In writing, the number words were rarely spelled out as words: in the hieroglyphic script, numbers were expressed using symbols, which also used a decimal system. There were seven signs, for 1, 10, 100 … 100,000, 1,000,000; each was written up to nine times in order to denote a number. The signs were placed in descending order of size. This was thus a very similar system to the Sumerian and Akkadian one, but without the complication of 60 as a base in addition to 10.

Egypt had a long tradition of calculation, of complex systems of weights and measures, and of scribes whose role and powers were very broadly comparable to those of Sumerian and Akkadian scribes, but were in many ways different. The scribes provided numerate expertise in contexts such as accounting, surveying, building and planning, and they are depicted in tomb decorations as well as physical models, in scenes such as supervising baking or brewing or inspecting goods of various kinds. The evidence is that over time some of the king’s power came to be transferred to the scribes who worked in his name.

The hieroglyphic script was invented late in the fourth millennium BCE, around the same time the Sumerians first wrote their script. In fact it is possible – though not certain – that Egyptians received the general idea of writing words down from Sumerian models or reports of them, though their symbols were all their own. Mainly written on stone, the hieroglyphs were originally a set of about a thousand signs for things, plus a further set of twenty-four signs for specific consonant sounds; the two were used in combination. They were later simplified and made cursive for use on papyrus in the script called hieratic. And over time the hieratic writing of the Nile delta region became sufficiently distinct to be called a new script, namely demotic. Even more cursive and distant from the original hieroglyphs, demotic was used for over a thousand years, from the seventh century onwards.

In these later scripts, the number symbols not only looked different from those of hieroglyphic but worked to a different system. In the hieratic script, groups of number symbols came to be combined into single symbols, so that instead of writing ‘1’ four times, say, you wrote ‘4’ just once. Thus, for each number from 1 to 9, there was now a distinct sign. The multiples of ten from 10 to 90 also had their own set of distinct signs. In demotic at least, the multiples of 100 had another set of signs, as did the multiples of 1,000. The demotic number symbols were widely used in commerce, administration and legal documents, throughout the period when the Ptolemies ruled Egypt. It was in demotic that Teianti’s tax receipt was written down; and the distinctive demotic number symbols are clearly visible there and in the other documents concerning her house and its previous owners.

Compared with the number systems that had preceded it in the region, this was something rather different: simpler in some respects but more complicated in others. Any number up to 9,999 could be written using no more than four signs, compared with the dozens of signs that some numbers needed in the hieroglyphic system. It was more compact, therefore, and faster to write. And a glance at a written number would tell you its approximate size in a way that a glance at a hieroglyphic number did not: a number involving four symbols was necessarily in the thousands, for instance.

But any attempt to see it as a straightforward improvement on its predecessors is liable to founder. The number of different signs users had to memorise was now relatively large. With nine different signs for the units, nine for the tens, nine for the hundreds and nine for the thousands, there were thirty-six different number symbols in use: more than the whole set of phonetic signs used to notate the consonants of the Egyptian language. (And this in a culture where a mistake could result in a beating: ‘woe to your limbs’ if you get things wrong, as one text about scribal life has it.) Furthermore, for the purpose of calculation, this was not a system whose symbols could be moved and accumulated intuitively like counters: it required you to memorise addition and multiplication tables, or to convert numbers into another representation – such as an abacus – in order to carry out even the simplest additions and subtractions.

Also, the system was not extensible in the way that a positional system – which reuses the same symbols for numbers of different sizes – can be. It had a fixed upper limit; there were no symbols for numbers above 9,000, and the largest number that could be represented was 9,999. If the number 10,000, or anything larger, needed to be written down, the rather cumbersome expedient was to write combinations of two lower signs, understood as being multiplied together. Like all ways of counting, in other words, the Egyptian demotic symbols were good for some tasks, but less good for others.

Teianti’s collection of documents ended up in a pot in a tomb of the nineteenth dynasty which had been reused as a private dwelling. How they got there is obscure: deeds were essential to showing one’s ownership of a property, meaning that collections of papyri like this would normally have been preserved with some care. Equally obscure is their excavation beginning in the nineteenth century CE, which has left the separate documents scattered among museums in the USA, England, France and Egypt. Teianti’s final receipt for tax paid is now in London, at the British Museum.

The Mesopotamian and Egyptian ways of counting with symbols had been some of the longest-lived the world has ever seen: a continuous tradition of 3,000 years and more, with change and modification, but recognisably interrelated and springing from the same original impulse to turn simple tally marks into something with more structure and hence more power. They lasted because the civilisations they served lasted; because they worked; and because the region had little contact with other parts of the world – the Indus valley, China – that might have shown them alternatives.

