliterations and translations based on [Powell 1976, 4321 the two tablets are as follows [l]:

Jestin 7937, no. 50

]

Q(HarulSlEarl 4(GeBuR2(GeBI 45,42,51 5(u) 1

de sila3:3 3 sila of

Zu-tag4 grain left on hand

I Jestin 1937, no. 671

rransliteration

s'e sila 7 3

guru7

16: 1 2-u ba-ti

l(s'aru)

iIdarl

Translation

Grain [in rations of] 7 sila, granary

each man receives

Workmen

45,36,0 [written on three lines]

In the above translation, as in the entire text, the Sumerian numerals are translated as normal place value sexagesimals: that is,

x, y,z:u,v means x . 60' + y . 60 + z + u - 60-l + v * 60-'.

In this connection one must remember that the place value notation is only attested half a millenium after the Fara texts were written. The numerals in these texts are not written according to a place value notation; instead they make use of special symbols for 1, 10, 60, 600, 3600, and 36000 (see Table 1). Thus, all numbers which are found in the Fara texts (certain simple fractions apart) can be translated into the form X,Y,Z, where X, Y, and Z are integers between 0 and 59 [2]. To remind us of the fact that the place value translation is anachronistic, even if in agreement with established practices, I have used a less orthodox notation in the transliterations.

22 Jens H$yrup

TABLE 1. The Numerals Used in the Fara Tablets, together with Their Corresponding Number Words.

0 =diS =I

0 =u = IO

0 = Ge.5 = 60 = I,0

ID = Ges'u = 600 q IO,0

0 = Sar = 3600 q l,O,O

0 0 = s'aru q 36000 q lO,O,O

Note. Observe that the signs as well as the words for 600 and 36000 are composed from 60 * 10 and 3600 * 10, respectively (see [Powell 1972a, 73.)

HM 9

These numerals are found on the two tablets discussed (see Figs. 1 and 2). The special symbol for 36000 used in No. 671 is probably a mistake; it has slipped in from the system of area notation [Powell 1972b, 2181.

TABLE 2. The Basic Capacity Units in Use in the Fara Tablets.

1 sila

1 ban = 10 sila ( = l(u) sila !

1 bariga = 6 ban = l,o sila I = l(Ge.3) sila )

7 qur-mab = 8 bariga = 8,o sila ( = 8(Ges') Sila )

Note. It should be noticed that this system differs somewhat from the systems in use 200 years later [Powell 1976, 423; Lambert 1953, 2071.


The sila is a measure of capacity (about 1 liter). Tablet 50 is then to be interpreted in the following way: The contents of one storehouse of grain are distributed to a number of men, each man receiving a ration of 7 sila. Consequently, 45,42,51 rations (i.e., 164571 rations) are distributed, and a remainder of 3 sila is left over. As pointed out by Lambert (11953, 2061 and a correction in [1954, 1501), this presupposes that the contents of a standard granary (or some specific granary which is dealt with in the exercise [3]) is known. Further, if the exercise is performed correctly we may conclude that the contents of the granary should be 40,O gur (4 (tjeh) gur), with the qur in question (the gur-matf) equal to 8‘0 sila in Fara (see Table 2 and [Lambert 1953, 206 f.]).

Tablet 671 would of course lead to a different result. How- ever, this tablet has obviously been written by a much less competent student, and since, furthermore, the resulting magnitude of the contents of the granary is much less simple, we may con- fidently follow tablet 50.

The first implication of this concerns metrology. The very simple structure of the granary led Lambert [1953, 2071 to include it in the list of metrological units used in Fara. Accord- ing to A. A. Vaiman, however, this is very doubtful since such a use of the term is attested nowhere else [Marvin A. Powell, private communication]. On the other hand, the use of the term guru7 used in two parallel, somewhat later royal inscriptions from Laga: to designate not the building but the quantity of grain which can fill the granary, seems beyond doubt. (Entemena, Cone A, 1121, II25, IV", and Cone B, corresponding places; [Sollberyer 1956, 37 f.; and Thureau-Danyin 1907, 38-411). Upon this point there is agreement between the otherwise wildly di- vergent interpretations of the passages in question [4]. Toward the end of the third millennium, the term meant 1 (.&r) royal gur of 300 sila each [Thureau-Danyin 1932, 391, that is, 1080000 sila, which is quite close to but yet different from our assumed Fara value. Thus, it might appear that the best conjecture is that the guru7 was not a real metrological unit, but rather a standard expectation concerning the contents of a physical storehouse, relatively unaffected by changing metrological conventions but always expressed in round numbers in the current metrology.

In any case, it must be regarded as justified to interpret the guru7 in our tablets as 4 (pe8u) qur, or, equivalently, as 5,20,0,0 (=1152000) sila (in the rest of the paper I shall assume this result). The problem dealt with on the tablets is therefore a formal division problem. It is probably the oldest such problem known, even though practical division problems presumably must have presented themselves at a much earlier date to the temple administrations. So, the method used to solve the problem is of some interest in connection with the problem of the development of Mesopotamian mathematics--not least because later Mesopotamian mathematics differs from all other mathematical tra- ditions by its use of reciprocals for divisions.

24 Jens H&-up HM 9

FIG. 2. Tablet 50. At bottom the lower right corner of the tablet ("3 sila of grain left on hand"). Photographs kindly supplied by Istanbul Arkeoloji Miizeleri Miidiirlii$i.

HM9 Sumerian Division Problem 25

Tablet 50 (Fig. 2) was first discussed at length by Genevisve Guitel [1963]. She proposed--mainly because this is a way in which the division can be performed and because of the visual impression offered by the arrangement of the result--that the scribe "ait utilise une m&hode absolument analogue a la pratique moderne d'une telle division," calculating the number of sila to a guru7 and expressing it as 32 h-u 153, and then performing a normal "long division" through the levels defined by the successive numeral symbols. As long as only a statement of the problem and a correct solution were known, nothing better could be done.

Powell [1976, 432 f.] discovered that tablet 671 contains the same problem, assuming a copying error on the part of Jestin. (Such an error is confirmed by the photograph.) This parallel, but erroneous, calculation was used by Powell as the basis of a new approach. He found Guitel's absolutely modern method suspicious, proposing instead the possibility that the Fara scribes used the only method of division attested in later Babylonian mathematics, namely, multiplication by the reciprocal. Powell correctly pointed out that in order to obtain the correct answer (that of No. 50), l/7 has to be calculated to four sexagesimal places. Since 1 guru7 = 5,20,0,0 sila, the procedure would be as follows:

(1) 5,20,0,0 - 0;8,34,17,8 = 45,42,51;22,40,

(2) 45,42,51 - 7 = 5,19,59,57,

(3) 5,20,0,0 - 5,19,59,57 = 3.

So, the number of men is 45,42,51, and 3 sila are left over-- just as stated in No. 50.

This calculation was carried out in the full sexagesimal system where it can, of course, be made. For the moment we leave aside the question whether such a calculation could be made by the Fara scribes.

Apart from historical continuity the main support for Powell's proposal is that it makes sense of the wrong result of No. 671: As stated by Powell, this wrong result is obtained if one uses 0;8,33 (=57/400) instead of 0;8,34,17,8, . . . for l/7; then the number of men is found to be 45,36,0, and no remainder is left. If no other explanation of this result could be found, this would support Powell's interpretation, and consequently the idea that an equivalent of the full sexagesimal system was in use in Fara. In any case, after Powell's parallelization of the two texts it should no longer be possible to neglect the error in No. 671 as a source of information [6].

26 Jens H&rup HM 9

Reverse

Obverse

Obverse, lower edge

Reverse, upper edge

FIG. 3. Tablet 671. Photographs kindly supplied by Istanbul Arkeoloji Miizeleri Miidiirl$ii.


There is, however, another way to carry out the calculation which not only makes the error committed on tablet 671 possible (as does the use of reciprocals), but also implies that this error is one of the most obvious of all possible errors.

This other method is suggested by the fact that quantities of grain greater than the gur were measured in gur and expressed by the standard numerals listed in Table 1, while smaller quantities were measured by subunits designated by a special series of symbols, a series which runs parallel to the series of standard numerals but which does not contain the gur and its multiples as round sexagesimal multiples of the smallest unit, the sila (see Table 2). We may therefore suppose that the first way in which the student would think of the granary would be as 4 (5reBu) gur. If we divide this number by 7, the result is 5,42 and a remainder of 6 gur. If we forget about the remainder and multiply 5,42 by the number of sila to a gur, we get 45,36,0-- just the result of No. 671!

