New York: Viking Penguin Inc. XVI, 503 p. $ 35.00 (1985).
Originally published in French as ”Histoire Universelle des Chiffres” (Paris, 1981), this history of numbers and number systems was compiled by a mathematics teacher who some day had to reply to the question ”How did numbers start?” Born in Morocco as a Jew, he knew Arabic and Hebrew, as well as French and English, and thus during six years of research was able to collect a huge amount of source material from a wide range of publications. It is the richness in documents from both primitive and advanced cultures, which makes this publication unique: there is hardly a page that does not exhibit some numeral symbols or number words, number systems, illustrations (drawn by the author), comparative tables, facsimile reproductions, etc. The author has organized the wealth of material in six major sections, comprising altogether 29 chapters, in a somewhat systematic way (although the historical point of view is never suppressed).
Section I ”Awareness of numbers” deals with rudimentary techniques, section II ”Concrete counting” covers finger counting and the use of notches, pebbles, strings, and the abacus in its various forms. Section III ”The invention of numerals” mostly concentrates on Roman, Babylonian and Egyptian number systems, while section IV is concerned with numerals and letters, i.e. the use of alphabets as number symbols (Hebrew, Greek, Arabic, Syriac, and other numeral letters). Section V, called ”Hybrid numeration systems”, confronts the additive and multiplicative principles and presents, among others, the Chinese numeration system. In section VI, ”The ultimate stage of numerical notation”, that is the place-value numeration systems are introduced: Babylonian, Chinese, Mayan, and finally the Hindu-Arabic positional systems and their history are described.
The author, of course, is indebted to some of the major works on this topic [e.g. Geneviève Guitel, Histoire comparée des numérations écrites (Paris, 1975), or the English translation of K. Menninger, Zahlwort und Ziffer. 3. Auflage in 2 Bänden (1979; Zbl 0406.01001). Menninger placed more emphasis on the cultural context and devoted a whole volume to number words. Guitel’s very scholarly and voluminous account stressed the systematic point of view, subdividing the written number systems into three main types and ten different sub-types. Although the author’s book includes a ”Bibliography of works quoted or mentioned in this book” (pp. 499-502), a number of authors mentioned in the text (such as Cantor, Formaleoni, Fregius, Kewitsch, Lehmann-Haupt, Löffler, Wallis) are not cited in this bibliography. And in many cases the sources of illustrations remain anonymous; this unfortunately diminishes the value of this treasure-house.