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Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and its Kin

Jens Høyrup
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Sources and Studies in the History of Mathematics and Physical Sciences
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Eleanor Robson
, on

Ever since the ‘three wise men from the east’ followed the star which heralded Jesus’s birth (Mt 2:1-12) the Babylonians — or Chaldaeans, or Magi — have been associated in the popular imagination with mathematics and astronomy. As Robert Recorde put it in The Ground of Artes Teachyng the Worke and Practise of Arithmetike (1543),

In that thinge all men do agree, that the Chaldays, whiche fyrste inuented thys arte [i.e., arithmetic], did set these figures as thei set all their letters. For they wryte backwarde as you tearme it, and so doo they reade.

Some 300 years later, British and French explorers began to rediscover the ruined cities of Babylonia along the banks of the Tigris and Euphrates rivers, in what was then a province of the Ottoman empire and is now southern Iraq. Those massive and sophisticated cities, which had flourished in the third to first millennia BCE, had lain unexplored and unlamented since their gradual demise in the very era that our scholarly trio purportedly made their westward trek to Bethlehem. Over the course of the later nineteenth century the ruin mounds yielded countless ancient artefacts from all periods of Babylonian history including many hundreds of thousands of clay tablets, varying in size from a postage stamp to a hefty hard-back book, and covered in wedge-shaped cuneiform writing. As the first decipherers of this script discovered, the languages it recorded included Babylonian and Assyrian (which we now consider as the southern and northern dialects of Akkadian, a sort of elderly aunt to the Semitic languages Hebrew and Arabic) and a hitherto unknown language, Sumerian, which turned out to have no surviving relatives at all.

Recorde, it transpired, had guessed wrongly at the direction of the Babylonians’ writing but was exactly right about their calculating prowess. For it soon became clear, as scholars worked their way through the cuneiform tablets in London, Istanbul, Berlin, and Philadelphia, that amongst the myriad administrative records of palaces, temples and households, the letters, the literature, and the royal annals, were significant numbers of astronomical and mathematical documents too. While the astronomical material was mostly very late — from the time of Persian and Hellenistic domination in the sixth to first centuries BCE — the bulk of the mathematics dated to the first centuries of the second millennium, the time of the great king Hammurabi, now known as the Old Babylonian period (c.2000-1600 BCE).

Cuneiformists such as Hermann Hilprecht, Vincent Scheil, and François Thureau-Dangin (whose names have long been forgotten outside the small world of cuneiform studies) struggled for years to publish and interpret Babylonian mathematics. But the great breakthrough came in the late 1920s when the great Otto Neugebauer turned his attention from modern to ancient mathematics. His painstaking hunt for new tablets, brilliant text editions and careful mathematical analyses culminated in the monumental three-volume work Mathematische Keilschrifttexte (Springer, 1935-37) and, with the Assyriologist Abe Sachs, Mathematical Cuneiform Texts (American Oriental Society, 1945). (Thureau-Dangin’s rival Textes mathématiques babyloniennes (Brill, 1938) was destined not to achieve the same classic status.) Neugebauer’s stature is such that his interpretations remained essentially unchallenged for half a century or more — until 1990, when JH began his programme of publications which culminated in the book under review.[1]

Marinus Taisbak reminds us in another MAA Online book review of David Fowler’s aphorism that “Greek mathematics is to draw a figure and tell a story about it.” [2] Here Jens Høyrup shows us that Old Babylonian algebra is at one level to not draw a figure but tell a story about it anyway. And, intriguingly, JH does this by paying more attention to words than to numbers. In best Babylonian style, let me demonstrate what I mean with a generic example. First a translation and interpretation of a typical problem in the style of Neugebauer and Thureau-Dangin; then JH’s new reading.

BM 13901, problem 1 [3] in the standard interpretation

I totalled the area and (the side of) my square: it is 0;45. You put down 1, the unit (?). You break in half 1. You multiply 0;30 and 0;30. You add 0;15 to 0;45. The square root of 1 is 1. Subtract 0;30 that you multiplied (with itself) from 1 and 0;30 is (the side of) the square.