Few other procedures lead to this result in a correspondingly simple way. Those I have been able to devise seem too artificial to be taken seriously into account [7]. Thus, a plausible explanation of No. 671 is that the scribe divided the number of gur by 7, forgot about the remainder, and multiplied by 8,0.

Truly, No. 671 was written by "a bungler who did not know the front from the back of his tablet, did not know the difference between standard numerical notation and area notation, and succeeded in making half a dozen writing errors in as many lines" [Powell 1976, 4321. However, several indirect arguments suggest that the able student who wrote No. 50 followed the same path as the bungler, though with greater success.

First, from all we know about the methods of Mesopotamian mathematics education, it consisted largely of working out specific problems, most often with many examples of similar type - It is not likely that the methods of the Fara school were at a theoretical level higher than those of the Old Babylonian scribal school--in other respects, at least, Old Babylonian teaching followed the very characteristic pattern which had been created many centuries before the Fara tablets were written (see [Falkenstein 1936, 46 f.1). Of course, teaching by means of examples does not prevent students from understanding the mathematical principles involved; but in most cases such an understanding will not inspire the average student (and a forti- ori not the dunce or the beginner) to invent a method radically different from the standard one. A student might err by dividing the number of gur instead of the number of sila, but then it is unlikely that he would almost rectify his error by multiplying by the number of sila to a gur.

Second, the above-mentioned custom of expressing large volumes in gur rather than in sila coincides with what appears to be the general method of thinking of large quantities. Certainly,

28 Jens H&yrup HM 9

the same applies to area measures: "The notation beginning with bur and its higher multiples runs exactly parallel to standard numerical notation" [Powell 197233, 1751. This general tendency--to count in terms of the greatest unit--makes a calculation in two steps much more natural than a conversion of the guru7 directly to sila, provided, of course, that the Sumerians could conceive of and perform the calculation correctly in two steps. (This is discussed below.)

Third, we may ponder the possible alternative strategies. Conversion to sila followed by a "long division," discussed above and in [6], appears to be incompatible with the result of No. 671. Another alternative is the one proposed by Powell.

It will appear from the discussion of No. 50 (above) that this calculation cannot be performed by the numerals which have come down to us from the Fara period (Table 1); even circumven- tion strategies such as the factorizations suggested by Guitel and Bruins will not do. This, however, is an objection of restricted value: For one thing, the numerals found on the material which has come down to us may very well be incomplete. While small numbers occur on most tablets from Fara, the s'ar and the garu occur so rarely that it may well be that still higher numerals either have not survived or have escaped notice completely [8].

It is less probable, although not impossible, that an alternative notation system, with genuine place value features, was already in existence; in fact, even in Ur III where the place value notation is attested this system is an alternative notation, used for marginal and intermediate calculations (see [Ellis 1970, 267 f.; Powell 1976, 420 f., 435 n. 61). But the existence of a real place value system is not a necessary pre- condition for the performance of division via multiplication by the reciprocal. Even in the Fara notation, as we know it, simple divisions may well have been performed by means of "multiplicative complements" followed by a shift of sexaqesimal "order of magnitude" [9].

Finally, we should remember that numbers can, for the purpose of calculation, be represented by means other than written nota- tions. Denise Schmandt-Besserat's recent discoveries 11977, 1978, 19791 of the pre-Sumerian use of small clay tokens to represent numbers, points to a possible basis for an abacus- like representation (requiring not necessarily an abacus board or frame). Moreover, archives from Nuzi from the second millenium B.C. show that material counters, probably related to the pre-Sumerian ones, were used for recording purposes even at this time [Oppenheim 19591 [lo]. On the other hand, in a problem discussed by Powell [1976, 426 f.] a student from c. 2200 B.C. is apparently groping after the basic idea of the extension as libitum of sexaqesimal notation; the way in which he loses track of the correct sexaqesimal place suggests that he was working with a mental construct and not with a notation or a material representation.


All in all, we must conclude that the existence in the Fara period of techniques permitting a solution of our problem in agreement with Powell's suggestion remains an unproven hypothesis. Therefore, the assumption that the scribes responsible for the two tablets followed the same path--that of a division in two steps--seems all the more likely.

