New York, NY: Wiley. xxii, 633 p. £ 32.50 (2000).

“The main aim of this two-volume work [see

Zbl 0969.68001 for Vol. II] is to provide in simple and accessible terms the full and complete answer to all and any questions that anyone might want to ask about the history of numbers and of counting, from the prehistory to the age of computers.” With this sentence begins the `Foreword’ of the present volume (neither the title page nor the imprint or the table of contents indicate that this volume is the first of a two-volume publication, as is said here). This exuberant claim is expanded in more detail in the `Introduction’ as follows (pp. xvii/xviii):
“$\ldots$ I think I have brought together practically everything of significance from what the number-based sciences, of the logical and historical kinds, have to teach us at the moment. Consequently, this is also probably the only book ever written that gives a more or less universal and comprehensive history of numbers and numerical calculation, set out in a logical and chronological way, and made accessible in plain language to the ordinary reader with no prior knowledge of mathematics.”
One does not need a great deal of imagination to see that it is impossible to fulfill such a claim. Even without being a specialist one will realize that the vast history of number systems and computational practices of the past, as developed in a great variety of cultures, must needs place the historian before numerous difficult and often unsolvable questions. Furthermore, the historical development seldom, if ever, conforms to a wholly logical structure. If the latter is at all possible, it will be a rational reconstruction of a rather haphazard natural growth of number words, number symbols, and methods of calculation; to arrive at it requires a good deal of simplification and abstraction from the actual historical process. To present both, the historical and the logical story, in all details in one work, without distorting at least one side of it, is virtually impossible.
Anyone familiar with the unfolding of a number system including its words and symbols in just one specific culture will appreciate the complexities of historical developments, and if honest, would hardly claim to be able to give “the full and complete answer to all and any questions,” let alone to do so for all of the developments that have happened elsewhere throughout the entire course of human history.
Before going into further details, it must be said that the present volume (this reviewer has not yet seen the second one) is an English translation of the author’s “Histoire universelle des chiffres”, first published in French in 1994 by Editions Robert Laffont, Paris. This in turn is the second edition of Ifrah’s one-volume book which appeared under the same title in 1981 (Editions Seghers, Paris). Several translations or re-editions of this first edition have been briefly reviewed previously in this Zentralblatt [see

Zbl 0589.01001;

Zbl 0606.01023;

Zbl 0686.01001;

