Counting with words and symbols

in the Fertile Crescent

Walk out of Africa, following the fertile land. It seems that always meant the Nile valley – the only way to cross the Sahara – and then the hospitable strip of the Levant between the Arabian Desert and the Mediterranean. Then cut through the Anatolian mountains and, turning right, follow the Tigris or the Euphrates, perhaps all the way down to the sea: the modern Persian Gulf. The route was important for humans, and for animals and plants, expanding out of Africa over tens of thousands of years. The whole route – now sometimes known as the Fertile Crescent – became an important site for human culture. It lies between three continents – Africa, Asia, Europe – and indeed between three tectonic plates, whose meeting and movement created the great valleys and mountains of the region: the Dead Sea Rift, the Tigris and Euphrates valleys, the Anatolian and Caucasus mountains. It is a region, consequently, of enormous variation: marshes, deserts, slowly shifting river valleys and mountains.

One branch of the story of counting flourished in the Fertile Crescent, from the beginning of writing around 5,000 years ago, through to the Hellenistic period more than three millennia later. In Sumer, Assyria and Egypt, from the earliest writing to the final flourishing of the Egyptian script under Greek domination, and from the world’s first city to the last centuries of the Pharaohs, counting was always present, used for administration, for boasting, for keeping track, for proving who owned what and who had paid whom.

The region was of course a patchwork of different cultures speaking different languages and doing things – including counting – in different ways. But it is fair to identify words and symbols as consistently the important features of counting in this place and time. This is a place where something more specific can be learned about number words: the oldest number words for which there is secure documentation come from Sumer, in Mesopotamia. And this was also one of the places where a crucial new item was added to the human repertoire of ways of counting: number symbols.

Sumer: Counting symbols

Squat on the ground in the sunlight, in the greatest city in the world. Shape a lump of clay into a flat tablet, to fit in the palm of your hand. Take a reed, scratch a drawing in the clay: a goat. Turn the reed on end and press holes into the clay: ‘dish, min, esh’. It means ‘three goats’. Five thousand years later, it will still mean ‘three goats’. The bright light of writing changes everything.

The eastern end of the Fertile Crescent, now sometimes called Mesopotamia, contains in itself much of the region’s diversity. Near the rivers, annual flooding makes the land fertile. Marshes skirt the coast. There are resources ranging from fish, clay and reeds, wild barley and wheat to wild sheep, goats, cattle, pigs and camels. The potential for hunting and fishing is great. People have been in the region for many millennia.

As the climate improved from its ice age low, this was one of the regions in the world – the first – where humans took control of their food supply by domesticating cereals: selecting, sowing, harvesting, storing. Settlement moved from northern Mesopotamia to the plains beside the rivers; by about 7000 BCE, villages dependent on agriculture existed throughout the Near East, wherever there was enough rainfall to sustain it, and the aeons-long dependence on hunting and gathering faded.

By late in the fourth millennium BCE, the plains in the south of what is now Iraq were becoming densely populated, supported by fertile soil and increasingly elaborate irrigation. Settlements became larger, and the first cities arose.

To its inhabitants, the greatest of these was simply ‘the city’: ‘Uruk’. It lay in the very south of Mesopotamia, on the Euphrates, near the Persian Gulf. It may have formed from two settlements on the two sides of the river, and by 3000 BCE or so it covered 600 acres or more; two and a half square kilometres. The population may have approached 40,000. The world had, literally, never seen anything like it.

Physically, it was dominated by a temple complex covering 75 acres: ceremonial areas set on platforms, some built of stone, their walls decorated with mosaics of white, black and red clay and visible for long distances over the plain. Culturally, the city dominated its surrounding villages, consuming much of the food they produced and setting up a specialisation of labour among producers: fishers, farmers, gardeners, hunters, herders; but also weavers, potters, metalworkers and other kinds of specialist. Timber, stone and metals all had to be imported, and long-distance networks of barter existed as a result.

All of this required organisation in various senses. There was a ruler and some sort of city assembly, though the working of these institutions in the fourth millennium are very obscure. To organise people and things on this scale required language, of course, and the inhabitants of Uruk and its area spoke a language called Sumerian, which has no known relatives and whose grammar and even set of sounds are not completely understood.

And it required counting.

Sumerian had a set of number words: dish or ash, min, esh, leemu, ya. This primary set from one to five was compounded to form words for six through nine: ya-ash becoming yash; ya-min becoming imin, and so on. Beyond that, Sumerians counted in a decimal system, with ten being hu, twenty nish, thirty ushu, and so on up to sixty: gi or esh. The limited evidence suggests that for larger numbers, 60 was used as a base, so that you would have said the equivalent of ‘sixty, ten, three’ for 73 and ‘two-sixty, forty, two’ for 162. Shar appears to have meant 3,600 (that is, 60 times 60) and (some scholars say) shar-gal 216,000 (that’s 60 times 60 times 60).