However, before this assumption is accepted two questions must be asked: Did the Fara scribes possess the technical com- petence to carry out the correct solution given on tablet 50 by a division in two steps? And, assuming this technical com- petence, is it reasonable to assume that they would have thought of using a division in two steps?

The result obtained in No. 671 will help us answer the first question: Assuming that the two-step interpretation of this wrong result is correct, 40,O is correctly divided by 7. (Although the remainder is dropped, we may assume that the remainder could be found by any competent scribe--if it was not yielded directly by the calculation, it could be found by multiplication and subtraction once the quotient was known.) Also, 5,42 is correctly multiplied by 8,0. the result being 45,36,0.

Would these skills suffice technically to solve the problem entirely? Probably in the following way: 40,O gur are divided by 7. The result is 5,42, with a remainder of 6 gur. The 5,42 are multiplied by 8,0 (the number of sila to a gur), and the result is 45,36,0 (men). So far we have done precisely what seems to have been done on No. 671. The remainder is converted to 6*8,0 = 48,0 sila; such a step does not explicitly appear in No. 671, but the ability to perform it is inherent in the metrological systems used by the Fara scribes in their accounting. What sense would be left to such systems if a unit could not be converted to a smaller one? Further, the conversion is concep- tually related to the multiplication by 8,0 in No. 671, even if this is not a simple conversion of units.

The remaining 48,0 sila are then divided by 7; this step is similar to the division of 40,O by 7 in No. 671. The result is 6,51 (men), which is added to the 45,36,0 (men) already found; this addition is not different from the addition probably inherent in the multiplication of 5‘42 by 8,0, and neither does it differ from the bulk of additions necessary in ordinary contemporary accounting. The result is 45,42,51 men, as stated in No. 50. Besides, 3 sila are left over, as also stated. So, the method and the abilities revealed in No. 671 indicate suf- ficient technical ability to produce the correct result of No. 50. Moreover, this analysis implies that the apparent result of No. 671 is not necessarily a wrong final result; it may be an intermediate result in a calculation which the student did not know how to complete, or which he was unable to complete on this tablet where no empty space was left (see Fig. 1 and 3).

30 Jens Hhyrup HM 9

Technical ability is one thing, how to use it another. Would the Sumerians have been able to conceive the idea of a division in two steps? Does this not require the explicit recognition that multiplication and division are interchangeable mathematical procedures? And is such a recognition to be expected at this early stage?

Of course we cannot exclude the possibility of this recognition. But a comparison with Middle Kingdom Egyptian mathematics suggests that it is not to be expected: Middle Kingdom mathematics constitutes a coherent and relatively well-developed structure [ill. Still, the vocabulary which it used until Demotic times to describe multiplication and division is fetched from counting [Peet 1923, 22 f.; Parker 1972, 7 f.]; the Egyptians seem not to have thought of these as independent procedures.

On the other hand, the Sumerians need not have thought of interchangeable procedures in order to have conceived the idea of a two-step division. The method can just as well be described quite intuitively, as follows: One man gets 7 sila. Since 1 gur is 8,0 sila, then 8,0 men will receive 7 gur. 7 gur are contained 5,42 times in a guru7 of 40,O gur, and 6 gur are left over. So, 8,0 men can be paid 5,42 times, and the 6 gur may provide for a supplementary number of men, etc. [12].

This pattern of thought is so close to concrete experience that it should be accessible to a mathematical culture in which problems such as No. 50 could be solved. However, it does not follow from the simplicity of the argument that precisely this form of reasoning was used, but only that a simple argument could produce the two-step calculation. In any case, we may say that the two-step-calculation is a well-supported hypothesis.

CONCLUSION

How much does this analysis of the tablets reveal about the general character of mathematical thought of the Fara period? More specifically, will it tell whether the concept of place value was already on the way? And further, can we be confident that the methods of the specific problems found on the two tablets reflect the normal customs of the time?

No traces of a concept of place value were revealed in the treatment of the problem. So, the tablets do not indicate that place value was already on its way. This absence of positive evidence is, however, inconclusive, because the problem dealt with on the tablets is peculiar in several respects:


The quantity to be divided is very large if expressed in sila.

