As the Sargonic empire crumbled under a combination of internal and external pressures, Mesopotamia reverted to a patchwork of minor city-states vying for importance. Nowadays, the best remembered of the rulers of these minor city-states is Gudea of Lagash, known for his many statues. Out of the jostling for power in southern Mesopotamia arose the kings who formed the Ur III empire (c. 2100-2000), one of the most centralized and bureaucratic states ever recorded. In his enormously influential article on Ur III bureaucracy, Steinkeller writes, "in terms of the resources concentrated and the level of governmental control exercised in their management, the Ur III state constitutes a unique phenomenon in the history of ancient Mesopotamia… never again did centralization reach such a high degree" [Steinkeller: 22].

The first king of the Ur III dynasty was Ur-Nammu. It is unclear exactly when or how far he extended his sway and his reign of 18 years seems to have been fairly peaceful. He engaged in a lot of temple building, including the famous ziggurat of Ur. The Ur III dynasty reached its zenith under the long reign of his successor, Shulgi. In the middle of his reign, Shulgi instituted a tremendous series of administrative, political and economic reforms. The system of weights and measures was reformed and a new calendar instituted; the writing system was changed, new administrative procedures were created and a huge bureaucracy developed [Steinkeller: 20ff]. From this point on, we have large numbers of records created by the vast administration and its need for continual, detailed accounting. We currently have around 100,000 Ur III tablets, mostly from a period of only 50 years. Of these, about half or less have been published (mostly just in copy or transliteration) and probably less than a tenth subjected to any concentrated study.

The overwhelming majority of Ur III tablets are economic. The huge numbers of scribes needed to maintain the bureaucracy must have been trained, and, given the nature of their society, probably trained in a standardized curriculum. Shulgi himself boasts of his skills in 'counting and accounting'. However, there are very few mathematical or educational texts. The paucity of sources and focus on practical matters has led Hoyrup to comment, "Ur III mathematics appears to have been strictly utilitarian in orientation" [1994: 22].

In contrast to the murky picture during the Sargonic period, there is clear evidence for Ur III scribes using the sexagesimal place value system [Powell 1976a]. But calculations were performed on 'scratch' tablets, which were then erased and re-used after the answers had been copied into the main document. By their nature, few of these rough work tablets have survived, and so we know little about how the calculations were performed. From the whole Ur III corpus, we have only one metro-mathematical problem text, where the student has to calculate the volume of a wall and the quantity of bricks needed to build it. The calculations on the back of the tablet use the sexagesimal place value notation [Robson 1999: 66]. Apart from this single tablet, two unusual reciprocal tables are often assumed to be from Ur III [Friberg 1987: 541]. However, the identification of these tablets as coming from Ur III rather than Old Babylonian depends on their unusual features, rather than archaeological context, leading some scholars to be more cautious. For example, Robson goes only so far as to state that they are "less certain of Ur III date" than the problem text [1999: 171].

These are the only documents that are clearly 'school mathematics'. Within the corpus of actual archival texts, we can discern some mathematical developments. Most of the Ur III documents are economic, and an important innovation was the use by the bureaucracy of theoretical work-norms for administrative purposes. That is, a foreman would be assigned a certain work-gang for a certain period of time, to perform specific types of work. The bureaucrats would decide how much work should be accomplished using standard conversions and norms. If the workers produced more than required, it was counted as a credit for the foreman; if less (which was more likely), then as a debit to be carried over to the next year. The plight of the workers was unenviable: "the expected labor performance was in all likelihood simply beyond the capabilities of the normal worker. Moreover, an incentive for the workers to produce more was nonexistent; their remuneration consisted of no more than the minimum amount of grain and clothing required to keep them able to produce" [Englund 1991: 280]. The foreman's lot was not much better. If he died while in debt, the state seized his property, including remaining family members, to pay off the balance [Nissen, Damerow and Englund: 54; Englund 1991: 267-268].

An additional insight into Ur III mathematics is given by a collection of field plans. In order to specify the amount of grain needed to sow in a field, or compute the harvest, administrators needed to know the area of a field. Fields were often of complicated shapes, so, to compute the area, scribes divided the field up into triangular and quadrilateral pieces. The area of triangles was given as half the base times the length of the side, implicitly assuming the side was measured off perpendicularly to the base (field plans were not drawn to scale). Quadrilateral pieces were usually computed via the 'agrimensor' formula - the area is the average length times the average width, but for some complicated cases, the areas are computed twice, first using one base and side, then the side opposite the base was used and the two results averaged.

The meager evidence at our disposal does not allow us to make a more detailed assessment of Ur III mathematics at this time. However, it is clear that at the close of the third millennium, all the pieces were in place to provide the background for the Old Babylonian mathematics that flourished during the next four hundred years. Indeed, Robson has noted that excepting the coefficient lists (with which she was most closely concerned) all other types of mathematical texts known from the Old Babylonian period were used in the third millennium [1999: 169].

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Mathematics

Last modified: 30 August 2003 Duncan J. Melville

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