N° 452
Algebra in cuneiform : introduction to an old Babylonian geometrical technique

Around 1930, it was discovered that certain Babylonian cuneiform texts contain calculations that agree with what turns up in the solution of second-degree equations. Since the meaning of most of the terminology had to be derived from the numbers contained in the texts, this led to a reading of these as numerically based algebra. This interpretation stood unchallenged until the author of the present book discovered around 1982 that it was incompatible the global structure of the terminology. As it turns out, two different and non-synonymous operations had both been understood as addition; two different subtractive operations had been conflated, and four different operations had been seen as one and the same multiplication. Instead, the structure points to a technique based on a geometry of squares and rectangles with measurable sides and areas. Avoiding such philological detail as would only be informative for readers that are familiar with basic Assyriology (yet with appendixes meant for these), the book analyses a number of texts in "conformal translation", that is, a translation in which the same Babylonian term is always translated in the same way and, more important, different terms are always translated differently. All of these texts are from the second half of the Old Babylonian period, that is, 1800-1600 BCE. It is indeed during this period that the "algebraic" discipline, and Babylonian mathematics in general, culminates. Even though a few texts from the late period show some similarities with what comes from the Old Babylonian period, they are but remnants. Beyond analyzing texts, this preprint gives a general characterization of the kind of mathematics involved, and locates it within the context of the Old Babylonian scribe school and its particular culture. Finally, it describes the origin of the discipline and its impact in later mathematics, not least Euclid's geometry and genuine algebra as created in medieval Islam and taken over in European medieval and Renaissance mathematics.