Where do figures come from? When did people learn how to count? Artless questions from students spurred Moroccan-born France-based mathematics professor Ifrah to this odyssey of origins. The result: a work of mammoth scholarship that traces mankind's tools and techniques of reckoning from Stone Age to modernity. Most math teachers make passing reference to the beginning of numbers in early society in the form of one-to-one correspondences: matching pebbles with numbers of sheep; cutting notches in sticks to record the passage of days--practical aids to the social economy. Ifrah enlarges on these themes with myriad examples worldwide, presenting hundreds of illustrations of ancient tablets, manuscripts and the like, with an occasional fresh insight of his own or other scholars. For example, he thinks it is far more likely that Roman numerals were not alphabetic symbols, but grew out of tallying: the ""I"" ""V"" and ""X"" would then derive from the almost universal tendency to mark off successive straight lines for 1 through 4, then draw an opposing line to indicate 5, and so on up to 10 where another cross line would set it off. As for reckoning, Ifrah guides a world tour of counting boards (the origin of the word ""exchequer"") and the various forms of the abacus. (It is comforting to know that the Romans were adept at using the abacus and did not have to directly confront the cumbersome problem of multiplying with their numerals.) Ifrah also makes side excursions into mysticism with the Greek, agnostic and cabalistic infatuations with associations between numbers and letters of the alphabet (the latter, a Phoenician invention). It is India that becomes the numerical ""jewel in the crown,"" however. Ifrah builds a strong case for the primacy of Indian invention, not only of place notation (e.g., a number is read as 4, 40 or 400 depending where it is positioned when writing out the number) but also of using a decimal notation with 9 symbols and--miracle of miracles--the zero. All this happened some 1,500 years ago, Ifrah documents, from whence the idea spread East and West, picking up the more familiar Arabic form of numerals along the way. The Babylonians, Chinese and Mayans share honors with the invention of the principle of place notation, but their systems either lacked a zero or exhibited other irregularities. Parts of this excellent compendium are repetitious; parts too detailed for the general reader. Occasionally, too, one might question a glib statement. On the whole, however, a superb job of omni-gathering, making this a fine reference volume in an area where little recently has been published.