Over the following millennium, it became usual for scribes to turn everything into the new positional notation when calculating. By the first millennium, it was in use in Babylonian astronomy, and its emphasis on the number 60 is the ultimate ancestor of the division of circles into 360 degrees, and hours into sixty minutes.

A question that arises naturally to a modern mind, and must have occurred fairly early to users of this notation, is what to do if there is, so to speak, a blank column. How do you express the number made of two sixties and three units but no intervening tens, for instance? Some sort of sign meaning ‘blank’, or just a blank gap on the clay, seems to be needed. The earliest surviving example of a solution to the problem is in a table of squares from Kish, dating to the seventh century BCE: it has a placeholder identical to the sign for ‘30’, performing the ‘blank’ function. Signs with this function would become common in later Babylonian arithmetical and astronomical number writing.

Cuneiform number symbols had come a long way from their origins, both in their appearance and their functions. Elsewhere in the Fertile Crescent, they were being put to very different uses from administrative record-keeping, or indeed astronomical calculation: they had become part of the tradition of royal self-advertisement, display and boasting.

Tiglath-Pileser I: Counting plunder

The city of Katmuhu, near Assyria; early in the eleventh century BCE. Two scribes survey the field of battle, the killed, the captured and the plunder. One uses a stylus to keep track, counting under his breath. Later the record will be written up and become part of the royal annals:

‘I captured in battle their king, Kili-Teshub, son of Kali-Teshub, who is called Errupi. I carried off his wives, his natural sons, his clan, 180 copper kettles, five bronze bath-tubs, together with their gods, gold and silver, the best of their property.’

As well as undergoing a range of developments in its original Mesopotamian location, cuneiform writing spread throughout the Near East. The system was exported early to other cities in Iraq and Syria; the evidence is fragmentary, but numerical symbols like those from Uruk have been found at a number of sites in the late fourth millennium, and it is possible that the original invention was a more widely distributed affair than the evidence now shows.

The later changes made cuneiform a flexible tool, capable for instance of being used for literary texts and letters. They made it capable of being used to notate languages other than the original Sumerian. Cuneiform would eventually be used to record about fifteen different languages across the Near East; at its peak, it was used from Turkey to Egypt and from Lebanon to Iran. In fact, cuneiform on clay would remain in use for over 3,000 years, making it the longest-lived writing technology the world has yet seen, its history longer than that which separates the present from the fall of Troy. The Uruk system of record-keeping was the basis for administrative systems over the whole of that period.

One of the languages most associated with cuneiform is Akkadian, a member of the Semitic language family, whose members also include Hebrew, Aramaic and Arabic. For reasons that are now very unclear, it replaced the unrelated Sumerian as a spoken everyday language in Uruk during the third millennium BCE. It went on to become a widely used language in the Near East, functioning as a common tongue for purposes such as diplomatic correspondence. In the dialect of Babylon, it was a major language of literature over the whole region; in the dialect of Assur it was the language of an important city and, at times, an empire.

Assur stood on the west bank of the Tigris, about 100 kilometres south of modern Mosul. It overlooked the river to the east from a rocky promontory. This area had a different climate from Sumer, with more rainfall, making agriculture easier and less dependent on irrigation. Stone was available locally and did not need to be imported; equally, timber and metal could be found close by. Conversely, the marshes and their characteristic products were absent. Assyrian merchants operated as far away as central Anatolia.

Ruled by a king and an assembly, Assur functioned a little like Uruk, with agricultural hinterlands providing cereal and animal foods and the city housing specialised crafts. Assur was established as a city by the middle of the third millennium, and it had periods of relative power and periods of collapse. From the mid-fourteenth century BCE, a series of ambitious kings made it an important political force in the region. Both diplomacy and military conquest played a part, with Tukulti-Ninurta I in the late thirteenth century – for instance – successfully attacking Babylon itself. He sacked the city, trod on the neck of its king ‘as though it were a footstool’ and took him in chains back to Assyria. There was campaigning in all directions at various times. On the diplomatic side, Assyrian power was eventually recognised by its regional rivals the Hittites, by the Babylonians, and even in Egypt; Assur was acknowledged to control the Tigris valley and the plains to the east, as far as the Taurus mountains to the north.

