THE UNIVERSAL HISTORY OF NUMBERS. By Georges Ifrah. Harvill pound;19.99.
Victoria Neumark is carried away by a dazzling, 600-page labour of love on the history of numerals
"I was born and brought up in the Countrie... I have, since my predecessours quit me the place and possession of the goods I enjoy, both businesse and husbandry in hand. I cannot yet cast account with either penne or Counters."
It is Montaigne speaking, sophisticated traveller, urbane intellectual, superb writer, successful aristocrat: a 16th-century Master of the Universe, to use Tom Wolfe's phrase from Bonfire of the Vanities. Yet, in an age where any old university might teach addition and subtraction but only the really top ones could aspire to multiplication and division, Montaigne felt no shame in admitting arithmetical skills inferior to today's Year 6s.
His "penne" would have been the newly current methods imported from India via the Muslim world; "counters", the time-honoured counting boards still in use at the end of the 18th century (the boards looked like chess boards, hence the British Exchequer).
Such vignettes sparkle throughout The Universal History of Numbers like fairy lights on a Christmas tree, turning Georges Ifrah's vast mass of material into a coherent symbolic structure. In focusing on the actual numerals rather than the story of mathematical thought, Ifrah has been able to link speculation about maths with as much cultural diversity as you can shake binomial expansions (first invented by the Chinese, centuries before Pascal) at. With Gallic gusto for the really hard questions, he moves easily from assertions about the cultural specificity of mathematical forms to celebrations of a "universal human aptitude" for such concepts as the zero (separately discovered by the Chinese, the Maya and the Babylonians, though none of them really knew quite what to do with it).
Georges Ifrah is a Jewish Moroccan teacher with "no special bent for mathematics". He dropped out of the French educational system to research his book, sponsoring himself through casual jobs and grants, "making a nuisance of himself" writing to eminent mathematicians and archaeologists. He is one of those oddball communicators, neither dull academic nor glib journalist, who can connect directly with readers' interest. His labour of love, a 600-page tome crammed with line-drawings, sample calculations, glossaries, ancient alphabets and writing systems and a 20-page densely printed bibliography, is more than a treasure-trove - it is a dragon's hoard of fascinating lore.
On any random spread you might move from the Hebrew mystics' play on the numerical values of letters in the Bible (known as gematria) to the widespread fear among the Chinese and Japanese of the number four (the Renault 4 bombed dismally in Japan for this reason) to Muslim magic and ciphering (where numbers and letters are used in each others' places) and on to the development of different forms of place value in India. Along the way, you will have found out how all alphabets are descended from the Phoenician traders of the third millennium BC; how our word for calculation comes from the Latin "calculus", meaning pebble, because people from fourth millennium BC Sumerians on used pebbles to count with; how it was impossible for the written Roman numerals ever to be used in actual arithmetic - the Romans used abacuses and writing in sand; and how Egyptians managed with only the two-times table. Oh, and don't forget how, if they had only not introduced a level of 60s into their vigesimal (20 as the base) system, the Maya would have streaked ahead as the mathematicians, since they had already mapped most of the heavens using only a jadeite tube and a wooden cross for focusing.
However, it is when it comes to extolling the Indian mathematicians who, probably as early as the fifth century of the current era (judging by the Jaina cosmological text known as The Parts of the Universe) invented the number system we use today that Ifrah reaches his peak.
And just why is that so totally wonderful, non-mathematicians may ask? Why is zero to nine so infinitely (so to speak) better than the Iqwaye counting on their hands, toes and elbows, the Greeks pressing the alphabet into service, the Etruscan notching his stick, or the Inca knotting coloured strings?
All of these were extraordinary, flexible recording devices, but they are inflexible, frustrating methods for operating arithmetic. Concrete methods cannot allow numbers to soar beyond basic principles of enumeration (there are this one, this one, this one); cardination (1, 2, 3); ordination (1st, 2nd, 3rd); and seriation (after 1 you add 1, then you get 2, then you add 1 and you get 3, and this does not vary, unless you have invariant place value, you get into trouble adding and subtracting, never mind multiplying and dividing).
While the reader's passion may not match the heights of Ifrah's, and attention may flag at trying out every bygone algorithm (step-by-step procedures named after the famous al-Khwarizmi, father of quadratic equations), it is possible to see, smell, almost taste Ifrah's intense satisfaction when, after all the civilisations from Ethiopia to Ecuador, from Cambodia to the Middle East have been surveyed, the Indian system unfolds and later arrives in Europe via the Arab empire.
Why are Indian-derived numerals so fantastic? They have no distracting visual link to any other symbol; there is an unalterable place value for each level above the base (hundreds, tens, units); and there is a zero to symbolise either nothing or a void in one level of quantity (either "nought" or, say, 102, where zero means "no tens"). This system is liberation.
The Christian Church, Ifrah concludes, fought "satanic" written or "Arabic" methods for centuries. Montaigne was not alone in being put off by the hideously complicated methods native to Europe. It was not until the French Revolution that everyone began to learn calculation. And so the frontiers of ignorance were rolled back. Not, perhaps, for the last time.