Mathematics in the third millenniumTo what extent the mathematical advances attested in Old Babylonian and Seleucid
documents existed in earlier centuries is difficult to know, principally due to a shortage
of sources. On the whole, we seem to see a steady development of complexity and
achievement.
Pre-Sargonic mathematicsPossibly mathematical exercises existed as early as the late Uruk period (Robson 2008 :
29 ). But, as Robson notes, “Up to the period around 2400 BCmathematics did not
have a very strong self-identity. Its terminology, subject matter, methodology, and
conceptualization were adopted directly from the culture of numerate bureaucracy
from which it developed, and which it directly served” (Robson 2008 : 51 ).
Although there is clear evidence in the administrative texts for well-developed
abilities to handle complicated accounting and metrological calculations, there are very
few examples of mathematical exercises. From the excavations at Fara are a cache of
approximately sixty school exercises dated to c. 2450 BC. Five tablets from here can be
securely identified as mathematical. One is a table of squares, four are problems for
the student to solve.^7 Archives from Ebla and Adab may yield a further five or six
mathematical tablets (Robson 1999 : 168 ; Robson 2008 : 32 ). On the basis of such scarce
evidence, we obviously cannot draw many conclusions, aside from the obvious one,
that the florescence of mathematical abilities evinced in later cuneiform archives had
mid-third millennium forerunners.
Sargonic mathematicsIt is with the Sargonic period that major achievements in mathematics begin to appear.
This may have been a new development or simply may appear to be new due to a
paucity of surviving evidence from previous centuries.
Administrative texts indicate a trend toward commodification of labor (Robson
2008 : 67 ) and a desire on the part of the administration to predict and estimate for
the future, as opposed to tallying up actual results (Robson 2008 : 69 ).
However, the clearest evidence for mathematical knowledge comes from school
texts, specifically, mathematical exercises, known from this period. Robson identified
twenty-two published Sargonic texts, mainly from Girsu (Robson 2008 : 55 ). Many are
problems: e.g. “ 4 , 3 nindan[is the] side. The front [is such that it encloses an area of ]
1 iku. Its front is to be found” (Powell 1976 : 425 – 6 ; Robson 2008 : 55 .)^8 Additionally,
there are model accounts, silver weights and a geometrical diagram (Robson 2008 : 55 ).
Powell described these as “the oldest well-defined group of cuneiform documents
showing an unequivocal interest in playing with numbers” (Powell 1976 : 422 ).
Another important mathematical skill that began to develop in the Sargonic period
relates to how to write numbers via positional notation.^9 Previous generations of
scholarship had considered the Old Babylonian period the apogee for early Babylonian
mathematics. One key element in this valuation was its use of positional notation. In
the words of Boyer, “The secret of the clear superiority of Babylonian mathematics...
undoubtedly lies in the fact that those who lived ‘between the two rivers’ took the most
felicitous step of extending the principle of position to cover fractions as well as whole
–– Calendars and counting ––