As at Uruk, resources were controlled through cuneiform accounting, which enabled them to be mobilised for building, irrigation and agricultural schemes. Like Sumerian before it, Akkadian had number words (isten, sina, salasat, erbett, xamsat …). Like Sumerian, it used a decimal system for smaller numbers and showed at least traces of a base-60 system for larger ones, with special words reported for 60 (sus) and 3,600 (sar) but also for 100 (meat) and 1,000 (lim).

To write numbers down, Assyrians used a hybrid system. For 1 and 10 they took over the symbols from the Sumerian positional system: a vertical wedge and a slanted one respectively, each repeated up to nine times as necessary. Beyond that, they used ideograms to denote 60, 100, 600 and 3,600. Over time, the 60s, 600s and 3,600s fell out of use and it became more usual to use a purely decimal system with just the signs for 10, 100 and 1,000. But a fully positional system was not consistently used.

They used their written numbers for accounting, for institutional record-keeping, much as the Sumerians had, although there were differences in detail. Akkadian-language accounting – even at Uruk itself – tended to be more interested in estimates and predictions with standardised round numbers than its early Sumerian equivalents; more interested, perhaps, in overview than in detail. The two styles could coexist in the same archive: ‘approximation tended to be used for calculating the assets and rights of the wealthy and the institutionally powerful; precision for the dues of the dependents.’ At Assur, numeracy also became a less exclusive property than at Uruk; it was used by merchants who trained less formally and depended less heavily on prestigious Babylonian models. And written numbers were also used by the Assyrian kings as part of their programme of public self-advertisement and propaganda: a quite different function for counting than any to be seen before.

From an early date, kings had had inscriptions carved onto stone monuments. From the time of Tiglath-Pileser I (1114–1076) they burgeoned into a whole new genre of self-glorification: the royal annals, remarkably detailed chronological accounts of royal and military actions. The first surviving copy of Tiglath-Pileser’s royal annal, from the fifth year of his reign, takes the form of an octagonal prism recovered from the temple of Anu-Adad at Assur. In it, he presents himself as pious, strong and tireless in the defence of his people, the protection of his land and the expansion of his territory:

I gained control over lands, mountains, towns, and princes who were hostile to Assur and I subdued their districts. I vied with 60 crowned heads and achieved victory over them in battle. I have neither rival in strife nor equal in conflict.

His was an eventful reign, with campaigns against the Arameans in the west as far as the Mediterranean, and into the mountains to the north aiming to secure trade routes. In the south he raided Babylonia, but his battle against Nebuchadnezzar was unsuccessful.

Numbers pervaded the text: there were nearly fifty different counts in Tiglath-Pileser’s annal, including the kings he had fought with (sixty), the lands he had conquered (forty-two), the times he had crossed the Euphrates (twenty-eight), the numbers of his soldiers and his prisoners. The numbers ranged in size from a few dozen up to tens of thousands: 20,000 men-at-arms fought with, 6,000 hordes captured, 30 chariots taken, 300 families received, 2,000 captives taken, 12,000 troops conquered … Tributes imposed on the conquered were quantified, as was the immense plunder removed from conquered cities: mules, donkeys, oxen, rams, gold, silver, white bronze, tin, carnelian, lapis lazuli, agate, jewels … Tiglath-Pileser was particularly interested in counting the items brought back to Assur as gifts for the temples there: twenty-five ‘gods’ (presumably cult statues), copper kettles in various numbers, and on one occasion five bronze bathtubs. In similar vein, the inscription included a list of the king’s hunting exploits, with a detailed enumeration of the wild bulls, elephants and lions killed or captured (plus, on the Mediterranean campaign, something called a ‘nahiru fish’ or ‘sea horse’: possibly a narwhal). The king thus spent his energies waging war on more than one front against the forces of chaos and darkness, on behalf of his people and at the command of the Assyrian gods.

