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.
The Greek world, of course, contained yet another kind of counter, one that was exchanged, handled and counted perhaps even more pervasively than the tokens and ballots of the political and legal instutions; that was counted, frequently, in both court speeches and public inscriptions. Coins.
The first coins in the world were struck in the early sixth century BCE in the vicinity of Ephesus, on the border between the Greek world and its neighbours to the east. They were an invention whose time had evidently come, and the making and use of coins spread through the Greek world and later the Greek empire, replacing (though not entirely) other ways of storing and exchanging value in the form of goods: cattle, cauldrons, corn, ingots and perhaps metal roasting spits. (The Greek word obol, the coin in which Philokleon was paid, embodied a memory of a time when goods were money, since its name literally meant a spit.)
Scraps of silver stamped with religious and civic emblems – turtles, owls, dolphins – and sometimes, later, with number symbols giving their date: these provided a new game of counters for Greeks to play, a new sphere in which to participate and a new way to measure your worth. They made wealth just as handy, tangible and countable as votes.
The Greek states, their institutions, and the concept of an all-purpose money developed hand in hand: nowhere more so than in Athens, whose currency and weight standards dominated the Greek world through the fifth and fourth centuries, and where talking and writing about finance – pay, trade, gifts, bribery, theft, tribute – became ubiquitous, creating new uses for the Greek number words and number symbols. When individual wealth had become a source of both complaint and excuse, it was not unusual for those pleading in the courts to go to some lengths to complain of their opponents’ wealth, to deny being wealthy themselves, evidently in the hope of playing on jurors’ presumed prejudices. When everything had its price, all a politician had to do to win over the public was lower the price of sardines (so it was said). When coins were everywhere, there was a need for whole new professions to handle them: bankers, money changers and coin testers.
When Philokleon received his three obols at the end of a long day of exchanging one counter for another, he could promptly go to the market and exchange them for goods. Items that had once been bartered against one another could now be conveniently counted on a common scale using silver counters. And not just everyday items like food or clothing. By embodying a universal standard of value, coins created equivalences between different spheres where none existed before: marriage contracts, court fines, gifts and pay, for instance, were all now in a sense interchangeable; they could be counted off against one another and placed into mutual correspondence. A fragment from the mid-fourth century reports that in the agora you could now buy ‘figs, marshals of the court, grape bunches, turnips, pears, apples, witnesses, roses, medlars, haggis, honeycombs, chickpeas, lawsuits, beestings, curds, myrtle, allotment machines, blue cloth, lambs, water clocks, laws, indictments …’ The strange alchemy of money seemed capable of transmuting anything into anything else.
Marcus Aurelius: Counting years
Rome, 176 CE. Emperor Marcus Aurelius has been away from the city for years, on the Marcomannic campaigns beyond the Danube. He addresses the Roman people on his return. They are entitled to receive a gold piece, an aureus, for each year he has been absent:
‘While he was saying, among other things, that he had been absent many years, they cried out, ‘eight,’ and indicated this also with their hands, in order that they might receive that number of gold pieces for a banquet. He smiled and also said ‘eight’; and later he distributed to them eight hundred sesterces apiece, a larger amount than they had ever received before.’
While systems of counting words developed to more sophisticated internal structures and with the capacity to represent ever larger numbers; while the use of counters burgeoned into the complexities of the Greek voting systems as well as turning into coins; and while number symbols attained something of their modern range and flexibility across the Near East and the Mediterranean world, what happened to counting on the fingers? This way of counting, too, is capable of development, change and sophistication. There have been many systems of counting gestures more complex than simply extending or folding one finger at a time to count up to five. One of the best-documented and longest-lived was that of ancient Rome.
Its origins are lost: the earliest firm evidence is from Rome in the third century BCE, and it is quite unclear whether it existed earlier in Greece or elsewhere. Though Greek has an expression for counting on the fingers, and Greek literature has the odd mention of sums so easy you can do them on your fingers, specific reference to the elaborate Roman system is lacking. An Egyptian origin has been suggested; a Persian connection is hinted at in some sources, but evidence is lacking.
How to count in gestures, from a manuscript of Bede’s book
British Library, Royal 13 A. XI, f.33v. By permission of the British Library.
The system provided hand signs enabling counting up to 99 on one hand, up to 9,999 using both hands. Invented apparently in a region where number words and number symbols counted in tens, it was not surprisingly a decimal system. Numbers from 1 to 9 used the little finger, ring finger and middle finger of the left hand, bent down either fully or partly and in various combinations. The multiples of ten from 10 to 90 were shown with the thumb and index finger of the same hand. For larger numbers, the same set of signs was used on the right hand to show the hundreds and thousands: the right thumb and index finger took the hundreds, and the right middle, ring and little fingers the thousands.