On the other hand, the gur probably presented itself as a natural intermediate step; it may be that the guru7 was spontaneously thought of only as consisting of a number of gur--just as we think of a foot only as consisting of 12 inches, not of 144 lines.

The divisor 7 is irregular; i.e., it does not divide 60 or any power of 60. As pointed out by Powell [1976, 4331, 7 may very well have been chosen for the exercise because it was irregular.

So, although analysis of the tablets suggests a mathematical mode of thought closely connected to current metrology and not yet familiar with the concept of place value, it does not necessarily imply that these were general characteristics of Fara mathematics. Metrology may simply have been used to circumvent the particular difficulties of this specific problem. Other calculations--in particular, the intermediate calculations used in the solution of the problem (40,O + 7, 5,42 l 40,0, etc.)-- may, but need not, have been performed metrologically; may, but need not, have been performed by means of nonwritten representa- tions of numbers [13]; and may, but need not, have been performed by place value-related reasoning. The wide variety of elementary mathematical techniques--many different from ours--which are known from other places and epochs [141 demonstrates that there are numerous possible ways to solve the same numerical problem.

ACKNOWLEDGMENTS

I am grateful to Olaf Schmidt, Aage Westenholz, and especially to Marvin A. Powell for reading and commenting upon previous versions of this paper; to Aage Westenholz for invaluable assistance in the treatment of the tablets; to the Istanbul Archaeo- logical Museum for permission to reproduce the photographs; and to the staff of the interlibrary loans of Roskilde University Library for their ever patient assistance.

NOTES

1. The reading .!&-tag4 proposed as a conjecture by Powell on the basis of Jestin's reproduction should according to Aage Westenholz (private communication) be consolidated by the photographs. According to later work by Powell [1978, 182 ff.], the reading tag4 should presumably be replaced by the reading taka. The interpretation "left on hand" is supported by this analysis.

32 Jens H4yrup HM 9

2. Here I have omitted all considerations of the various quasi numbers used in various metrological systems (e.g., [Powell 1971; 197233; Friberg 19781; most of them are structurally similar to the quasi-sexagesimal system shown in Table 1.

3. Many things, not least of which is the enormous number of rations, point to the tablet's being in fact a mathematical exercise and not an administrative calculation.

4. Thus, according to Thureau-Dangin [1907, 391 and Sollberger and Kupper [1971, 721, Cone A, IV, 11, means "1 sar guru7s"; according to Barton [1929, 631 and Lambert 21956, 143 n. 11, the meaning is "1 s'ar [qurs, i.e., one] guru7." The difference amounts to that between realistic and imaginary quantities of grain. Still, everyone agrees that definite quantities of grain are meant.

5. Such factorizations have often been used in other historical contexts when there was the problem of a restricted numeral system, e.g., Ancient Egypt [Sethe 1916, 91, Shang China [Needham 1959, 133, Ancient Rome [Friedlein 18681, and the early Latin abacists as well as vernacular mathematical treatises of Medieval Western Europe [Bubnov 1899, 203-209 and passim; Henry 1882, 67 f.]. So, even the Sumerians may very well have hit upon the trick--indeed, a similar way of looking at things is suggested by the multiplicative structure of the number words 9e8-u and s’ar-u and the corresponding written symbols (see Table 1).

6. Yet this was done by Bruins [1978] in a critical abstract of Powell's paper. In principle, Bruins went back to Guitel's interpretation, with the difference that either he factorized 5,20,0,0 as 5,20 gar, or assumed the use of the full sexagesimal system for integers. Bruins' argument for this use of long division was based partly on the writing of the result of No. 671 in three lines. He neglected, however, to note that this writing is unsystematic; .%aru and xar are separated, while ges'u and ges' are written together. He also assumed (advancing it as a fact) that for "the division by irregular numbers a table of multiples is made"-- apparently an extrapolation from the much more recent Old Babylonian multiplication tables.

7. One is the deliberate use of 57/400 as an approximation to l/7. Since such an approximation is clearly not used in No. 50, I would discard it. The other possibilities all consist in the measurement of the guru7 in terms of units other than the gur-maa : 576 sila, 720 sila, 960 sila, 1440 sila, or 2880 sila.