The profusion of numbers in the royal inscriptions begs the question of how all the artefacts in question were counted – particularly for the larger numbers – and by whom, as well as who chose which items to count and which to describe with vaguer quantifiers like ‘many’, ‘all’ or ‘as many as there were’. When the king’s inscription did enumerate soldiers or plunder, the implication was presumably not that the king himself literally counted them. Who did?

Fortunately, imagery from later reigns can shed some light on the question. Several Assyrian royal reliefs show scribes involved in the counting process. One depicts an official dictating to a pair of scribes, on the roof of a building; one scribe has a tablet and stylus, the other a parchment or perhaps a papyrus. The same relief shows two soldiers chopping up a statue and others weighing seized goods: the whole scene gives an impression of some attention being paid to the quantification of what was captured. The scene with a pair of scribes is common in other such depictions:

Two scribes counting booty, in an Assyrian relief.

Nineveh, South-West Palace, Iraq; British Museum. Lanmas / Alamy Stock Photo.

Typically a clean-shaven scribe holds a papyrus or parchment, next to a bearded one with a hinged wooden writing board, which opened out to reveal a waxed writing surface that could easily be erased for re-use (though other combinations of scribes are also known). The scribes are more often depicted pointing their styluses at the plunder or bodies they are counting than actually using them to write with.

In other words, the scribes were depicted in the very act of counting, using a stylus to help keep track of the sequence of objects while counting (in words? mentally?), ready to record the outcome in the form of number symbols. It is likely, then, that every important military action was followed by officials whose role included counting the plunder and collecting information about it.

These images of scribes on the battlefield give the impression – and were certainly intended to give the impression – that the items enumerated in royal inscriptions were really counted; that the numbers were neither mere guesses nor inventions. Most of the numbers are indeed plausible in their general size, though the larger ones were certainly rounded. Indeed, the Akkadian system of number symbols made it rather easy to round a number by simply leaving out the symbols for the lower denominations; most numbers in royal inscriptions in fact have only their first one or two digits specified. But they were not systematically distorted, as far as anyone can tell. The counting was real.

The monuments to Tiglath-Pileser I stood at the beginning of a long-lived tradition of Assyrian royal inscriptions, which continued down to the seventh century BCE, through periods of both eclipse and triumph for the aspirations of the Assyrian kings. Numbers always remained an important part of the inscriptions, and their uses expanded to include the duration of events, the size of cities built or conquered, their numbers of columns and the heights of their walls. In 722 BCE the Assyrian inscriptions reported the deportation of 27,290 people from a western neighbour, the kingdom of Israel.

The Assyrian royal inscriptions were placed in a range of locations. They were carved into stone stelae – free standing monuments – soon after the end of a military campaign and erected on its very spot. They were written on tablets, prisms or cylinders of clay and buried in the foundations of building projects. Their audience was in one sense the Assyrian court, the same circle of officers and scribes who produced the texts. In another sense it was the gods, or posterity, or the future kings who might dig up the foundation inscriptions, bringing the great deeds of their predecessors back into the light to check and even modify the record. For their purpose as propaganda, the use of numbers gives a sense of solidity to the information presented which mere words would not have done. They amplified the fact that, say, Tiglath-Pileser subdued ‘a lot’ of cities or brought home ‘several’ of their cult statues, imparting some precision – though admittedly without any possibility of verification – to what the text celebrated.

More than that, the royal inscriptions constituted both a new use of text and a new use of counting, and they illustrate the power of number words and number symbols to shift the way the world is described, perceived and categorised. In Sumer, number symbols made ten sheep into a single item, sixty containers of grain into a single possession, or the transfer of 3,600 vessels of oil into a single transaction. Tiglath-Pileser and his scribes and readers created a world in which conquering ten cities, subduing sixty rulers or carrying away five bronze bathtubs was – for a king – a single act.

Teianti: Counting coins

Egyptian Thebes; 28 February, 274 BCE. Ink scratching onto papyrus.

