numbers.... Accuracy in approximations was relatively easy for the Babylonians to
achieve with their fractional notation, the best that any civilization afforded until the
time of the Renaissance” (Boyer 1991 : 27 ). The small Sargonic corpus of mathematical
texts seems to display the germ of positional notation, as noted by Powell in 1976 :
“Astonishing as it may seem, the conclusion forced upon us by these texts is that
calculations involving the conceptual framework implicit in Old Babylonian place
notation were already being performed in the Sargonic period... it is well to keep in
mind the wise admonitions of O. Neugebauer... who has repeatedly stressed that the
written record constitutes an insufficient witness to the real nature of Babylonian
mathematical thought” (Powell 1976 : 427 ).
However, the full development of positional notation cannot be demonstrated until
the next era, the Third Dynasty of Ur.
Mathematics in the Third Dynasty of UrOur knowledge of mathematics in the Third Dynasty of Ur is mixed. While we have
an (over-) abundance of archival material, which allows us to see mathematical and
metrological knowledge in practice, we have almost no school tablets from this period
due to accidents of preservation and discovery. The known corpus of Ur III mathe-
matical texts is therefore ridiculously small – identified by Robson as eight tablets, three
of them accounting exercises (Robson 1999 : 171 , 2008 : 59 ). From the extant evidence,
two new mathematical developments manifest themselves in the Ur III period: first,
reciprocal tables, which were useful in allowing a scribe to divide easily to calculate
volumes, building calculations and workdays (Robson 2008 : 82 ). Second, and more
importantly, we have clear evidence from a tablet dating to the reign of Amar-Sin that
sexagesimal place value notation had been developed.^10
NUMERACY AND SCRIBAL TRAINING IN THE THIRD
MILLENNIUM BCThe tablets that do survive, however, allow us a window into how a scribe for a “great
institution,” that is, a temple household or one run by the state, used and understood
numbers.
Scribal training did not produce merely men who could read and write: the scribal
school trained boys for a number of jobs, including many important administrative/
accounting positions. Numeracy was critical for these positions and hence had an
important role in the school curriculum. Most of what we know about schools comes
from the early second millennium, that is, the Old Babylonian period. Interestingly,
many Old Babylonian teachers held administrative positions too (Robson 2002 : 118 ).
Old Babylonian school dialogues, often taking the form of jibes or insults hurled by
one schoolboy against another, stress that a scribal graduate had to master a number
of subjects.
You have written a tablet, but you cannot penetrate its meaning; you have written
a letter [but] that is all you can do.^11
Do you know multiplication, reciprocals, coefficients, balancing of accounts,
administrative accounting, how to make all kinds of pay allotments, divide property,
and delimit shares of fields?^12–– Tonia Sharlach ––