Indirectly, the use of five and ten in the set of counting words suggests an interest in the fingers used as a set of five, as in some – but by no means all – other languages and language families. It is very likely that Sumerians counted on their fingers at times, but there is no direct evidence of the fact: no hand prints like those from Cosquer. Equally, it is possible that they sometimes made tally marks, but none have been found from early contexts.

More probable is that counters were used in the Mesopotamian world. Unmarked pebbles cannot be confidently identified as counters in the archaeological record, but from as early as 8000 BCE, small clay objects appear at sites in Syria and Iran, and one attractive interpretation is that at least some of them, at least some of the time, functioned as counters. Thousands have been found all over the Near East from the Mediterranean to Persia, and it seems very likely that those with more elaborate shapes or those pierced for – perhaps – stringing, had other uses and meanings besides number (these become more common after about 4000 BCE). Yet, as with beads from much earlier periods, they witness at least to the presence in the human environment of objects whose manipulation could form an ordered sequence and come to be used to count. Exactly when and how frequently that happened in this case seems likely to remain obscure.

By the late fourth millennium, the demands of running long-distance trade and managing the supply of food and materials to the world’s largest city prompted a variety of experiments in information storage. Wet clay could be used as a seal, much as wax was at other times and places – say around the neck of a sack or the cover of a jar – and could bear the impression of a stamp for extra security. Simple stamping had already appeared by 6000 BCE. More sophisticated than a simple flat stamp was the cylindrical seal, which was rolled across the wet clay to provide an impression of its design. These first appeared in the mid-fourth millennium; they could be made of shell, bone or stone and they were small – rarely more than 3cm long – making their carving a matter of considerable skill and time, since the scenes carved on them were sometimes elaborate and detailed. Presumably the owner of the seal could be identified from the design by those who needed to know. The increasing use of seals speaks of a society in which memory, word of mouth, and trust were no longer adequate tools to identify and protect property, an effect presumably of that society’s size and complexity.

A different experiment in record-keeping also used clay, but in the form of hollow spheres or envelopes, into which small clay tokens could be placed before the envelope was closed. Very possibly the size, shape and number of the enclosed tokens represented types and numbers of real-world objects, and it is possible that this technique related directly to the use of tokens to signify the same things in other contexts. The outside of the envelope could be marked with a seal or with symbols corresponding to the number of tokens inside: only a handful of examples are known, and it is not clear how common this was or exactly how it worked. It is, in fact, the only direct evidence for clay tokens denoting numbers in the region. These clay envelopes were in use for perhaps a millennium, from about 3700 BCE on.

All of these clay-based recording systems still relied on people’s memories to some degree; their capacity for recording details was limited, as was their ability to prevent tampering. It is tempting to arrange them into a developmental sequence from the more simple to the more sophisticated, but the archaeological contexts – sometimes far apart – in which they are found do not provide enough chronological detail to confirm such a story. Perhaps the better reading is that the later fourth millennium, and particularly its final two centuries, were a period of innovation and experiment, at various places and in various directions.

Number symbols from Uruk.

Metropolitan Museum of Art, 1988.433.2. Public domain.

Yet another possibility was to take a free-standing flat tablet of clay and mark it with impressions denoting the things to be recorded: the impression of a cylinder seal or a stamp seal; a quick stylised sketch of an object (a goat, a sheep) done with the sharp end of a piece of reed; an abstract sign functioning as your personal ‘mark’; a few quick stabs with the thick end of the reed to count how many (goats, or whatever). Left in the sun, the clay would dry and become a durable record of the information it contained. It could be moistened again and modified, though tampering with the text in this way would surely be fairly easy to detect; the whole tablet could be dunked in water prior to erasure and reuse; or it could just be thrown away once its purpose had been served. Artefacts like these began to be made perhaps as early as 3500 BCE, and the simpler ones contained just a single unit of information: say one or more stylised signs plus a set of reed-stabs denoting a number. At first, the number marks were simple tallies; one example from a Syrian site has twenty-two marks with no sign of any internal grouping.

By about 3200–3100 BCE, this method had become distinctly more elaborate and sophisticated. The flat tablet of clay remained (typical sizes were from perhaps 5 to 15cm across, but both larger and smaller were possible), as did the reed stylus. So did the convention of scratching signs with the thin end of the reed and impressing number notations with the thick end. The number of signs had burgeoned to several hundred – nearly a thousand in some estimates – and the origin of some of them as pictograms had become obscured; a high proportion were now to all appearances wholly abstract, their meaning needing to be learned and handed down from person to person.

Furthermore, the surface of the clay was now divided up with lines into boxes, each holding a separate unit of information. The order of signs within each box still seems to have been quite arbitrary, but the arrangement of boxes on the tablet was itself perhaps starting to be standardised, the relationship between one box – one unit of information – and another a matter of conventions understood by those who used the clay tablets.

