"This work is an enormously significant contribution to the history of mathematics. No other work surveys the vast landscape of Mesopotamian mathematics from a position of the modern understanding of the past, incorporating the latest scholarship and yet still managing to be so accessible to nonspecialists. Robson's book is an outstanding guide that can be consulted by anyone interested in the field."--Duncan J. Melville, St. Lawrence University

"A very significant contribution to the history of ancient mathematics, and to the history of mathematics in general. I anticipate this book will be very, very useful to readers outside the field and general readers, because it is very clearly and incisively written, it gives clear indications about where to find the primary sources, and it summarizes previous historiography very effectively. There is no comparable book."--Serafina Cuomo, Birkbeck College, London

"A truly exciting and highly readable intellectual history of ancient Iraq. Eleanor Robson'sMathematics in Ancient Iraq is destined to become a classic. The book will have a wide readership among people interested in Near Eastern archaeology, the origins of writing, the history of education, and the history of science. This is truly a magnificent read."--Gary Urton, Harvard University

"This is a wholly original work, the first to integrate all the recent work from the history of mathematics in this area with archaeological scholarship. The result is a remarkably rich portrayal of mathematics in ancient Iraq. The breadth of coverage is striking, both in chronological terms but also in broad intellectual ones. The book is very well written and handsomely illustrated."--Jeremy Gray, The Open University

"This work is an enormously significant contribution to the history of mathematics. No other work surveys the vast landscape of Mesopotamian mathematics from a position of the modern understanding of the past, incorporating the latest scholarship and yet still managing to be so accessible to nonspecialists. Robson's book is an outstanding guide that can be consulted by anyone interested in the field."--Duncan J. Melville, St. Lawrence University

"A very significant contribution to the history of ancient mathematics, and to the history of mathematics in general. I anticipate this book will be very, very useful to readers outside the field and general readers, because it is very clearly and incisively written, it gives clear indications about where to find the primary sources, and it summarizes previous historiography very effectively. There is no comparable book."--Serafina Cuomo, Birkbeck College, London

"A truly exciting and highly readable intellectual history of ancient Iraq. Eleanor Robson'sMathematics in Ancient Iraq is destined to become a classic. The book will have a wide readership among people interested in Near Eastern archaeology, the origins of writing, the history of education, and the history of science. This is truly a magnificent read."--Gary Urton, Harvard University

"This is a wholly original work, the first to integrate all the recent work from the history of mathematics in this area with archaeological scholarship. The result is a remarkably rich portrayal of mathematics in ancient Iraq. The breadth of coverage is striking, both in chronological terms but also in broad intellectual ones. The book is very well written and handsomely illustrated."--Jeremy Gray, The Open University

Mathematics in Ancient Iraq

Princeton University Press

Chapter One

The mathematics of ancient Iraq, attested from the last three millennia BCE, was written on clay tablets in the Sumerian and Akkadian languages using the cuneiform script, often with numbers in the sexagesimal place value system (1.2). There have been many styles of interpretation since the discovery and decipherment of that mathematics in the late nineteenth and early twentieth centuries CE (1.1), but this book advocates a combination of close attention to textual and linguistic detail, as well as material and archaeological evidence, to situate ancient mathematics within the socio-intellectual worlds of the individuals and communities who produced and consumed it (1.3).

