Reviewed by Lis Brack-Bernsen/University of Regensburg
Mathematics in Ancient Iraq: A Social History. By Eleanor Robson. Princeton, N.J.: Princeton University Press, 2008. 472 pp. $49.50 (cloth).
In this monumental book, the author includes all 957 cuneiform tablets (published before 2007) that in some way or other are concerned with mathematics in a broad sense: accounts, metrological lists, arithmetical lists or tablets, calculations and diagrams, and all kinds of mathematical exercises (word problems or model documents) as well as mathematical astronomy. By so doing, Eleanor Robson for the first time traces the origins and the development of mathematics in the ancient Middle East from its earliest beginnings in the fourth millennium b.c., continuing through six epochs to the mathematical and astronomical texts written during the late first millennium b.c., when cuneiform writing was gradually abandoned. The book witnesses a new and inspiring approach to the vast field of cuneiform mathematics. It does not replace former works but it presents their results from a new point of view, introducing mathematics as developed parallel to the development of writing and societies—as an integral and powerful component of cuneiform culture. The investigation includes linguistic considerations combined with analysis of the contents, methods, and concepts behind the mathematical tablets. Curriculum and teaching in the scribal schools during the different periods are analyzed, and new insight is gained from analysis of scribe families, mentioned in colophons. In addition, Robson treats the clay tablets as archaeological objects, where material and form give important information, not to speak of the whereabouts they were found and which other tablets were found near them (for the lucky cases where location and circumstances of the excavated tablets are known). [End Page 131]
When cuneiform mathematics was deciphered for the first time around 1930—a pioneering work that deserves our respect and admiration—only the internal, mathematical content was (and could be) considered. In this first approach the mathematical content of (mostly Old Babylonian, OB from now on) mathematical texts were reproduced in modern algebraic notation. Algebraic translations of the texts were treated further, and it became evident that OB mathematics was versatile and, for example, able to solve second-degree equations. In the 1980s, Jens Høyrup started analyzing the language used in “algebraic” texts from Old Babylonian time. He pointed at the fact that two different words were used for addition, and he succeeded in demonstrating that some geometrical figures had guided the “algebraic” calculations. It was not some known algebraic formulas or identities that determined the operations, but rather a geometric cut-and-paste technique that was utilized for many different mathematical exercises.1
Instead of concentrating on mathematics from the important OB period, from which 712 mathematical cuneiform tablets have come down to us, the author has taken a wider view of mathematics and numeracy and asked new questions. Seeking to trace mathematical thinking in the Mesopotamian culture, she has analyzed tablets from all periods of Mesopotamian history, searching for mathematical thoughts and practices within its social, religious, cultural, and historical context.
Chapter 1 gives an excellent and general introduction to the topic—a great help to all newcomers not familiar with Assyriology. Chapters 2–4 and 6–8 examine the six epochs into which the author has structured the cuneiform mathematics. Each chapter begins with background information and then presents texts and figures, representative for the period, followed by its arrangement into sociohistorical context. Three concentric circles or domains are analyzed: the inner zone is the scribal school with its teaching and methods, the middle zone is the sphere of practical work (utilizing techniques learned in the schools), and the outer zone goes beyond mathematical practices and includes for example ethno-mathematics or reliefs and Sumerian hymns. For each time period, the conclusions drawn from the material presented in the chapter are repeated in condensed form at the end of the chapter. [End Page 132]
History is not “what happened” but an interpretation of the past, arisen in the brain of the historian. The questions asked and the point of view taken by the historian determine the answers one gets. Jens Høyrup explained the fact that UR III mathematics (as compared to OB) was quite dull and mostly concerned with numeration and book-keeping from a political perspective: the despotic emperor Šulgi suppressed mathematical invention and freedom.2 Robson characterizes the UR III period as the epoch of standardization and approximation in which the sexagesimal place value system became the means of all calculations—while the new overwhelming mathematics of the following OB period is explained by the ideology of kingship: piety and justice exemplified through righteous and fair measurements. These virtues were transferred to the scribes and are seen by Robson as the prime motor for the blossoming of precise mathematics in OB times. She underpins this interpretation by all the Sumerian hymns praising these virtues of goddesses frequently copied in OB school exercises—and by the mathematical equipment “rod and ring” as symbols of kingship. The hymns and symbols are also found in the UR III period, but Robson uses them to explain OB mathematics by a political cultural ethic. I could imagine that mathematical interest, curiosity, and the joy of inventing and of solving problems may have led to the blossoming of OB mathematics, after the cut-and-paste method had been invented. Could it be that it was the other way around: that the OB kings adorned and described their ruling power by referring to the successful mathematics?
Apart from many numerical tables and texts with calculations or mathematical exercises, which are well explained in the text, the book contains a lot of figures and photos of tablets, many very informative and useful tables, an index of all tables mentioned, and an extensive bibliography and subject index. With such a comprehensive study, it is no wonder that printing errors occur—sometimes at places where it, quite disturbingly, may make it hard to follow the calculations or the interpretation of a text. But the book is a very significant contribution to the history of mathematics. It is well written, solidly founded and argued, and easy to understand. It is a fine and important addition to the literature on Babylonian mathematics, and it will be very useful [End Page 133] to readers from both inside and outside the field. The book is warmly recommended to everyone who is interested in mathematics and its history, in ancient cultures, or in science seen as an integrated part of culture, and to the broader public of historians of early science or Mesopotamian culture.
Footnotes
1. See Jens Høyrup, “Algebra and Native Geometry: An Investigation of Some Basic Aspects of Old Babylonian Mathematical Thought,”Altorientalische Forschungen 17 (1990).
2. Jens Høyrup, “How to Educate a Kapo, or Reflections on the Absence of a Culture of Mathematical Problems in UR III,” in Under One Sky: Astronomy and Mathematics in the Ancient Near East, ed. J. M. Steele and A. Imhausen, pp. 121–145, Alte Orient und Altes Testament 297 (Münster: Ugarit-Verlag, 2002).