Köbel began by describing the process of turning a surface into a counting board by drawing lines and, optionally, labelling them; he seems to have thought pre-marked boards were the exception rather than the rule. On Köbel’s board the lines were horizontal to the user rather than vertical, a ninety-degree rotation compared with the Greek boards. But the sequence of lines denoting fives and tens was exactly the same as before. As with the Greek and Roman counting boards, beyond the ones line there was a set of further lines for fractions, reflecting the denominations of currency. Like some of his contemporaries, Köbel suggested the very slight innovation of putting a star next to the thousands line to help its quick recognition. He also suggested that, rather than labelling the lines using writing, you simply lay a coin of the appropriate value at the end of each one. One of the beauties of the counting board was that no essential modifications of its form or its procedures were needed in order to calculate with non-decimal systems of currency, weight or measure: you could simply assign, say, ounces to one line and pounds to the next, or inches, feet and yards, or any other system that was required.
Scene at the counting board, from Köbel’s title page.
Deutsches Museum, 1987-178. CC BY-SA 4.0.
Köbel’s procedures always started with the smallest denominations. For the ones in your number, set out counters on the ones line; for the tens, counters on the tens line, and so on. To add another number to this first one, start with the ones again and place extra counters corresponding to the ones in the new number; then for the tens and for the hundreds. If at any point you find you have placed five counters on a ones (or tens, or hundreds) line, you must remove them and substitute a single counter in the adjacent fives (or fifties, or five hundreds) line. If there are two counters on a fives (fifties, five hundreds) line, remove them and substitute a single counter on the tens (hundreds, thousands) line.
Subtraction was scarcely more difficult, although in order to have counters to take away from a given column you might briefly have to substitute five ‘ones’ counters for a single ‘fives’ counter: or the equivalent elsewhere on the board. To multiply or divide, though, it was necessary to have memorised the multiplication table, and to work with fractions you had to understand the system of coin denominations in operation.
Köbel also mentioned that it was possible to divide the board into more than one section by adding vertical lines, gaining the possibility of representing two or more numbers simultaneously, perhaps for the purpose of keeping track of a complicated calculation. Other writers advocated another slight change: using the spaces between the lines, rather than the lines themselves, to place the counters. Functionally this was a trivial shift, making no real difference to the function of the counting board; but it changed the physical look of (some) boards into a set of – possibly different-coloured – bands. If this was combined with the lengthways division of the whole board, the whole would start to look something like a chess board. The special form of board used for English public accounts was indeed known as the ‘exchequer’, a word from the same root as ‘chess’.
Meanwhile, Blanche’s innovation – counters specially struck for use on the counting board – showed a similar, and surprising, longevity. It could have been a mere fad, the toy of a particular queen with a particular interest in administration and accounts. But it was not. Jetons were struck for every subsequent French monarch, from the thirteenth century until the Revolution in 1789. From the royal court itself, the use of specially struck jetons spread to the various departments of the royal household and beyond: in due course the orders of chivalry, the assembly of French clergy, the households of the queen and the dauphin, and the treasury in its various departments all produced their own series of jetons. Provincial administrations and personages had their own jetons struck, and it seems that not only the royal mint in Paris but mints around the country – Tours, Dijon, Toulouse, Grenoble, Arras, Laon, Valenciennes – were also producing jetons at the peak of their popularity. They were made by the same simple process – hammer striking – as early coinage.
Early examples were in brass, but by the fourteenth century silver was in use, and subsequently gold, meaning that jetons had real value as bullion. The custom of giving a set of one hundred jetons to certain officials each year, nominally for use in their work, took on the character of a supplement to their salaries: something like an annual bonus. It may have been possible either to sell the jetons back to the treasury for their value as metal, or perhaps to commission their transformation – melting, casting and striking – directly into coin. One of the things that makes jetons intriguing is how closely they superficially resembled coins while always remaining distinct from them. Their shape and size and the choice of metals made them coinlike in all but their detailed design. And their designs tended towards the celebration of notables and their exploits, with a proliferation in later centuries of portraits, blazons and mottoes: just like coins. Some bore as part of their design a statement that they were not coin of the realm, but the practice of passing them off as money nevertheless led to the long-lived proverb ‘false as a jeton’.
The peak of jetons’ range and complexity in France came under Louis XIV in the seventeenth century; in the 1680s, the jetons struck for the central institutions of the French government numbered 1,200 in gold and a staggering 48,000 in silver. The issuing of jetons continued sporadically for another century, up to about 1780. But their close connection with the royal courts and institutions meant that they entirely ceased to be produced after the Revolution, and that the overwhelming majority of jetons extant at that point – with their royal and noble portraits and inscriptions – were swiftly melted down.
But for the centuries between Blanche and the last Louis, there seems no reason to doubt that actual computation in royal households, financial departments and elsewhere in the government was carried out using the jetons struck in such numbers for that purpose: that jetons really were preferred as counters, even though functionally they represented no improvement whatever over the use of pebbles or coins for the purpose.
Jetons also became popular outside France. There were important centres of their manufacture and use among the merchant and banking community of northern Italy – the Lombards who exported their goods and their practices to almost every corner of Europe – and subsequently at Nuremberg, whose ‘reckoning pennies’ came to be well known and widely exported throughout the sixteenth and seventeenth centuries. Some were self-referentially stamped with images of an accountant seated at a board using jetons.