1.1 THE SUBJECT: ANCIENT IRAQ AND ITS MATHEMATICS

Iraq-Sumer-Babylonia-Mesopotamia: under any or all of these names almost every general textbook on the history of mathematics assigns the origins of 'pure' mathematics to the distant past of the land between the Tigris and Euphrates rivers. Here, over five thousand years ago, the first systematic accounting techniques were developed, using clay counters to represent fixed quantities of traded and stored goods in the world's earliest cities (2.2). Here too, in the early second millennium BCE, the world's first positional system of numerical notation-the famous sexagesimal place value system-was widely used (4.2). The earliest widespread evidence for 'pure' mathematics comes from the same place and time, including a very accurate approximation to the square root of 2, an early form of abstract algebra, and the knowledge, if not proof, of 'Pythagoras' theorem' defining the relationship between the sides of a right-angled triangle (4.3). The best-known mathematical artefact from this time, the cuneiform tablet Plimpton 322, has been widely discussed and admired, and claims have been made for its function that range from number theory to trigonometry to astronomy. Most of the evidence for mathematical astronomy, however, comes from the later first millennium BCE (8.2), from which it is clear that Babylonian astronomical observations, calculational models, and the sexagesimal place value system all had a deep impact on the later development of Old World astronomy, in particular through the person and works of Ptolemy. It is hardly surprising, then, that ever since its discovery a century ago the mathematics of ancient Iraq has claimed an important role in the history of early mathematics. Seen as the first flowering of 'proper' mathematics, it has been hailed as the cradle from which classical Greek mathematics, and therefore the Western tradition, grew. But, as laid out over the course of this book, the mathematical culture of ancient Iraq was much richer, more complex, more diverse, and more human than the standard narratives allow.

The mathematical culture of ancient Iraq was by no means confined to the borders of the nation state as it is constructed today. The name al-'Iraq(Arabic 'the river shore') is first attested about a century after the Muslim conquests of the early seventh century CE, while the lines on modern maps which delimit the territory of Iraq are the outcome of the division of the collapsing Ottoman empire amongst European powers at the end of the First World War. The mathematics of pre-Islamic Iraq, as it has been preserved, was written on small clay tablets in cuneiform writing. Because, as argued here, mathematics was an integral and powerful component of cuneiform culture, for present purposes it will be a useful first approximation to say that cuneiform culture and mathematical culture were more or less co-extensive. The core heartland of the cuneiform world was the very fl at alluvial plain between Baghdad and the Gulf coast through which the Tigris and Euphrates flow (figure 1.1). It was known variously in antiquity as Sumer and Akkad, Babylonia, Kardunias, or simply The Land. The Land's natural resources were primarily organic: reeds, small riverine trees, and other plant matter, but most importantly the earth itself. Alluvial clay was the all-purpose raw material par excellence, from which almost anything from sickle blades to monumental buildings could be manufactured. Equally, when judiciously managed the soil was prodigiously fertile, producing an abundance of arable crops (primarily barley), as well as grazing lands for herds (sheep and goats but also cattle). Even the wildest of marshlands were home to a rich variety of birds and fish and the all-purpose reeds, second only to clay in their utility. What the south lacked, however, were the trappings of luxury: no structural timber but only date-palms and tamarisks, no stone for building or ornamentation other than small outcrops of soft, dull limestone, and no precious or semi-precious stones at all, let alone any metals, base or precious. All these had to be imported from the mountains to the north, east, and west, in exchange for arable and animal products.

The centre of power shifted north at times, to northern Iraq and Syria east of the Euphrates, known in ancient times as Assyria, Subartu, Mitanni, or the land of Assur. Life here was very different: rainfall could be counted on for wheat-based agriculture, building stone was abundant, and mountainous sources of timber and metal ores relatively close to hand. Conversely, the dates, tamarisks, and reeds of the south were absent here, as were the marshes with their rich flora and fauna. Overland trade routes ran in all directions, linking northern Iraq with the wider world.

The fluid peripheries over which these territories had at times direct political control or more often cultural influence expanded and contracted greatly over time. At its maximum extent cuneiform culture encompassed most of what we today call the Middle East: the modern-day states of Turkey, Lebanon, Syria, Israel and the Palestinian areas, Jordan, Egypt, and Iran. Chronologically, cuneiform spans over three thousand years, from the emergence of cities, states, and bureaucracies in the late fourth millennium BCE to the gradual decline of indigenous ways of thought under the Persian, Seleucid, and Parthian empires at around the beginning of the common era. The history of mathematics in cuneiform covers this same long stretch and a similarly wide spread (table 1.1).