The number signs themselves had also undergone some development by now, and had acquired something of the same complexity as the non-numerical signs. To count discrete objects, the system mimicked the structure of the spoken Sumerian number words, with symbols for 1, 10, 60, 600, 3,600 and 36,000. For 1, the symbol was a small triangle; for 60 a larger triangle; and for 3,600 a large circle. The symbol for 10 was a small circle, and that symbol could be added inside the signs for 60 or 3,600 to yield 600 or 36,000. Each sign was repeated as many times as needed, and signs for larger numbers conventionally came to the left, smaller to the right.

To count particular commodities such as grain, land, beer or sheep, though, there were different systems, with not only different sets of signs but different relationships between them. To count processed grain, for instance, there were symbols for ½, 1, 10, 60, 120, 1,200 and 7,200; for areas the symbols stood for 1, 10, 60, 180, 1,800 and 10,800. Presumably these were based on conventional measuring instruments such as rods and bowls and their relationships, the written symbols reflecting the fact that, say, there were ten small bowls in a large bowl, and so on.

In all, there were at least twelve different systems of number signs, perhaps as many as fifteen. Some symbols were specific to one system, but others were reused across more than one, making them ambiguous unless the reader paid close attention to the context. A small round impression could stand for ten discrete objects, but also for eighteen large units of land, thirty small units of grain …

The marks on the tablets from Uruk were probably the world’s first writing. If there were any earlier attempts to represent human language through marks on surfaces, they have not survived, or archaeologists have not yet found them. Thus, from Uruk’s period of apparently concurrent, competing ways to record information and signify the world more durably than speech and more reliably than memory, this was the innovation that ended up being most long-lived and important; although clay tokens, seals and stamps would also continue in use for centuries more. Admittedly, this very earliest writing performed only some of the functions that would later be associated with written scripts. Neither the order of the words nor their specific grammatical form was represented, for instance. The signs, indeed, were not an attempt to record the spoken language completely; the tablets should be compared to invoices and receipts, not to pages of prose.

Equally, the numerical marks from Uruk are the world’s first number symbols: more complex and abstract than one-to-one tallying, and clearly deriving their structure from the number words that preceded them. Using Sumerian words, you said the number of tens then the number of units; or the number of sixties, then of tens, then of units. In Uruk’s symbols you did much the same, making a mark for each sixty, each ten, and each unit. When speaking in Sumerian of, say, amounts of grain, you reported the number of large bowls followed by the number of small bowls; in symbols you made a mark for each large bowl then a mark for each small bowl. So the symbols followed, apparently quite closely, the way quantity was talked about in Sumerian.

Around 5,000 tablets have been recovered from Uruk, where after their useful life was over they were repurposed as building rubble in the temple complex. They apparently date from soon after the system of symbols was invented. They were used within what was evidently the complex domestic economy of the city, in which it was of real importance to know how many units of different commodities had been transferred from owner to owner or from location to location. Clay is not a very flexible medium, and its limitations shaped how the process could work: it would harden in the time it took to move a sheep from one side of the city to the other, say, so there could be no running accounts built up over days or weeks. Indeed, if a set of tally marks or a set of raised fingers relates to the activity of counting rather as smoke does to fire, more structured notations leave a wider gap between the act of counting and the trace it leaves behind. First you count (or indeed calculate), then you write a notation that records the result. There was no question, then, of a scribe traversing the city with a set of wet clay tablets in a satchel, witnessing transactions and writing them down in real time. It is likely that most records were made after the fact and that there was an element of unreality about them: that their purpose was to establish responsibility for certain assets rather than necessarily to report movements of goods and people that had literally happened.

That said, the clay tablets speak of a concern to illuminate even the smallest details of the city’s economic life: information about animals, barley, people and land was reported with painstaking accuracy. They speak of a culture in which counting (and measuring) had become a critically important part of the economy. Some were later summarised into secondary accounts reporting totals and subtotals, say of temple offerings over a period of years. It is possible that the information was used – was intended to be used – for planning, such as the apportioning of rations or of labour.

Who did the writing, and the reading, and the planning? The number of recovered clay tablets is enormous, but marking clay takes little time and it would not have occupied more than quite a small number of people on a full-time basis. The Uruk term for them was umbisag, denoting an accountant or scribe: the world’s first numerate professionals. It was a new role, a person who neither made goods nor wielded political power but who stood between the two, managing the production and movement of goods on behalf of the city’s rulers. They were few, and they had skills and therefore power that the vast majority – including quite probably the elite rulers themselves – did not. In the seventh century BCE, nearly two and a half thousand years after the invention of writing, an Assyrian king could still boast of being able to read and write as though that was unusual. As historian Marc Van De Mieroop puts it, ‘people must have realized how important documents were, how they protected them against accusations of theft or careless loss. The scribe had thus certain esoteric powers, a control over people’s lives that was expressed in ways most of them could not control.’

