Next we come to the Early Dynastic (ED) period, stretching over the bulk of the third millennium, from around 2900 to 2350. The ED period is conventionally broken into ED I, ED II and ED III sub-periods on an archaeological basis. This was an immensely important stage of the development of civilization, but unfortunately we have few remains and there are huge gaps in our understanding, especially for the earlier ED I and II periods.

During the first half of the third millennium, the water-levels continued to drop in southern Mesopotamia and the land dried out [Nissen 1988: 129]. At first this decline in the water-levels had drained marshland, allowing the great increase in population and productivity in southern Babylonia during the Late Uruk period. In contrast, the continued disappearance of water in the Early Dynastic period forced more and more of the population into fewer settlements, concentrated more closely on the major rivers. By the late ED period, as much as 80 or 90 percent of the populace lived in cities [Kuhrt: 31; Nissen 1988: 131]. As Hans Nissen states, by this time, "small settlements out in the countryside had almost ceased to exist."[1988: 130]

These city-states certainly did contend at times, probably mostly over water, but they also engaged in trade, not only with each other, but over a wide area of Mesopotamia and beyond. Our evidence is so fragmentary that it is difficult to make general statements about the political and social systems during this 500-year period, especially about the role of the king and the local religions. For an analysis of the history of the period and discussion of the problems interpreting the evidence, see [Kuhrt: 27-44] and [Nissen 1988: 129-164].

There are only a few substantial collections of tablets from the ED period: there are several hundred from Ur in ED II; large groups from Šuruppak (modern Fara) and Abu Salabikh (ancient name unknown) from around 2500, and a big administrative archive from Girsu at the very end of ED III. Writing developed tremendously in this half-millennium. The number of signs was reduced to about half and sign-shapes became more regularized and abstract. Also, there was a greater use of syllabic signs which in turn led to writing reflecting the grammar of the spoken language (Sumerian) and thus to its being used for a wider range of purposes, including literature and history.

Included in these collections of tablets are many economic documents which allow us to see changes in systems of writing and metrology, but do not give much indication of the mathematics underlying the numbers recorded. There are very few ED metro-mathematical texts. Each of these provides a unique and valuable glimpse into the mathematics of the time. A few texts from Šuruppak provide examples of complex arithmetical problems, the first multiplication table, and the first geometrical exercise.

The texts from Šuruppak were excavated by a German expedition in 1902-3 although many of the tablets were only published in the 1920's and 1930's (see Deimel and Jestin). Among them are several that may be considered mathematical exercises with greater or lesser confidence [Powell 1976a, n.19]. These texts clearly indicate that mathematics was being taught in some structured way around 2500, although we do not have details of how, or where, such instruction took place. Remarkably, two of the tablets, TSŠ 50 and TSŠ 671 concern the same problem. A loose translation of the problem is: A granary. Each man receives 7 sila of grain. How many men? That is, the tablets concern a highly artificial problem and certainly present a mathematical exercise and not an archival document. The tablets give the statement of the problem and its answer (164571 men - expressed in the sexagesimal system S since we are counting men - with 3 sila left over). However, one of the tablets gives an incorrect solution. When analyzing these tablets, Marvin Powell commented famously that it was, "written by a bungler who did not know the front from the back of his tablet, did not know the difference between standard numerical notation and area notation, and succeeded in making half a dozen writing errors in as many lines." [1976a: 432] Of course, errors help us understand procedures in ways that correct computations often cannot. Noting that 7 is the only irregular divisor under 10, Powell proposed that the tablets showed evidence of a place-value system and computation of reciprocals of irregular numbers, "the two answers obtained to the problem are explicable by a single hypothesis, but only if one assumes multiplication by the reciprocal." [1976a: 433] Powell's interpretation rests upon the two tablets using different approximations to the reciprocal of 7. In the case of the tablet with the correct answer, this approximation must have been correct to four sexagesimal places.

Re-analyzing the tablets, Høyrup [1982] proposed an alternative interpretation that did not require the students to have any understanding of reciprocals of irregular numbers, or an abstract place-value system. Høyrup's interpretation stays much closer to the underlying metrology (essentially it involves first dividing by seven and then changing the units using a multiplicative factor), although it does require thinking about some numbers in a sexagesimal system. It is hard to see how one should have written down some of the intermediate calculations in the prevailing quantity systems. Høyrup also does not explain the choice of 7 as the divisor. Høyrup cautiously concludes, "although analysis of the tablets suggests a mathematical mode of thought closely connected to current metrology and not yet familiar with the concept of place value, it does not necessarily imply that these were general characteristics of Fara mathematics."[1982: 31] A more recent analysis has suggested that the problem be viewed not as a division problem, but as one involving repeated addition [Melville 2002].

Another text from Šuruppak, contains the oldest known multiplication table, in fact, a table of squares. It purports to give the areas of square fields with sides of lengths from 10 (x60 ninda), decreasing in steps of 60 ninda to 1 (x60 ninda) on the obverse, and from 5 (x10 ninda), to 1 (x10 ninda), decreasing in steps of 10 ninda on the reverse, with a final entry of 5 ninda. The lower part of the reverse is broken; there is room for more entries. Apart from the actual calculations involved, the tablet displays some interesting characteristics. First, the organization of the computations in smoothly descending size is in contrast to the way later Old Babylonian multiplication tables are laid out; these always proceed from the smallest to the largest. As Powell noted, "real metrological tables are organized according to the "list" principle and have an entirely different format." [1976a: 430] Secondly, during the 500 years from the archaic period to the time this tablet was written there had been a steady simplification and rationalization of the many archaic metrological systems, as well as an expansion in their size. The lengths are now written in the standard sexagesimal system, while the resulting areas are written in the old area system. Damerow and Englund observe, "The number of basic numerical sign systems had been reduced to the two applied on this list, namely the sexagesimal system and the area measure system." [Nissen, Damerow and Englund: 139] They also note that in the head of the table, the first side of the field "is qualified with the word 'sag,' meaning 'head,' " terminology that would be used widely in Old Babylonian mathematics.

The last significant find from Fara is a small fragment of an exercise text containing a square inscribed with four circles. There is no writing on what we have of the tablet, so we cannot know what problem was being dealt with, but the tablet provides our first evidence of geometrical instruction [Powell 1976a: 431; Friberg 1987: 540].

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Millennium Mathematics

Last modified: 30 August 2003 Duncan J. Melville

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