Archaic Mathematics

    The oldest substantial collection of sources we have comes from the Uruk IV and Uruk III (Jemdet Nasr) periods. Thus, these tablets date from around 3200 to 2900, and together are called protoliterate. There are about 6000 tablets and fragments in this collection, about 5000 from Uruk, the rest from other sites, most notably Jemdet Nasr, which is further north, not far from Babylon.  About 85% of these are economic administrative tablets, and most of the rest are lexical lists. There are a few possibly metrological-mathematical texts, but they are hard to interpret.

    All of the tablets that we have from this period were either found in ancient rubbish tips (most of the Uruk tablets were used as fill at the massive Eanna temple precinct in Uruk), or else were illegally excavated and sold on the antiquities market.  In neither case is it possible to establish any useful archaeological context for the tablets. We do not know how they were used, how they were stored, for how long they were kept, by whom they were written or read, what their relationship was to the goods they are about, or any details of their function.

    The Late Uruk period saw a great increase in the size of the cities, of which Uruk was the largest. As the climate changed in the mid-fourth millennium, with water levels dropping and much land that was previously marsh becoming inhabitable, there was a surge in population in southern Mesopotamia. The population may have increased by as much as ten times during the Late Uruk period [Nissen 1988:67].

    In the cities, there appears to have been a well-developed social hierarchy. There were complex trading networks and there was an explosion of art and architecture of which the most famous remains 5000 years later are the giant temple complexes.  The construction and maintenance of these huge temple complexes, and other massive structures such as the city walls, must have required a great deal of economic control over labor, materials and rations, and a large supporting bureaucracy. The Late Uruk period saw a number of attempts to find a suitable means of economic record-keeping. Finally, some time around 3200, writing developed. At first it was used almost exclusively for economic purposes. The earliest tablets list quantities and types of commodities and possibly a signature or title of the responsible official. The typical entry reads "quantity X of commodity Y", for example "3 sheep" or "5 bowls of barley". There are no sentences, there are no verbs, there is no grammar. The first writing was not a representation of spoken language, and, indeed, we do not know what language is behind the first protoliterate tablets. A modern similar example is the icons on our computer screens. A printer icon gives no hint of the language of the user. Nor, more mathematically, does the symbol '2'. Beginning in the Jemdet Nasr period, though, we begin to see traces of Sumerian, and it has often been assumed that these tablets are proto-Sumerian.

     An example of an archaic tablet is shown below. This tablet is almost certainly from the Uruk III period. It was purchased on the antiquities market, so its provenance is unknown, but it may be from Larsa [Englund 1996:14-17].

   Economic documents are typically ruled into sections or cases, with one piece of information in each section. Major subdivisions may be shown by a double-ruled line, as in this example. The tablet concerns the disbursement of barley rations to a collection of workers(?). Each register on the tablet contains a numerical sign or signs indicating the quantity of barley and a personal or functional identifier. These latter signs can be read, but their meaning is unclear. For our purposes, what is most interesting is that the last row can be read as a total of the disbursements in the previous row. If we make this assumption, then we see that 11 of the wedges make one circle and 5 wedges. Thus, we have established a metrological relationship between the symbols: one circle is equal to six wedges. We do not need to understand the absolute size of the units in order to understand their relative sizes. Nor do we need to be able to read the rest of the tablet, nor understand its precise economic function. In fact, this particular tablet has some modern historical importance, as the first example that Friberg gave in order to establish the identity we have shown, in place of the accepted ratio of 10 wedges to a circle that had previously been assumed [Friberg 1978:I, 8-9; II, 26-27]. This analysis was part of Friberg's major re-evaluation of archaic metrology that began the unlocking of the numerical relationships displayed in these difficult tablets.
   As this example illustrates, archaic numeration is tied to metrology. There was as yet no abstraction of the concept of number. In fact, it is better to think of these archaic signs as 'quantity' symbols rather than 'numerical' symbols. Around 1200 archaic signs are known [Nissen, Damerow and Englund : 25], and of these about half can be read, although, as we have seen, their meaning may be obscure. About 60 of these are quantity signs, which can be arranged in about a dozen series, depending on what they are measuring [Nissen, Damerow and Englund: Chapter 6]. The conventional way to represent these metrological systems is in a 'factor diagram', due to Friberg. The signs are arranged in ascending order of size from right to left, with arrows showing how many of the smaller unit makes one of the large unit. An English example would be
An archaic example is the sexagesimal system used for recording most discrete items.
For more on these systems see the Sumerian metrological systems page.
   Each of these series was used in specific contexts. The S system was used to count most discrete objects, such as sheep or people, and is the forerunner of the later abstract sexagesimal system. However, when counting discrete ration objects, such as cheese or fish, one would use a different, bisexagesimal, system B.  The the signs and their relationships were the same for these two systems when the quantities are small, but diverged for large quantities. Both of these systems used the 10 to 1 ratio between the wedge and circle. A third system, the ŠE system, was used to measure quantities of grain. This was the system uncovered by Friberg that has the 6 to 1 ratio between the circle and wedge. Hence, some symbols occur in multiple settings where they may take on differing relationships, and some symbols are reserved for particular systems. In order to develop a clear picture of the mathematics of the period it is necessary to sift through a large number of tablets and let the numerical relationships between the metrological signs gradually emerge (see [Nissen, Damerow and Englund: 25-27] for a brief discussion of the methodology involved).
   Almost all of our evidence for mathematics in the protoliterate period comes from such economic documents, although there are a few tablets that may be metrological exercises. From these economic tablets, we can determine the way quantity information was recorded, that is, the metrological numeration systems, but we cannot say anything about how the computations were performed. The most common operation is of course addition in finding a total of a disbursement or accumulation. Although large quantities may be recorded, these quantities are recorded using large units. The factors between successive units are small and there is no evidence at this stage for calculations being carried out in a base unit and then translated into the correct set of units. All that is required is the ability to add small numbers in given units. An arithmetically more complicated procedure is the requirement to disburse rations over a period of time, so necessitating a multiplication of the rations for a smaller unit of time. We do not know how these calculations were performed.

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Last modified: 30 August 2003
Duncan J. Melville

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