Even during its first centuries, the Sumerian system of marks on clay was changing and developing. Perhaps as soon as there was a community committed to sustaining this new way of recording information, that community set about, both by accident and by design, to refine the system. It became conventional to arrange the signs in lines read from left to right and to place them in the order in which they would be spoken; the lines themselves were arranged in columns whose number depended on the size of the tablet. It was soon realised that impressing the stylus into the clay was quicker and more attractive than scratching lines, and so the older curved lines were replaced by straight ones. The scribe would now press the head of the stylus into the clay and then push it down to the side, producing a characteristic wedge-shaped mark. The signs, already mostly stylised, became completely so, and consisted of one or more wedges in various configurations. (It could now properly be called a cuneiform, ‘wedge-shaped’, script.)

The number signs developed in the same way, so that numbers from two to nine were now shown by a single wedge repeated the appropriate number of times; the number one by a slightly more elaborate single T shape, and ten by a one-sign rotated to the left. The structure of the system remained the same, though, and a range of different systems for counting and measuring different things continued in use. There was some reform during the course of the third millennium. Gradually the systems seem to have been reduced to five: one for lengths, one for areas, volumes and bricks, one for liquid capacities, one for weights, and one for discrete objects. They continued to have various bases, in which multiples of sixty were important but not all-pervasive. Lengths, for instance, had thirty fingers in a cubit, six cubits in a reed and two reeds in a rod.

Perhaps more important than these changes in the appearance of the script were moves towards recording a larger proportion of the words actually spoken. Many Sumerian words had only a single syllable, which meant that the sign for a word could also be used to represent its sound. That created the possibility of spelling out other or longer words, including for instance people’s names, on a phonetic basis. Thus the sign for ‘reed’, sounding ‘gi’, was reused for the word ‘to return’, which happened to sound the same. A single sign could end up having as many as twenty different meanings, and you had to know the language well to interpret the script.

The system of receipts and summaries continued in use, and up to 90 per cent of the documents excavated from the third millennium are of that quantitative type. The remaining 10 per cent include an increasing proportion of Mesopotamian literature: the world’s first written myths, set in the world of Gilgamesh and Utnapishtim. They also contain lists of words, compiled by and for trainee scribes. The longest contain thousands of words, and give a vivid sense of the difficulty of becoming proficient in the system of cuneiform writing.

Techniques of calculation, similarly, had to be learned and practised at the cost of considerable effort. Some clay tablets were used for artificial calculation exercises, such as invented accounts, or problems like how many workers a given quantity of grain could feed or how much land was represented by a given survey.

Addition is fairly easy with a system like the Sumerian ones; you merely need to write down the complete set of signs represented by the two quantities to be summed, and if any sign occurs more times than is allowed you do some replacement, setting a 10 in place of ten ones, a 6 in place of six ones, depending on the system being used. Subtraction is not much harder. But multiplication and division require some work, as indeed they do with any set of number symbols. For all the calculating that was involved in producing these accounts, traces of written calculation – rather than its end results – are very rare indeed in the early cuneiform record, and it is likely that a non-written technology was used to calculate: an abacus with beads on strings, perhaps, or a board on which counters moved. None has survived; neither has any depiction of one.

Finally, the later third millennium saw an intriguing innovation in the cuneiform way of writing numbers. Scribes began to simplify their calculations and the conversions between different systems, by taking quantities expressed in the traditional systems and rewriting them all in the system normally reserved for counting discrete objects: the one that went 1 – 10 – 60 – 600 – 3,600 – 36,000. The convention that the signs were written in descending order of size was now long established, and the scribes further simplified the notation by abandoning the traditional range of differently shaped signs for the larger values; instead they just used the ‘one’ sign – an upright wedge – for 1, 60 and 3,600, and the ‘ten’ sign – a slanted wedge – for 10, 600 and 36,000, relying on the order in which the signs appeared to convey which number was intended. You would now read \\||\\\|||| as two 600s, two 60s, three 10s and four 1s (equal to 1,354).

These were understandable simplifications motivated by practical needs. Scribes, having written a quantity in the new notation, could calculate with it efficiently, and then convert the result back into whatever traditional system was appropriate; this new notation was not used for record-keeping. Some surviving clay tablets include tables specially for multiplication and division with this 60-based system, namely tables of number pairs whose product is sixty.

The earliest datable calculations in the new notation are from 2039 BCE. A system that relied on position rather than symbol shape to denote numbers in this way was a flexible tool; it was, interestingly, an indefinitely extensible system, with no particular upper limit on the size of the numbers that could be expressed.