None of these occurs in the rather meager evidence for Fara metrology, although all are simple multiples of the gur-mab, of the gur of 240 sila (also found in Fara), or of the gur of 144 sila known from later Laga: (see [Powell 1976, 423; Lambert 1953, 205 f.]). Measurement in terms of any one of these mag- nitudes is not likely to have occurred, unless that magnitude was (unknown to us) a metrological unit. If this were the case,


however, we would, in principle, be brought back to the division in two steps, but using another gur rather than the gur-ma;.

8. A preliminary and elementary statistical analysis suggests that the existence of unnoticed higher numerals is improb- able, but far from impossible. Of the first 173 tablets in Jestin's randomly organized collection [1937], about 162 contain numerals or metrological symbols (the precision is limited because of Jestin's not always quite reliable reproductions). "1" is present on about 145 tablets, "10" on about 75, "60" on about 27, "600" on about 21 "3600" on about 10, and "36000" on about 3. This fits beautifilly to a straight line in a log-log diagram. By extrapolation, the next numeral of the series ("216000") should be expected to occur on approximately l/2 to 1% of all tablets, if it existed. On the other hand, the only argument for the use of log-log analysis is that it works between "1" and "36000".

9. It is possible, perhaps even plausible, that one of the motives for the introduction and general adoption of the full place value system was the fact that it permitted the generali- zation of the principle of multiplicative complementarity. (I have discussed this in another connection in [H$yrup 1980, 19; 84 ff., nn. 37, 38, 431.)

10. Surely, one should not conclude too much from the Nuzi find. Nuzi was a Hurrian city and need not have inherited its seemingly rather primitive administrative techniques through Sumer. True enough, mid-third-millenium clay tokens are found in Ur, Kish, and a number of other cities ([Schmandt-Besserat 1977, 9 f., 14, 201; and Schmandt-Besserat, private communication) . However, using the published background references for Ur and Kish [Woolley 1934; Mackay 19291, I have found nothing connecting these finds with any specific use, except, on one hand, an interesting similarity between the tokens and the men and dice used in board games (cf. [Woolley 1934, I, 175-178; II, pl. 95, 98, 1581 and, on the other hand, the fact that all third-millenium tokens seem to belong to categories with a numerical, rather than a conceptual, significance. In any case, the very rich token system of the fourth millenium again becomes very simple (Schmandt-Besserat, private communication). In a recent publication, Stephen J. Lieberman 119801 suggests that the tokens were used as calculi in an abacus tithout a counting board. Lieberman introduces some interesting considerations involving the use of the two different ways to write numbers ("curviform" and "cuneiform": see the drawings in [Powell 1972a]) in later Sumerian accounting practice but he does not decide (nor even mention that it is a problem) whether the hypothetical abacus was analogous to the later place value system (this is almost claimed on p. 3421, or it was isomorphic with the system of curviform numerals as shown in Table 1 (this isomorphism is implied by the arguments on pp. 344 f.).

34 Jens H$yrup HM 9

11. The cumbersome character of the Egyptian unit fraction calculations should not be taken as evidence for lack of mathematical coherence. Further discussion of the character of Egyptian mathematics may be found in [H$yrup 1980, 31 ff.].

12. I am told by Marvin Powell that A. Vaiman of the Lenin- grad Hermitage has mentioned a similar method of thinking to him (private communication).

13. One possibility is the use of some abacus-like representation, for example, one based on Schmandt-Besserat's clay tokens. Another is the finger reckoning of the Ancient and Muslim world, which the Muslims thought of as "arithmetic of the Byzantines and the Arabs" [Saidan 1974, 3671, and which Saidan conjectures to have descended from Greco-Babylonian man- ipulational practices. (Still, an Egyptian cubit rod, repro- duced by Karl Menninger [1958, 231, carrying pictures of finger positions instead of the corresponding numbers suggests that Egypt may be a more plausible origin for Greek and Muslim finger reckoning than Mesopotamia.)

14. A few examples should be mentioned, all belonging to unsophisticated mathematical cultures:

The Ancient Egyptian multiplication by duplations and division by filling-out;

the "Russian peasant multiplication" described by Plakhovo [18971, which is related to, but yet different from, the Egyption method;

the awkward but very down-to-earth division prac- ticed on the Medieval "Gerbert" abacus (see, for in- stance, [Smith 1925, 134 f.] or [Yeldham 1926,'42 ff.]).

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36 Jens Hbyrup HM 9

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