Nevertheless, the archaeology of the region bears witness to both their rise and their fall. Both the scripts and their number symbols vanished from use during the first millennium CE. Cuneiform and Egyptian hieroglyphics became dead scripts, which no living person could read. In Egypt, the demotic script was replaced by a Greek-derived alphabet known as Coptic, and the demotic number symbols were abandoned in favour of Greek symbols imported – like the Ptolemies and their settler subjects – from the other side of the Mediterranean.

Yet the story of these systems of number symbols does not end here. Although the symbols were replaced, the structure of the demotic system persisted and passed into the Greek world, where it would travel widely and enjoy a long life even down to the present. There, though, it would have to compete with other systems of number symbols: and indeed with entirely different ways of counting and imagining numbers.

4

Counter culture

from Athens to the Atlantic

From the crossroads that is the Fertile Crescent, follow the coast of the Mediterranean: north and west. Through Anatolia, and on into Europe.

A very different branch of the story of counting sprang up in this part of the world. Everyone knows that the Greeks loved mathematics, that their philosophy privileged numbers and geometry and that their language of mathematical theorem and proof is still in use today. But how did they count? And, for that matter, what did they count? And what – if anything – of their counting practices did they bequeath to the later European traditions of the Roman Republic, the Roman Empire and the Latin Middle Ages?

There were number words and number symbols in Greece and throughout Europe from the classical period onwards; some were apparently based on Egyptian models. But the way of counting most closely associated with Greece, Rome, their empires and their successors is one that recalls the very first evidence for wild number lines: counters. The techniques of counters and counting boards dominated European experiences of counting and calculation from the fifth century BCE for more than two millennia, fading out as late as the seventeenth or eighteenth century. Efficient, practical and versatile, they would eventually fall victim to a new emphasis on writing and printing; and therefore on number symbols. But their story is a fascinating one, and it starts in Greece.

Philokleon: Counting votes

Athens, 325 BCE. A middle-aged man – call him Philokleon – rises early and makes his way to the law courts, at the northeast corner of the agora. Others have risen earlier still, and the ten entrances are already staffed and busy, would-be jurymen buzzing about like wasps. Up to ten courts sit on any one day, and each typically has either 201, 401 or 501 jurors, so there is quite a crowd. Philokleon uses the entrance marked with the name of his tribe, where he finds ten boxes lettered from alpha to kappa. He brings out a little boxwood plaque that identifies him as a member of this year’s jury pool; it is stamped with a letter as well as his name and tribe, and he places it in the corresponding box: say box beta.

Once all the potential jurors from this tribe have come in and given up their plaques, the presiding archon signals slaves to pick up the boxes and shake them. The archon then picks out from each box one plaque, whose owner will both serve as a juror this day and perform some administrative functions.

The administrators (Philokleon is not picked) place all the plaques in a machine called a kleroterion: a tall marble slab with many slots, arranged in five columns; the plaques go into the slots in random order. Once they are in, the archon signals for a die – in fact probably a ball – to be released from the mechanism at the side, rather like some modern lottery machines. If the ball is white, the plaques in the top row of the kleroterion are picked for jury service this day; if it is black, they are not. A second ball is released to determine whether the plaques in the second row will be picked, and so on. Philokleon’s plaque is in one of the rows picked today, so now he knows he will serve as a juror. He gets his plaque back. Next he has to find out in which court he will sit.

For this purpose, the court staff have already assigned a letter label to each of the courts, from lambda on. Philokleon and the other jurors go to a basket full of (probably fake) acorns and pick one out; each acorn has a letter on it. Say Philokleon gets mu, so he’s assigned to court mu today. He gives up his plaque again to the archon, who puts it in a chest marked mu; he shows his acorn to another official, who gives him a staff whose colour matches that of the court to which he is assigned: say the blue one. Philokleon heads into that court.

Meanwhile, the magistrates are using the kleroterion again to decide who will preside over which court. Balls marked with the courts’ colours are placed in the machine and are matched with balls naming the magistrates, who head to the courts as well.

Once everyone is assembled in the blue court, the game of tokens, counters and random selection is far from over. Philokleon gives up his acorn and staff to another official, who gives him a token marked with a letter of the alphabet, telling him in which section of the court to sit. Next, the box of plaques is used once more, to choose ten jurors to perform further administrative functions: to control the water clock, count the votes, and supervise payment at the end of the day. Only now can the spectators enter and the trial itself begin.

Once the pleas and evidence have been heard, a herald announces that the vote will be taken. There is no designated time for the jurors to confer, but in practice Philokleon can chat with his neighbours as they wait for yet another exchange of counters. Each juror gives up his seating token and claims in return two psephoi: literally pebbles, but in fact little brass wheels with an axle protruding on each side. Each walks to the voting urns, placing his ‘guilty’ ballot in one and his ‘innocent’ ballot in the other. The shape of the ballots, and possibly some sort of cover for the urns, makes it possible to be quite discreet about this if you want to. Philokleon puts his ‘guilty’ ballot in the bronze, ‘valid’ urn, the other in the wooden ‘discard’ urn. He receives in exchange a pay token.