Zbl 0758.01017].
In the meantime, a vivid discussion about the merits and shortcomings of Ifrah’s “Histoire universelle des chiffres” has been conducted both on the Internet and in the French journal “Bulletin APMEP” (Bulletin de l’Association des Professeurs de Mathématiques de l’Enseignement Public), nos. 398 (Avril-Mai 1995) and 399 (Juin 1995). This journal asked six scholars (specialists on the cultures and number systems of China, Egypt, Mesopotamia, India, the Arabs, and the Maya) to evaluate critically the chapters they were competent to judge. It published their comments in the issues just mentioned with highly unfavourable results for the overly ambitious author, a former school teacher. The chief rebukes were: 1) Ifrah is inclined to present hypotheses as established facts; 2) often, where hypotheses are labeled as such, he makes unfounded generalisations; 3) the way Ifrah uses his sources (scholarly articles, archaeological findings, etc.) is doubtful in a number of cases -- sometimes he presents a one-sided selection, or he offers interpretations that do not correctly reflect the statements and opinions of the authors, or he may simply have misunderstood something; 4) sources are even fabricated by mingling material from widely different areas or time periods in order to substantiate exaggerating claims.
One such claim, to give an example, is the unfounded assertion that an abacus was used for practical computations in numerous early cultures. This might be advanced as a reasonable hypothesis, but as long as there is no (or insufficient) archaeological evidence it ought not be presented as an established fact. However, among the material presented here -- including new chapters that have been added in this second edition -- there is, for instance, a “reconstructed” Babylonian abacus, complete with illustrations, and instructions for calculations as they (in the authors’s opinion) were carried out, without the least shred of evidence that such a device ever existed, since no example survives in either the tangible archaeological or written record of an abacus from ancient mesopotamia.
An issue of major concern is of course the question of how our present decimal place-value number system, with its unique symbols, developed and succeeded in being accepted virtually everywhere throughout the world. Basically, it is well known that Europe received it from the Arabs who in turn had become familiar with it by some source (or sources?) from India. But exactly how, when, where and by whom the various steps of this century-long development were taken, is still mostly shrouded in darkness. The author claims to “be able to tell the story much more rigorously and to track the invention of the Indian system very closely indeed”. Under the revealing subtitle “Proof of the Event” he summarizes the method that, so he believes, entitles him to put forward such a surprising claim (p. 367). In his own words:
“In the previous chapter we offered a classification of written numbering systems that are historically attested, and through it we drew out a genuine chronological logic: the guiding thread, leading through centuries and civilisations, taking the human mind from the most rudimentary systems to the most evolved. It enabled us to identify the foundation stone (and, more generally, the abstract structure) of the contemporary numeral system, the most perfect and efficient of all time. And it is precisely this chronological logic of the mind which shows us the path to follow in order to arrive at a historical synthesis. A synthesis intended to show just how the invention of the numerals actually “worked”, and to place it in its overall context, in terms of period, sequence of events, influences, etc.”
The steps “to prove that India really was the cradle of modern numeration” are enumerated by Ifrah as follows:
“1. To show that this civilisation discovered, and put into practice, the place-value system;
2. To prove that this same civilisation invented the concept of zero, which the Indian mathematicians knew could represent both the idea of an “empty” space” and that of a “zero number”;
3. To establish that the Indians formed their basic figures in the absence of any direct visual intuition;
4. To show that the early forms of their symbols prefigured not only all the varieties currently in use in India and in Central and Southeast Asia, but also the respective shapes of Eastern and Western Arabic figures as well as the appearance of those figures used today and their various European predecessors of the same kind;
5. To prove that the learned men of that civilisation perfected the modern system of numeration for integers;
6. Finally, to establish once and for all that these discoveries took place in India, independent of any outside influence.”
To sum up: first the author constructs, by drawing from historical evidence that is taken from sources of very different origins, a “genuine chronological logic” of the development of number systems. He then uses this construction as the guiding thread for his “historical synthesis” of the formation and spread of the decimal place-value system -- the things he wants to explain or rather to prove “once and for all” (regardless of any future discoveries of historical sources that may yet come to light?). What is surprising is that Ifrah is quite aware of the pitfalls threatening the historian’s work. He even discusses some of them (e.g., on pp. 365-367). But he does not seem to see that he himself is a victim of them when he constructs his own chain of reasoning.
A typical statement is the following: “Considering the quantity and extreme diversity of the information contained in this chapter, it would seem appropriate to present a summary of all the historical facts which have been established concerning the discovery of zero and the place-value system.” All arguments that hitherto were put forward by the author, with some caution, as plausible or highly probable, are then summarized as historical facts, so that in the end not the slightest uncertainty about his hypothesis remains.
The above caveats must suffice for the general characterisation of the book. In comparison with the first edition, it has been greatly expanded: numerous tables and diagrams have been added (often drawings by the author), the text has become less concise, more wordy and perhaps thereby, the author believes his conclusions are more persuasive. Long lists of number names in various languages are presented, and even a 70-page “Dictionary of the Numerical Symbols of Indian Civilisation” is included. The 27 chapters of this volume of almost square format $(24\times 23 \text{cm})$ and printed in two columns deal with the following main themes: early numbering and counting, tally sticks, numbers on strings, the development in Mesopotamia (6 chapters), in Egypt, Greece, Rome, alphabetic number systems, China, the Maya, and (what was sketched above) the origin, rise and spread of the Hindu-Arabic decimal place-value system. The extensive bibliography is subdivided into works in English and those in other languages. The index of nearly 20 pages runs through three columns per page. A comparison of the table of contents with that of the first English edition of 1985 shows a great similarity; it is therefore not possible to make a well-founded guess about what the second volume will cover.
It is sad to see how the author has spent so many years of hard labour to amass such a vast store of information, but that he obviously failed to seek advice or dismissed the counsel of experts in the various disciplines relevant to understanding the complexities of the history of numbers in any universal sense.
Even worse is his pretentious approach in which facts and fiction are intertwined. Unfortunately, despite its additional length gained largely through the insertion of an abundance of tables and schemes, the second edition may appear even more convincing; the author’s more persuasive and insistent language would seem to guarantee its popular success.
This is an old dilemma: scholars hesitate to write a “universal history” because they do not feel competent for such an all-embracing undertaking; an interested layman has fewer scrupels and perhaps more fantasy, and fills the gap in the book-market by telling a more or less plausible or even a fantastic story. This catches the attention of the general public, and the story presented here becomes part of the general “Bildung” or folklore.
The great majority of popular readers may henceforth rely only on this book for information on the development of numbers and number systems. For years it may even serve as the principal source for lecture material, articles, and possibly yet other books, the author’s lax standards of accuracy notwithstanding.