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Mathematical Tablets from Tell Harmal

Carlos Gonçalves
Publisher: 
Springer
Publication Date: 
2015
Number of Pages: 
141
Format: 
Hardcover
Series: 
Sources and Studies in the History of Mathematics and Physical Sciences
Price: 
89.99
ISBN: 
9783319225234
Category: 
Monograph
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Duncan J. Melville
, on
08/11/2016
]

As the discipline of Mesopotamian mathematics has matured, building upon the pioneering work of the first translators and interpreters, so the concerns of research have changed. The questions we ask, the answers we seek, and the interpretations we give are not the same as those of preceding generations. Along with this evolution of the research agenda, a steady flow of new text publications has improved readings of difficult or doubtful signs on often damaged cuneiform texts and enhanced the slow penetration of an often difficult technical vocabulary, built on distinct, ancient categories of thought that are only slowly being understood.

Together, these developments mean that the time is ripe for new editions of previously-published texts, treated in a modern fashion, and that task is what Carlos Gonçalves has undertaken in the book under review. Tell Harmal is the modern remains of the ancient city of Šaduppûm, located near Baghdad in what was, in the period that concerns us, the kingdom of Ešnunna. The site was first excavated by Taha Baqir from 1946 to 1949, with later campaigns in the late 1950s and early 1960s, as well as more recent excavations in 1997 and 1998. Altogether, some 2000 tablets have been recovered, with the mathematical group all being discovered in the early excavations of the 1940s.

The twelve tablets chosen by Gonçalves for this new edition were originally published, not always satisfactorily, by Baqir in a series of articles in the journal Sumer in 1950 and 1951. In the immediately ensuing period they attracted the attention of von Soden and Bruins, who provided improved and/or alternate readings and interpretations of some passages, and some of the texts have been treated by more recent commentators (Gonçalves includes a complete publication history of each tablet). The tablets themselves are (presumably still) in the Iraq Museum and not available for direct collation, and so one of the criteria for selection of the texts for this volume is that the original publication included hand copies and photographs of the tablet. For this reason the large mathematical compendium published by Goetze in 1951 had to be excluded.

Gonçalves opens with a brief description of the site of Tell Harmal and the archaeological context of the mathematical tablets, followed by a succinct summary of the present understanding of Old Babylonian mathematics covering terminology, techniques (e.g., cut-and-paste geometry), systems of measurements, and the distinction between quantities (called here measure numbers) and abstract numbers in Chapter 2. This chapter, in particular, will provide those looking for a brief introduction to Mesopotamian mathematics with a rapid, accurate, and up-to-date summary of current understanding.

Chapter 3, Conventions, describes the editorial choices Gonçalves made in preparing and interpreting his edition of the texts. This chapter is much more technical than Chapter 2 and, while there is some nice exposition of the minutiae of readings and interpretation that experts will enjoy, the chapter could be comfortably skipped on a first reading by non-experts and referred to as necessary. In Chapter 5, the author pulls back from the specific texts he has discussed to take a wider view of the place of Old Babylonian mathematics in the history of mathematics and in the overall assessment of Old Babylonian intellectual culture. Chapter 6 gives the vocabulary of the texts published in this volume with line references, but also some discussion of the semantic range of various terms.

The core of the book is Chapter 4. Gonçalves devotes one section of the chapter to each of the twelve tablets under consideration. The texts form a coherent corpus (ten of them were found in the same room) and all date from about 1800–1770 bce. Each text is presented in transliteration (with Sumerograms), transcription (in Akkadian), and translation, followed by both philological and mathematical commentary. The texts vary in size, orientation, and state of preservation. However, they are all problem texts, stating one or more problems and providing a solution procedure.

Across the twelve tablets, the problems cover a wide variety of topics, from those in plane geometry dealing with triangles, subdivided or striped triangles, rectangles, and quadrilaterals (possibly trapezia) , as well as a nice ‘broken reed’ puzzle about measuring a field, to problems involving excavations, brick transportation, wall construction, harvests, and market exchange topics. Gonçalves walks the reader through a clear discussion of each problem, including cases where the state of preservation or just plain missing data means that interpretation must remain conjectural, while also providing details of divergent readings among other scholars for those wishing to probe the topic more deeply.

The book is designed to be read with a variety of levels of interest, diligence, and expertise. Much of Mesopotamian mathematics appears in technical publications with a daunting apparatus and ferocious learning curve. Gonçalves’s book is decidedly more user-friendly. Those looking for a quick orientation to the study of Old Babylonian mathematics and maybe a nice problem or two for a class will find what they need in Chapter 2 and a skim through parts of Chapter 4. Those with a deeper interest but not yet much background will find an excellent introductory text that will guide them through a coherent group of texts that span a fair amount of the terrain of Old Babylonian mathematics and provide pointers for more advanced study. Additionally, this book brings together for the first time and gives a modern and unified interpretation of this important corpus of Ešnunna texts. The author is to be commended for a fine and useful publication.


Duncan J. Melville is a Professor of Mathematics at St. Lawrence University.

See the table of contents in the publisher's webpage.