Sargonic Mathematics

   For most of the third millennium, Mesopotamia was characterized by fairly small city-states vying for power and influence. In the last third of the millennium there arose two large empires.  The first of these was the Sargonic (c. 2350-2200), named after Sargon of Akkad. According to tradition, Sargon first came to power in Kish. He made Akkad (probably hitherto a small village) into a new capital of the world's first empire. The Sargonic kings were Semitic-speaking and their language is named Akkadian after their capital. The hundred and fifty years of the Sargonic Dynasty, and especially the long reigns of Sargon and his grandson Naram-Sin, always loomed large in Mesopotamian mythology. However, we have a frustratingly small amount of primary source information from this period and much of what we do know comes from much later documents. There have been few stratigraphically detailed digs from the Sargonic period. In particular, despite its role as the center of an empire, and despite evidence that after its period of glory it persisted into Hellenistic times, the present whereabouts of Akkad are unknown.

    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.

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Last modified: 30 August 2003
Duncan J. Melville

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