Teianti, daughter of Djeho and Tamin, bought a house (actually two houses) five years ago. She has just finished paying off the tax on the purchase. The scribe Esioh son of Djechensertais writes the receipt for her:

‘There are 6 kite, making 3 staters, making 6 kite again which Teianti daughter of Djeho has paid as the 10% of the price of the house of Pabuche son of we-R‘ and of the house of Teihor daughter of Harsiesi, making 2 houses in all, which [they] sold being the 10% for the scribes of the bailiffs of Thebes. Written by the scribe of the land of Thebes Esioh son of Djechensertais, the scribe of the phylae, in year 9 Typi (day 1) of Pharaoh Ptolemy son of Ptolemy.’

Heir to an immense history, Egypt was by the 270s BCE ruled by a Greek-speaking dynasty whose founder, Ptolemy Soter, had been a general under Alexander the Great. There had been a Greek presence in Egypt since the seventh century BCE; the country was ruled by the Persians for periods in the fifth and fourth centuries. With its thousands of square kilometres of fertile land beside the Nile, Egypt was always a tempting prize to its neighbours. Its own urban tradition stretched back into the fourth millennium, with thousands of smaller towns and villages in addition to the great cities of Memphis and Thebes.

Teianti’s tax receipt

British Museum, EA 10529 (recto). ©The Trustees of the British Museum. All rights reserved.

Ptolemy I set up a literate administration run by scribes, centred on the new Greek cities. Many thousands of surviving documents testify to the details of local administration such as taxation and law enforcement. They also contain the annual land surveys that followed the Nile’s yearly flood, reporting on the crops in the fields and enabling the crown, at least in principle, to estimate its income for the year.

Greek settlement was mainly in the Fayum, the former marsh near the Nile where drainage and irrigation made it possible to reclaim land and conjure into being a whole new province. By the later third century BCE a sixth of its population was Greek, including both military and civilians; there were villages where Greeks were the majority.

Yet in Thebes, 800 kilometres up the Nile, society remained overwhelmingly Egyptian: its temples were ancient and wealthy, as were the families of priests that served them. Here, the Ptolemies did little building work. Away from the main streets you would have seen mud-brick homes and shops lining narrow streets; most had courtyards and wells, with space to store animal feed. Such buildings functioned both as family homes and as sources of wealth, to use as security against loans or as dowries. Sometimes, collections of papers have survived, vividly recording networks of family and financial ties, of legal and social obligations.

The house that Teianti would eventually buy stood to the east of the river, in the city’s northern district and in the vicinity of the Great Shrine of Amen. Its ownership is documented in a cache of papyri which trace it for more than fifty years, starting in the year 324 BCE: just two years before the Greeks took Egypt from the Persians. The first known owner, Djufachi, was a carpenter of the temple of Amen. In the ninth year of Alexander the Great, he apportioned his fairly substantial property – a set of buildings around a courtyard – among his children, leaving a section of the north wing beside the road to the younger two, Petechons and Phib. For a period, the two brothers lived in a single household, but they subsequently divided their property into two separate dwellings, part of which passed, through Petechons’ marriage agreement, to his widow Taesi. Taesi used the house as security for a loan and subsequently sold it to the holder of the loan, an unrelated man named Pleehe who held the title ‘Lector of the Ape’. It passed from him to his wife Teihor, who in turn sold it to Teianti.

Teianti’s tax receipts survive for this and another property bought around the same time – 10 per cent was levied on such transfers of property – as well as a document showing that she then leased it to her sister. The papyrus trail ends in the year 274 BCE with her payment of the final instalment of the tax, a relatively careful and elaborate document in which the scribe took the trouble to give names in full, to spell out the nature of the tax being paid and to give the sum involved in both Egyptian kite and Greek staters. Its impression of business and bustle is surely not deceptive about this property; it had now been home to several different households within the world of Theban temple officials, and had changed hands five times in as many decades.

The language of this and similar documents from the same period is Egyptian, in a late form directly descended from the Egyptian of the previous 3,000 years. It is an Afro-Asiatic language, a member – at some distance – of the same group as the Semitic languages like Akkadian. The Egyptian language contained number words in a purely decimal system, with terms for the numbers from 1 to 9 and for the powers of 10 all the way up to 1 million. Those elements were used in combination (subject to some further complications) to specify numbers up to that limit.