The ballots are now tipped out of the bronze urn and counted by inserting them into a board with five hundred holes in it, putting the ‘guilty’ votes at one end and the ‘innocent’ at the other. The herald announces the result. If there is a question as to what penalty or damages to apply, a second vote has to be taken: the jurors receive their staffs or seating tokens again, hear the arguments, give up the seating tokens in exchange for ballots, vote again … Once all of these procedures are over, each juror finally receives a pay token.

The – perhaps long – day ends with a final exchange of tokens, as the jurors are paid. The box of plaques is brought in, they are picked out at random and each juror in turn gives up his pay token in exchange for his plaque and three obols (coins), concluding the day’s flow of tokens. (Or not quite. One Athenian playwright has a joke in which the court runs out of small change and gives two jurors one larger coin between them, which they take to the fish market to exchange for smaller ones.)

Athenian ballot tokens.

Wikimedia Commons. CC BY-SA 2.5.

The Greeks had a set of number words, inherited from their proto-Indo-European ancestors. The Proto-Indo-European hoinos, duoh, treies, kwetuor of perhaps 5,000 years before became the classical Greek hen, duo, tria, tettara in a development that is – as these things go – relatively easy to reconstruct. It was a purely decimal system: there were words for the numbers from 1 to 10, and larger numbers were built up from those by a combination of multiplication and addition: thus 43 would break down as ‘four-ten-three’ just as it does in English, although the separate elements were in some contexts so distorted as to be hard to recognise. Greek had some irregularities and alternatives in the -teens: for 18 and 19, particularly, it was possible to say what amounted to ‘two from twenty’ and ‘one from twenty’. There were words for 1,000 (chilios) and 10,000 (myrios, a word originally meaning innumerable). Beyond that, there were no standard expressions.

Greek was also relatively well supplied with alternative series of number words to answer questions like ‘which in a sequence’ (first, second or third), ‘how many times’ (once, twice or thrice), ‘into how many parts’ (there’s no real equivalent in English), or to express ideas like single, double and triple or twofold, threefold and fourfold. They used their spoken numbers throughout their culture: most enduringly in the work of Greek historians, poets and orators, who variously enumerated casualties, gifts, costs, debts, embezzlements and much more. Philokleon could have heard, a little earlier in the century, the orator Demosthenes’ celebrated attack on his guardians and their handling of his father’s estate:

Gentlemen of the jury, my father left two factories, each of them a decent-sized business: thirty-two or -three sword makers worth five or six hundred drachmas apiece, the least of them worth not less than three hundred, from whom he got an income of three thousand drachmas per year free and clear; then sofa makers, twenty in number, who were security for a loan of four thousand drachmas, who brought him twelve hundred drachmas free and clear and about a talent of silver lent out at twelve per cent, from which the interest every year came out to more than seven hundred drachmas …

There is some question as to how much of this kind of thing even the most alert jury could follow. In fact, the uses of spoken numbers in Greek rhetoric ranged from pitiless forensic clarity with clear, correct and repeatable calculations, to much vaguer invocations of large numbers for the sake of their emotional appeal.

The Greeks also had a set – three sets, in fact – of number symbols. It was possible just to use the Greek letters, in the alphabet order inherited from the Phoenicians: alpha, beta, gamma … The books of the Iliad and the Odyssey, and of other classic Greek works, are to this day numbered using this system.

There was, second, a system established at least by the fifth century BCE which used letters as abbreviations for certain number words: M for myrioi (10,000), X for chilioi (1,0000, H for hekaton (100), Δ for deka (10). A simple vertical stroke like a tally mark was used for the ones. Each symbol was repeated as many times as needed. Long strings of the same symbol were avoided by using Π to stand for pente (5), wrapping it around an M, X, H or Δ as needed. (There is a widespread, though by no means universal, tendency for systems of number symbols to avoid ever needing more than four repetitions of the same symbol: ultimately perhaps a response to the limit of subitising, the fact that groups of more than four cannot be recognised at a glance but have to be explicitly counted.)

This system had the same structure as the Roman numerals, though the family relationship between the two systems is not completely clear: a common Etruscan ancestor may have been involved. In the Greek world it was widely used, particularly for notating sums of money. But from about the third century BCE it was edged out by a more concise alternative. This used all twentyfour of the Greek letters, plus three extra symbols, to signify the numbers from 1 to 9, the multiples of ten from 10 to 90 and the multiples of one hundred from 100 to 900. To extend it into the thousands, the symbols from 1 to 9 were reused with a special mark before them. Structurally identical to the Egyptian demotic system, it was quite possibly a translation of that system into Greek letters; it could have been encountered by Greek traders in the cities of the Nile valley. Despite needing a little more effort to learn than the other Greek systems, it became the dominant way of writing numbers in Greek. It had the advantages of nearly always using fewer symbols than the other system, of showing at a glance roughly how large the number was – how many digits it had – and being relatively easy to read out.

