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.

Blanche of Castile: Counting with silver

To return to ‘counter culture’. How did Greeks and Romans perform calculations, or handle more complex sets of numbers than simple counts?

The Greek and Roman systems of number symbols – like the Near Eastern systems that preceded them – were better suited to recording the outcome of a count than to performing it, and like most ways of writing down numbers they were particularly poorly suited to arithmetical calculation. Instead, for counting and calculation – as opposed to long-term record – people turned again to their counters, and to a key element of their ‘counter culture’: the counting board.

The counting board, taken broadly, is a technology that seems to have been independently invented more than once, and in some of its forms it has left evidence that is frustratingly scant or ambiguous. It is generally assumed that the scribes of the ancient Near East used a board of some kind for their calculations, but no such boards or even descriptions have survived. In Greece, though, the counting board was a central technique for computation, and it is both mentioned in literature and found in archaeological sites. Wherever numbers needed to be added, subtracted or otherwise reckoned with, from the courts to the banks to the practice of engineers, surveyors and architects, the counting board was used.

It consisted of very little. Any flat surface would do, together with a set of objects to use as counters. A set of lines would be drawn or engraved, optionally labelled, and the counters placed on them and moved as necessary. It has been well said that this was more a state of mind than an artefact. The name abax was used both for counting boards and for gaming tables and other items of furniture, including sideboards and even trenchers or plates. If it is correct, as often claimed, that it relates to a word in the Semitic languages meaning dust (abq), its first meaning may in fact have been a dust-covered writing board.

Around thirty surfaces marked for use as counting boards have survived from the ancient Greek world, though it is not always easy to be certain whether they are counting boards or game boards (or were used for both functions). They range from large marble tables to repurposed roof tiles; at the time, in fact, the majority of counting boards would surely have been wooden, and have therefore not survived. The counters could have been ordinary pebbles or even coins.

The lines, whether labelled or not, were used in a basically decimal system: there was a ones line, a tens line, a hundreds line, and so on. There were also, closely matching the older Greek way of writing numbers, a fives line, a fifties line, a five-hundreds line, and so on. Some boards interspersed these so that the values of the lines were 1 – 5 – 10 – 50 – 100 – 500; others simply used the top halves of the ones, tens and hundreds lines for fives, fifties and five hundreds. The effect, in either case, was that you never needed to deal with a large group of counters on any one part of the board: if the units line got to five counters you replaced them with one on the fives line; if the fives line got to two counters you replaced them with one on the tens line, and so on. There was a much-used ancient quip, that kings move courtiers around like pebbles on the counting board, making them now worth more, now less. Groups of more than four counters on one line were avoided, just as repetitions of one symbol more than four times were avoided in the Greek and Roman systems of number symbols: and perhaps for the same reason, that such groups lie beyond the human ability to count at a glance. Many Greek boards, finally, boasted a special set of lines for fractions, whose different values followed the denominations of the coinage system.

The counting board was just as ubiquitous in ancient Rome as it had been in Greece, and has left still more sparse evidence: a handful of references in literature, a few (five) archaeological finds and no actual description of how it was used. The word calculus literally means a pebble in Latin, but became transferred to mean a reckoning, computation or calculation; its descendants today provide words in several languages with similar meanings: calculate in English, calcolare in Italian, calcular in Spanish.

Physically, the Roman counting boards that have been found differ strikingly from the Greek examples. They are made of bronze, they are small – around eight by twelve centimetres – and their counters are permanently bound to them. That is, the ‘counters’ take the form of metal studs held in narrow slots in the metal surface of the board, bringing the whole device close to what is normally meant by the – vexingly ambiguous – English word ‘abacus’. The arrangement of the slots is that of the more compact Greek boards, with the column for fives above that for ones, for fifties above that for tens, and so on. This structure of fives and tens of course closely matched that of the Roman numerals, making it easy to translate a number from board to writing or vice versa: exactly one written symbol was required for each counter in play. (Roman numerals of the ancient period generally represented 4 as IIII: the convention of writing it as IV is a later one.)

A third-century bas-relief showing two Romans moving counters on a table is the best evidence for the existence of the counting board in the later Roman Empire; it is generally assumed that its use continued widely until the fall of the Western empire in the late fifth century. Thereafter, though, evidence – both literary and archaeological – falls silent until a clutch of texts from the tenth and eleventh centuries show that the counting board was alive and well in the Latin Middle Ages. There was a vogue at that time for learned or at least quasi-learned expositions of the counting board, its operations and the uses to which it could be put in learning about number theory, geometry or even logic. Some involved modified forms of the board and counters, but the whole phenomenon probably amounted to an attempt to rescue for respectability and learned study a device which had never been out of the hands of ordinary people; or indeed the hands of their rulers.

Paris, in the 1240s. A learned clerk sits at his counting board and reckons, showing his queen how much has been raised in revenue, how much spent, what achieved and how much yet to be achieved in a lavish monastic building project. Her name is Blanche, and she is in her fifties. Keen-eyed, alert, interested, and fluent in French, Spanish and Latin, she quizzes him closely, following the reckoning carefully. Possibly she checks the account by doing her own calculations on the clerk’s counting board or on one of her own. Specially struck counters bearing the royal arms fly under their fingers.

Blanche of Castile was born in 1188, and lived one of the more adventurous lives of the period. She was the granddaughter of Henry II of England and his queen Eleanor of Aquitaine; Richard I ‘the Lionheart’ was an uncle, as was King John of England. But on her father’s side, she was a Castilian princess, the daughter of Alfonso VIII.

Blanche of Castile.

The Morgan Library and Museum, MS M.240, fol. 8r. incamerastock / Alamy Stock Photo.

At the turn of the century, aged twelve, she was married to Prince Louis, heir to the French throne; she never returned to Castile. Louis acceded to the French throne in 1223 but died after only three years of rule. Blanche wept so hard and so long that her courtiers feared for her sanity. There followed three decades during which joint rule with her son, another Louis, alternated with periods when Blanche was for all practical purposes the sole ruler of France. For the eight remaining years of Louis’ minority, during his illness in 1244–5 and after he left on crusade in 1248, her power was complete.

At all times, the dowager queen wielded diplomatic power, sat in judgement and organised some of the major events of court life. Her contemporary, the scribe and scholar Matthew Paris, reckoned her the greatest lady on Earth. A recent biographer calls her one of the most imaginative and successful rulers of medieval Europe, ‘remarkably adept at ensuring the people did what she wanted’, both at settling high-level diplomatic disputes and dealing with the everyday business of government: