Computational Techniques and Computational Aids in Ancient Mesopotamia

Abstract

Any history of mathematics that deals with Mesopotamian mathematics will mention the use of tables of reciprocals and multiplication in sexagesimal place-value notation—perhaps also of tables of squares and other higher arithmetical tables. Less likely there is a description of metrological lists and tables and of tables of technical constants. All of these belong to a complex of aids for accounting that was created during the “Ur III” period (twenty-first c. bce). Students’ exercises from the Old Babylonian period (2000–1600 bce) teach us something about their use. First metrological lists then metrological tables were learned by heart. These allowed the translation of real measures into place-value measures in a tacitly assumed basic unit. At an advanced level, we see multiplications, where first two factors and then the product are written in sequence on a clay tablet for rough work. Problem texts show us more about the use of the metrological tables and the tables of technical constants. Neither genre allows us to see directly how additions and subtractions were made, nor how multiplications of multi-digit numbers were performed. A few errors in Old Babylonian problem texts confirm, however, that multiplications were performed on a support where partial products would disappear once they had been inserted—in a general sense, some kind of abacus. Other errors, some from Old Babylonian period and some others from the Seleucid period (third and second c. bce), show that the “abacus” in question had four or five sexagesimal levels, and textual evidence reveals that it was called “the hand”. This name was in use at least from the twenty-sixth c. bce until c. 500 bce. This regards addition and subtraction from early times onward, and multiplication and division in Ur III and later. A couple of problem texts from the third millennium deals with complicated divisions, namely divisions of large round numbers by 7 and by 33. They use different but related procedures, suggesting that no standard routine was at hand.