The lost world of the ancient Middle East was rediscovered by Europeans in the mid-nineteenth century (table 1.2). Decades before the advent of controlled, stratigraphic archaeology, the great cities of Assyria and Babylonia, previously known only through garbled references in classical literature and the Bible, were excavated with more enthusiasm than skill, yielding vast quantities of cuneiform tablets and objets d'art for Western museums. The complexities of cuneiform writing were unravelled during the course of the century too, leading to the decipherment of the two main languages of ancient Iraq: Akkadian, a Semitic precursor of Hebrew and Arabic; and Sumerian, which appeared to have no surviving relatives at all.

In the years before the First World War, as scholars became more confident in their interpretational abilities, the first mathematical cuneiform texts were published. Written in highly abbreviated and technical language, and using the base 60 place value system, they were at first almost impossible to interpret. Over the succeeding decades Franois Thureau-Dangin and Otto Neugebauer led the race for decipherment, culminating in the publication of their rival monumental editions, Textes mathmatiques babyloniens and Mathematische Keilschrifttexte, in the late 1930s. By necessity, scholarly work was at that time confined to interpreting the mathematical techniques found in the tablets, for there was very little cultural or historical context into which to place them. For the most part the tablets themselves had no archaeological context at all, or at best could beat tributed to a named city and a time-span of few centuries in the early second millennium BCE. The final reports of the huge and well-documented excavations of those decades were years away from publication and nor, yet, were there any reliable dictionaries of Akkadian or Sumerian.

After the hiatus of the Second World War, it was business as usual for the historians of cuneiform mathematics. Otto Neugebauer and Abraham Sachs's Mathematical cuneiform texts of 1945 followed the paradigm of the pre-war publications, as did Evert Bruins and Marguerite Rutten's Textes mathmatiques de Suse of 1961. Neugebauer had become such a towering figure that his methodology was followed by his successors in the discipline, though often without his linguistic abilities. Cuneiformists put mathematical tablets aside as 'something for Neugebauer' even though he had stopped working on Babylonian mathematics in the late 1940s. Since there was almost no further output from the cuneiformists, historians of mathematics treated the corpus as complete. In the early 1950s the great Iraqi Assyriologist Taha Baqir published a dozen mathematical tablets from his excavations of small settlements near Baghdad, but virtually the only other editor of new material was Bruins, who tended to place short articles in the small-circulation Iraqi journal Sumer (as did Baqir) or in Janus, which he himself owned and edited. All attempts at review or criticismmet with such vitriolic attacks that he effectively created a monopoly on the subject.

Meanwhile, since Neugebauer's heyday, other aspects of the study of ancient Iraq had moved on apace. The massive excavations of the pre-war period, and the more targeted digs of the 1950s onwards, were being published and synthesised. The monumental Chicago Assyrian dictionary gradually worked its way through the lexicon of Akkadian, volume by volume. The chronology, political history, socio-economic conditions, and literary, cultural, religious, and intellectual environments of Mesopotamia were the subjects of rigorous, if not always accessible, scholarship. In the course of the 1970s and '80s attention turned to much earlier mathematical practices, as scholars led by Marvin Powell and Jran Friberg found and analysed the numeration, metrology, and arithmetic of the third millennium BCE, from sites as far apart as Ebla in eastern Syria and Susa in southwestern Iran. Denise Schmandt-Besserat began to formulate her mould-breaking theories of the origins of numeracy and literacy in the tiny calculi of unbaked clay that she had identified in prehistoric archaeological assemblages all over the Middle East.

