# Reassessing an ancient artifact

The famous Mesopotamian clay tablet known as Plimpton 322 has tantalized historians of mathematics ever since its discovery more than 60 years ago. Scholars have considered the tablet to be an anomalous mathematical exercise well in advance of its time. They have variously interpreted the cryptic columns of numbers, written in the wedge-shaped script called cuneiform, as a trigonometric table or a sophisticated scheme for generating Pythagorean triples. A Pythagorean triple is a set of three whole numbers, a, b, and c, such that a2 + b2 = c2.

Now, Eleanor Robson of the Oriental Institute at the University of Oxford in England offers an alternative explanation of the tablet’s purpose. The tablet served as a guide for a teacher preparing exercises involving squares and reciprocals, she suggests. Robson also pinpoints the tablet’s date to within 40 years of 1800 B.C. and says that it probably came from Larsa, a Mesopotamian city about 100 miles southeast of Babylon.

Previous historians had typically failed to consider the tablet’s cultural context and relied on later mathematical developments to infer its purpose. For example, the concept of angle measurement, which is essential for a trigonometric table, was not developed until nearly 2,000 years after the tablet was made. New scholarly approaches to Mesopotamian mathematics, however, combine historical, linguistic, and mathematical techniques to address questions such as, How did Mesopotamians approach mathematical problems, and what role did these problems play in their society? “We need to understand the document in its historical and cultural context,” Robson says. “Neglecting these factors can hinder our interpretations.”

By comparing Plimpton 322 with other ancient tablets, Robson established that its style is consistent with temple records and documents of about 1800 B.C. in Larsa. Scrutiny of various mathematical tablets revealed the importance of computational methods based on reciprocals (1/x) and squares (x2) of numbers. Robson also found examples of student exercises that consisted of problem lists, each one registering essentially the same problem with slightly different numbers.

Such evidence enables modern mathematicians to view Plimpton 322 “not as a freakish anomaly in the history of early mathematics but as the epitome of Mesopotamian mathematical culture at its best,” Robson says. “It’s a well-organized, well-executed, beautiful piece of mathematics.” Robson describes her findings in a report scheduled for publication in Historia Mathematica.