The kings of the Sargonic dynasty attempted to control the former city-states by appointing rulers from either the royal family or people who owed primary allegiance to the king rather than the local people. A major administrative reform was the introduction of standardized year-names. However, there is little evidence during the Sargonic period of the vast bureaucracy that developed later in the Ur III period. During the Sargonic period much of the documentation is in Akkadian, although there are many texts still written in Sumerian, and administrative and economic documents are known from a wider variety of places than during previous periods. For more on the history of this period, see [Hallo and Simpson: 51-65] or [Kuhrt: 44-55] and the references in these works.

Our knowledge of the mathematics of the period must be gleaned from economic documents (which show that 'numbers' were now written in cuneiform) and a total of only a dozen or so metro-mathematical tablets. One of these is a geometric problem concerning a partitioned trapezoid, a forerunner of a class of problems well-known from the Old Babylonian period and later. The rest form a group of problems concerning fields in the shapes of squares and rectangles. One typical example is the following: 3600 + 5×60 nindan minus 1 'seed-cubit' is the side of a square. Its area: 2 šar-gal, 2 šar-u, 4 bur-u, 9 bur 5 1/8 iku, 5 1/2 sar 1 gin 2/3 še is found.

It is difficult to draw certain conclusions from such a small amount of evidence. However, closely analyzing these texts, Powell [1976a, 1976b] and Whiting [1984] have forcefully argued that these problems betray the usage of a place-value system and the abstract 'sexagesimal' scientific system known from the Old Babylonian period. The abstract sexagesimal system may have arisen from the abstract generalization of the weight unit gin to indicate one-sixtieth of any unit in a metrological system [Whiting: 61 n. 6]. Powell has buttressed his arguments by a study of the Sargonic metrological reforms, which he claims were intended to facilitate calculation in the new sexagesimal system [1976b: 99]. Whiting notes, "sexagesimal notation was being used to perform calculations in the Old Akkadian period and … instruction in these techniques was being carried out at Lagash/Girsu and probably at Nippur" [Whiting: 66], while Powell observes, "In the Akkad/Ur III period… length measures were defined to relate systematically to area, volume, capacity and perhaps to weight. This scientific system struck deep roots and was incorporated into the mathematical text-book tradition" [1987: 458].

Friberg [1987] has commented on the occurrence of what he terms 'wide-span' numbers in Sargonic texts. These are numbers using both very large and very small units (as in the example quoted above). The effect is that the student must demonstrate great technical facility with the computational system. There is an emphasis on virtuosity rather than depth.

Further refinements of our understanding of Sargonic mathematics will have
to await new discoveries, but it is certainly clear that many of the mathematical
techniques, problems and concerns we know from the Old Babylonian period
had their origins in the third millennium.

Go up to Third Millennium
Mathematics

Last modified: 30 August 2003 Duncan J. Melville

Comments to dmelville@stlawu.edu