Throughout the period of the classic Roman authors, the finger-counting system was well enough known to serve as the basis for jokes and double meanings. Quintilian, in the first century CE, mentioned that orators were expected to be fluent in the finger numbers so as not to give a poor impression of their training. Juvenal, a little later, was able to imply that someone was very old by saying that he counted his age on his right hand. Catullus in the previous century could say ‘give me a thousand kisses, then a hundred, then another thousand, then a second hundred …’ and be understood by at least some of his readers to be providing instructions that led to a middle-finger gesture on the right hand (yes, it was already rude in those days; the finger itself was known as the impudicus, the rude one). Up to two dozen Latin authors might be cited for similar hints or discussions of the gestures accompanying particular numbers, from Plautus in the third century BCE to Cassiodorus nearly a thousand years later.
The gestures were used to show a count to others, or to keep track of a count for yourself; mention was made of their use in the law courts, the arena, the public assembly and the church. Background noise and distance between people wanting to communicate would have made them useful in those situations and many others, and they persisted alongside spoken and written numbers because they were good for different things.
More vivid than textual references are the surviving depictions of number gestures. Counting-house scenes or pictures of negotiation about prices – with the participants showing each other numbers in gesture – appear in carvings, reliefs and mosaics. What seem to have been gaming pieces, with the hand gestures depicted on one side of them, turn up in archaeological contexts: a full set apparently showed the numbers from 1 to 15.
More symbolic uses also occurred. The legendary second king of Rome dedicated a statue of Janus which displayed the number 365 on its two hands: the number of days in the year, apt for the god of time and duration. A late Roman tradition placed pictures of the hand signs on sarcophagi, showing the number 52: the number of weeks in the year and a symbol of the cycle of life.
Several ancient sources refer not just to counting on the fingers in this way but also to performing calculations with numbers in the same form. But no surviving source explains in any detail how it worked. It would have been possible in principle to memorise the addition and multiplication tables in the form of hand gestures, but it is perhaps more likely that the fingers were used primarily for simpler computations, or to keep track of intermediate stages in calculations being performed mentally.
The Roman finger-counting system was hugely long-lived. It survived the end of the Western Roman Empire and was still in widespread use well into the Middle Ages. In medieval contexts, it provided a way to give gestures and postures an additional layer of meaning: to insert numbers and number symbolism into texts and pictures. A second-century Gnostic text found at Nag Hammadi contains an exposition of the New Testament parable of the lost sheep which makes use of the fact that the ninety-nine sheep – who never go astray – are counted on the left hand, but the full count of one hundred sheep – after the stray has been recovered – passes to the more auspicious right hand. Similar exegetical moves can be found in the sermons of several Christian writers including St Augustine, addressing congregations of no remarkable education who were nevertheless expected to see the point at once, to be familiar with the Roman finger-count.
In fact, like most things that ‘everyone knows’, the finger-counting system was seldom explained or described in writing: at least until much later, when it was in danger of dying out and had become more specialised knowledge. The earliest complete descriptions of the system – and therefore the basis for interpreting allusions and uses of it from the Roman period – date from the late seventh and early eighth centuries: a short pamphlet called the ‘Roman computation’ whose earliest copy probably dates to 688, and a better-known discussion by the Venerable Bede, monk of Jarrow in northern England, from a few decades later. Bede addressed the finger-counting system in the first chapter of his book on calendar calculations, The Reckoning of Time. Writing in Latin for his fellow monks, he praised the system and implicitly admitted that by this time it faced some resistance: it was ‘very useful and easy’, he said; one should not ‘despise’ it or ‘treat it lightly’. Presumably some did.
While the gestures Bede described seem to be the same as those used by earlier Latin writers, with just a few minor alterations over the centuries, both he and the ‘Roman computation’ pamphlet described a way to extend the count beyond 9,999 that has no parallel in any earlier source, and that may have been a medieval innovation. For the ten thousands you would place the right hand at various spots on the body in turn: head, throat, breast, side, stomach, groin, thigh. The left hand was then similarly employed to count the hundred thousands, while one million had a special gesture of its own with the two hands held at the sides of the face. The details differ from one copy of Bede’s text to another, and it is unclear how much this extended system was ever used in practice.
Bede’s book – and in fact separate copies of its first chapter – circulated widely, and a large number of copies survive to this day. Many were illustrated to show what the finger signs should look like. Despite this, evidence of the system in actual use in the Latin world becomes sparse after Bede’s time. By the time late medieval and Renaissance writers – such as the mathematicians Leonardo of Pisa in the thirteenth century and Luca Pacioli in the fifteenth – described the system in their books, it seems to have been of merely antiquarian interest. The same is true of printed versions of Bede’s Reckoning of Time, which appeared from 1525 onwards.
Meanwhile, though, the system was also in use outside the Latin world, in the other inheritors of parts of the Roman Empire. A fourteenth-century Greek author at Smyrna described a version of the system. His treatise also exists in an Arabic version, and Arabic and Persian texts contain references to the system down to about the seventeenth century.
During the same period, there is also visual evidence for a continuing interest in the number signs in Europe. Frescoes and altarpieces continued, occasionally, to incorporate figures making the hand signs for numbers of significance to the scene in which they stood, down to the twelfth century if not later. The ability of viewers to interpret the gestures could eventually no longer be relied on, and the practice dropped out of use.