From the seventeenth century onwards, jetons gained siblings in other kinds of counters with overlapping functions and physical characteristics. The habit of giving away sets of jetons as a new year bonus to employees or a birthday gift to friends spread to institutions such as religious establishments, university faculties and learned societies, which took to giving jeton-like tokens to their members in exchange for attendance at meetings or the performance of other duties. Lead or brass were often used (these tokens are sometimes distinguished from jetons by the term meraux). As quasi-money which could in some circumstances be – illegally or semi-legally – used as small change, or might be legitimately handed over to the issuing authority in return for money, they have some overlap with jetons proper. The difference was that there was never any suggestion that they might be used for calculation on a board.
Further afield still from the central purpose of jetons, there are references to their use for marking points at games, a safer alternative to putting actual money on the table during play. In due course, tokens were made for this specific purpose, sometimes in special shapes so that they could not easily be confused with other tokens or coins: the ancestors of modern-day poker chips. And the reluctance of certain governments – notably the English – to strike small-denomination coins led to several periods when merchants produced their own, in copper, brass or lead, counters that served the dual functions of small change and advertising. Tens of thousands of different types were produced, and they were manufactured on an industrial scale. There came to be a fad for collecting these ‘merchants’ tokens’, leading some makers to issue commemorative or decorative tokens specifically designed for collection. Some collectors even had their own tokens specially manufactured to give to their friends.
Although there are certainly separate categories of object here, it also seems certain that between jetons and their siblings there was a long history of influence, imitation and – surely in some cases – crossover use. There is a sense, then, that by the seventeenth century the counter-saturated culture of classical Athens had been recreated in northern Europe, with a range and complexity of tokens in circulation as great as anything ever seen anywhere. And, consequently, that for most people, a primary experience of number and counting came through the fingers, from counters and their manipulation.
By this time, though, a new leaf was being turned as far as counting and calculating in Europe were concerned. Blanche’s clerks used Roman numerals only to record the outcome of their calculations; but new techniques of calculation on paper had become ever more visible since the high Middle Ages, and it was during the sixteenth and seventeenth centuries that they at last took over from counters and counting boards. Today the number symbols are called the Arabic numerals: but their ultimate source was India.
5
Number symbols from India
Number symbols were a crucial addition to the human repertoire of ways of counting, when they were invented in Sumer and, concurrently or soon after, in Egypt. But a majority of the people alive today are familiar with one set of number symbols above all: the ones that look like this – 0 1 2 3 4 5 6 7 8 9 – and that obey a special set of rules about their positions and their values. They are particularly associated with calculation, but their spread is certainly one of the main branches on the story of human counting.
They were first used in texts in Sanskrit in the first millennium CE, but theirs has been a remarkable journey from language to language and from country to country. By the twelfth century they had a history of hundreds of years in South Asia and were well established in the Arabic-speaking world as well. And they were just beginning to appear in texts written in Latin, too. Over the following centuries they would conquer Europe and become a key element of modern accounting, of data-gathering and statistical analysis: techniques at the heart of modernity.
Was there anything inevitable about the rise of the Indian number symbols? Surely not: South Asia has other systems of number symbols with the same structure, that show no sign of spreading across the world. Happenstance, as in almost any historical process, played a leading part. Are they the last word in human counting? Again, surely not: they have not yet survived as long as the cuneiform or the hieroglyphic number symbols, and in the last few decades they have lost a great deal of ground to binary codes as a way of counting, of representing and manipulating numbers. There seems no reason to suppose that the process of growth and change in human counting is at an end. Their importance, and their interest, lies instead in the myriad places they have been and things they have done over the last thousand years and more. And their story begins in India.
Bhaskara II: Brahmi numerals
South Asia is a region where language, culture and counting have deep roots. Indo-European languages have been spoken in the Indian subcontinent from at least the second millennium BCE; the most prestigious was Sanskrit. Urbanisation in the first millennium and the exchange of goods and ideas with India’s neighbours – Greece, Mesopotamia and China – contributed to a thriving and numerate culture.
The earliest surviving texts in an Indian language are the Vedas, in an archaic form of Sanskrit, dating from perhaps 1500–500 BCE; around 500 BCE, Sanskrit was codified and effectively frozen as a learned language. Its descendants include many languages and dialects spoken today, including Hindi, Gujarati and Bengali.
Evidence for counting in the region includes a tradition of wooden tallies which survived into the twentieth century, and references in ancient texts to a counting board on which clay counting pieces took different values depending on their position. Tallies and boards, like the Sanskrit language itself, adopted a decimal system. Inherited from Proto-Indo-European, like those of the European languages, the words were eka, dva, tri, catur, panca … (Similarities with number words in Greek and Latin were in fact one of the first pieces of evidence to be collected showing the relationship of those languages with Sanskrit.) Unlike many languages, Sanskrit has a wholly regular decimal system of number words, with a separate word for each new power of 10 (contrast English, where only ten, hundred, thousand and the higher powers of 1,000 have distinct words; other powers of 10 are named using compound terms like ‘ten thousand’).
Counting words occurred in some of the Vedic hymns (‘You, radiant Agni, are the lord of all offerings; you are the distributor of thousands, hundreds, tens of good things’). In some cases long sequences of powers of ten were incorporated:
Hail to a hundred,
hail to a thousand,
hail to ayuta [ten thousand],
hail to niyuta [hundred thousand],
hail to prayuta [million],
hail to arbuda [ten million],
hail to nyarbuda [hundred million],
hail to samudra [billion],
hail to madhya [ten billion],
hail to anta [hundred billion],
