We all know or heard of Plimpton 322, the (almost) four thousand years old Babylonian tablet, part of a collection of cuneiform clay tablets found in an archeological site in Southern Iraq and bought by an archeologist-turned-dealer, E. J. Banks, who in turn sold some of them to the New York publisher G. A. Plimpton in 1922, and who ultimately donated them to Columbia University, where these tablets are now kept. Otto Neugebauer and his Assistant Abraham Sachs, (*Mathematical Cuneiform Texts*, American Oriental Society, 1945) interpreted the numbers of this tablet as *Pythagorean triples*, that is, non-trivial integer solutions to the equation \(x^2+y^2=z^2\), more than a thousand years before the Greeks found the Pythagorean theorem. Was the scribe a mathematician? What was his motivation to list that many Pythagorean triples?

The interpretation of O. Neugebauer, with all its romantic charm, has been challenged by more down-to-earth interpretations: either a table of values of trigonometric functions (R. C. Buck (1980), A*m. Math. Monthly ***87**, 335–345) or as list of exercises by an ancient colleague (E. Robson (2002), *Am. Math. Monthly* **109**, 105–120). As it is usually the case, making these interpretations is not straightforward. The tablet in question is a little bit damaged, for example: a part of it is missing and some symbols are not that clear to discern. Moreover, of the four-column table contained in this tablet, only the rightmost one is clear: It is a numbering of the rows, from top to bottom, with the numbers 1 to 15.

From left to right, the second column apparently has the word *width* at its heading, and the third column has the word *diagonal* at its heading. What Neugebauer observed, with these words as hints and assuming the translations are correct, was that the corresponding entries \(d\), \(w\) satisfy, with a few exceptions, that \(w^2-d^2\) is an integer square \(e^2\). That couldn’t be a coincidence! Finally, the entries in the last column are just the rational numbers \(d^2/e^2\). To make things more interesting the corresponding Pythagorean triples are *primitive*, i.e., \(w\) and \(d\) have no common factors.

A discussion of these ideas, with a possible explanation for the few mentioned exceptions, can be found in W. Casselman’s page. Which brings us to the book under review, one of those lovely Dover reprints of classical texts that otherwise would be available only as expensive used or collectible items.

The book is based on lectures by O. Neugebauer, given in 1949 and addressed to an interested audience not especially trained in mathematics, astronomy or ancient languages. Chapter two is mostly devoted to the discussion of Plimpton 322 within the frame of Babylonian mathematics. Chapter one could be thought of as preliminary to Chapter two, since it explains the Babylonian sexagesimal (base 60) system of numeration and the sophisticated arithmetic of this civilization. Chapter three discusses the sources and decipherment of Babylonian mathematical cuneiform texts, including some points on methodology.

Babylonian astronomy is discussed in Chapter five, again with authority since Neugebauer is also the author of the classical monograph *Astronomical Cuneiform Texts* (First edition, 1953, Second Edition, Springer 1983). Neugebauer emphasizes that, judging from a few surviving ephemerides, Babylonian astronomers obtained mathematically sophisticated descriptions of astronomical phenomena. Neugebauer argues that this sophistication is mainly mathematical since the recorded observations were relatively few.

Chapter four, devoted to Egyptian Mathematics and Astronomy, contrasts the very few contributions of Egyptians to mathematics, contrasted with the advanced level of the Babylonian contributions. Thus, Egyptian predictions of astronomical phenomena are practically null except for the Hellenistic period. Neugebauer describes their contributions as mainly observational.

The final chapter is devoted to the influence of Babylonian and Egyptian mathematics and astronomy on Greek science. I especially liked the initial quotation from Hilbert, regarding the measure of a scientific development by the number of previous works made superfluous. The towering Greek accomplishments collected in Euclid’s *Elements* or Ptolemy’s *Almagest *hide the contribution of all their predecessors, as the author reminds us.

This is a highly readable book, non-technical and accessible to everyone. The narrative flows quite nicely; as exciting as an ancient puzzle whose eventual solution keeps the reader turning pages.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx

## Comments

## Very good, but read on

I agree completely with this review, but note that the lectures were given in 1949. Historians have continued their research, so read more, especially Robson's Mathematics in Ancient Iraq and Katz's Sourcebook.