This alphabetic system of number symbols was used on public inscriptions throughout the Greek world, on coins and for purposes such as sections in texts or – later – pages in books. It survived until the fall of the Byzantine empire, and it could still be seen in the page numbers and section numbers of early printed Greek books in sixteenth-century Europe. Like the Roman numerals, it is still occasionally used to number pages today.

In ancient Greece, its most visible use was in public inscriptions, ranging from tribute assessments, records of loans, accounts of military expeditions, inventories and other accounts of institutions, to accounts of the auctions of confiscated property. This public, numerate documentation was a particularly Athenian habit, associated with Athenian democracy and its emphasis on – its display of – public accountability. Indeed, it came to an end with the tyrannies of the late fourth century BCE that ended Athenian democracy. Even if few perused the inventories and the accounts in detail, any citizen could do so in principle; some were even set out in column format as though to make them easier to read.

The accounts carved on stone and erected in public places were of course copies of accounts that also existed elsewhere, sometimes in fuller, more detailed versions, written in charcoal on whitened boards and stored in private archives. And they were the outcome of a process, more private still, of counting and arithmetic for which neither the ancient Athenian public nor modern historians have much direct evidence. Indeed, it does not seem that the Greeks used their number symbols for the dynamic processes of counting and arithmetic, to which purposes those symbols were very poorly suited. For that, they had another way.

Ancient Greece has well and wittily been called a counter culture, a culture in which most people’s primary experience of and ideas about number – their ways of counting – involved not words or written symbols but small, graspable, manipulable objects: counters. (A world in which you could say ‘number’ and mean ‘collection’ or ‘group’, in turns of phrase that survive to this day: ‘I want to be of that number’.) As Philokleon’s day at the law courts illustrates, counters were everywhere in ancient Greece. Made of wood, clay, lead or bronze, or even of silver, they were used for games and for divination, they were used as tickets to political institutions or the theatre, and they may have been used in the distribution of grain.

And they were used for the workings of democracy. A system whose basic idea is majority decision-making must lean rather heavily on its practices of counting; must define them clearly and establish procedures to ensure they are transparent, reliable and trustworthy. There are rumours of a ‘shouting vote’ in neighbouring Sparta, while in the Athenian assembly a mere show of hands was used for most decisions. But in the law courts, decisions were more strictly quantitative, and the way Athenians selected, sorted and assigned individuals, as well as finally counted them, was to match them one-to-one with counters. Out of Athens’ no more than perhaps 20,000 eligible citizens (those who were male and over thirty), a normal court day saw 2,000 of them engaged in the elaborate exchange of lots, tokens, counters and ballots. There were around two hundred court days in the year. The most important counter, the psephos or pebble, came to stand for the whole procedure, giving its name to the decision-making process and its outcomes, and even to the law courts themselves. One of Aristophanes’ jokes about the subject has an obsessive devotee of the law courts waking up with his fingers stuck in the counter-holding position. The word was further transferred to the simple meaning of ‘number’, and it survives to this day in the term ‘psephologist’ for someone who studies voting patterns. Some Athenians took the plaques identifying them as jurors literally to their graves, arranging to be buried with them just as an earlier generation of Greeks had been buried with their swords and armour.

The complicated procedures of the late fourth century had of course developed from earlier, simpler ones. Originally, decisions were made by placing actual, literal pebbles in one of two open piles, by placing shells in urns, or even by making marks on wax tablets. The kleroterion was the culmination of procedures that had originally amounted simply to pulling a marked object (a pebble, a potsherd or a black or white bean) out of a vessel. The direction of change was towards greater secrecy for the individual, and greater transparency – less possibility for collusion and cheating – in the various random choices that had to be made.

The court’s procedures, as they evolved, were also used for other things. The kleroterion was employed for other choices by lot, such as the annual selection of magistrates and the various other political officers: over a thousand individuals needed to be selected in total, and the machine could make the procedure much easier and more reliable. Surviving kleroteria are of various sizes with various numbers of vertical and horizontal rows, tailored to particular selection processes. There was even a process called synklerosis in which two of the machines were used together, to match random candidates to random offices.

Similarly, the use of counters to vote was adopted in the Athenian Council (of five hundred members) for cases where exact numbers were important, with a procedure probably similar to that in the courts. A bizarre variation occurred when the Council voted on the expulsion of a councillor, for which it used not pebbles nor bronze psephoi but olive leaves as counters. Meanwhile, if and when the full assembly of Athenian citizens chose to vote on the banishment of a citizen (‘ostracism’), it used potsherds (ostraka) as counters.