Nevertheless, it would be no exaggeration to say that between them, Neugebauer's renown for scholarly excellence and Bruins's reputation for personal venom seriously stifled the field of Babylonian mathematics until their deaths in 1990. It is perhaps no coincidence that 'Algebra and navegeometry', Jens Hyrup's seminal work on the language of Old Babylonian algebra, was also published in that year, signalling a paradigm shift away from the history of Mesopotamian mathematics as the study of calculational techniques and their 'domestication' into modern symbolic algebra. Hyrup's work was in effect a discourse analysis of Mesopotamian mathematics: a close scrutiny of the actual Akkadian words used, and their relationship to each other. In this way he completely revolutionised our understanding of ancient 'algebra', showing it to be based on a very concrete conception of number as measured line and area. An interdisciplinary project based in Berlin developed further important new methodologies in the early 1990s, leading to the computer-aided decipherment of the complex metrologies in the proto-literate temple accounts from late fourth-millennium Uruk which had resisted satisfactory interpretation for over eighty years. Uruk also provided new sources from the other end of the chronological spectrum, as Friberg published mathematical tablets from the latter half of the first millennium BCE.

In the past decade, large numbers of new mathematical tablets have come to light, both from excavation and from renewed study of old publications and large museum collections, and are now attested from almost every period of cuneiform culture. The published corpus now comprises over 950 tablets (table B.22). Still the largest body of evidence, though, is the pedagogical mathematics-exercises set and solved, metrological and mathematical tables copied and recopied-from the early second millennium BCE or Old Babylonian period. This currently accounts for over 80 percent of the published sources, not far short of 780 tablets. There are fewer than sixty known mathematical tablets from the whole of the third millennium, on the other hand (about 6 percent), and just over twice that number from the millennium and a half after 1500 BCE (some 13 percent). Thus the main focus of attention is still therefore on the large body of Old Babylonian material.

With some exceptions, the new generation of scholarship has taken a long time to filter through to the wider historical community. Cuneiformists have been put off by technical mathematics, historians of mathematics by technical Assyriology. Thus mathematics tends to be ignored in general histories of the ancient Near East, and even though it has an inviolable place at the beginning of every maths history textbook, the examples found there are for the most part still derived from a few out-of-date general works. Neugebauer's The exact sciences in antiquity, first published in 1951, was justly influential, but Van Der Waerden's derivative Science awakening (first English edition 1954) and later Geometry and algebra in ancient civilizations (1983) are both deeply Eurocentric and diffusionist. All in all it is time for a new look, from a new perspective-which is what this book sets out to do.

1.2 THE ARTEFACTS: ASSYRIOLOGICAL AND MATHEMATICAL ANALYSIS

Perhaps the most important methodological thread running through this book is that although mathematics is most immediately the product of individuals, those individuals are shaped and constrained by the society in which they live, think, and write. In order to understand the mathematics of a particular people as richly as possible, historians need to contextualise it. This approach is especially important for comprehending the mathematics of ancient Iraq, where anonymous tradition was prized over named authorship and we are more often than not completely unable to identify the work or influence of individuals within the written tradition. But context, crucial though it is, has to be paired with scrupulous attention to the mathematical, linguistic, and artefactual details of the tablets themselves. In order to demonstrate this, on the following pages a typical example, in the standard style of primary publication, is used to explain the basics of the media, script, numeration, and language of the sources, and to exemplify the usual methods of decipherment, interpretation, and publication. The final section demonstrates some of the different ways in which contextualisation can add layers of meaning to the interpretation of individual objects.

The primary publication of a cuneiform tablet should normally comprise at least a hand-copy (scale drawing) and transliteration, and often a photograph and translation as well. The sample tablet, 2N-T 30 (figure 1.2), has been partially published twice before: once as a rather blurry photograph, and once as a transliteration and translation based on that photograph. The hand-copy presented here is also based on that photograph, and on personal inspection of the tablet in Baghdad in March 2001.

(Continues...)

Excerpted from Mathematics in Ancient Iraqby E. Robson Copyright © 2008 by Princeton University Press. Excerpted by permission.
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