Ur III Mathematics
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].
Go up to Third Millennium
Mathematics
Last modified: 30 August 2003
Duncan J.
Melville
Comments to dmelville@stlawu.edu