Or, in modern symbolic notation, \[x^2+x=0;45.\] So, using the quadratic formula, \[x=-0;30+\sqrt{0;30^2+0;45} = 0;30.\]

BM 13901, problem 1, in JH’s reading

The surface and my confrontation I have accumulated: 45' is it. 1º, the projection you posit.
The moiety of 1º you break, 30' and 30' you make hold each other.
15' to 45' you append: by 1º, 1º is the equalside. 30' which you have made hold
in the inside of 1º you tear out: 30' the confrontation.

It should be immediately apparent that where Neugebauer’s generation were understandably concerned to interpret Old Babylonian algebra in relation to modern mathematical practice, JH aims to recover, as far as possible, the original thought processes behind it. He does this through close analysis of the language of Old Babylonian mathematics, for which he uses a technique he calls ‘conformal translation’. This involves consistently translating Babylonian technical terms with existing English words or neologisms which match the original meanings as closely as possible. For instance, the Akkadian word mithartum, literally ‘thing which is equal and opposite to itself’, becomes a ‘confrontation’ instead of a ‘square’ and the two different verbs of addition are distinguished, following their non-technical usages outside mathematics, as ‘to accumulate’ (kamârum, also ‘to pile up’) and ‘to append’ (wasâbum, also ‘to add on, increase’). Most crucially, JH translates the noun wasîtum, literally ‘thing which comes out’, as ‘projection’; the Neugebauerians, by contrast, uncomfortably translated ‘coefficient’ or ‘unit’ after its (apparently redundant) role in the calculation. When expressed in these very concrete terms, Old Babylonian algebra becomes not arithmetical but geometrical and metric: concerned not with abstract numbers but with measured lines, areas, and volumes.

JH’s ‘conformal’ translations are not easy to follow; I confess it took me a good five years to fully grasp what his intentions were. One complicating factor is that the translations ‘conform’ syntactically as well as lexically; in other words, they follow Akkadian word order as closely as possible. What is more, they also obey the line-breaks, as in a poem. However, as JH’s analysis is all at word level, and there is nothing particularly special about Old Babylonian mathematical syntax (all grammatically correct Akkadian prose has subject-object-verb word order), one loses nothing of his intentions by rearranging his translations into grammatically correct English. On the contrary, they become much easier to follow, thus:

I have accumulated the surface and my confrontation: it is 45'. You posit 1º, the projection. You break the moiety of 1º. You make 30' and 30' hold each other. You append 15' to 45': 1º is the equalside of 1º. You tear out 30' which you have made hold from the inside of 1º: the confrontation is 30'.

Although no diagrams of this geometrised algebra survive, it is possible to reconstruct them simply by following the instructions on the tablet. Let’s take our example sentence by sentence, drawing the pictures as we go.

I have accumulated the surface and my confrontation: it is 45'.

This first statement sets up the problem. Unusually, it is not followed by a question explicitly asking for the length of the square (‘confrontation’). But, if we are in geometric mode, how can we add a line (the side of the ‘confrontation’) to its area (‘surface’)? Let us follow the first instruction:

You posit 1º, the projection.

In other words, we make the line into an area by giving it unit width (the ‘projection’). In this way we can add ‘broad lines’ to areas and, analogously, ‘thick surfaces’ to volumes. (In fact, the concept of ‘thick surface’ is ubiquitous in Old Babylonian mathematics, the standard unit of volume being defined as 1 rod square in the horizontal plane by 1 vertical cubit, where 1 cubit is approximately 50 cm and 1 rod is approximately 6 m). So, let us draw the surface as a grey square with mystery length s, and the projected, or broadened ‘confrontation’ as a rectangle of length s and breadth 1 next to it:

According to the initial statement, their combined area is 45'. Now we can get going without further ado:

You break the moiety of 1º.
You make 30' and 30' hold each other.

In other words, break the ‘projection’ in half. Then let the two pieces form the sides of a square (marked with dotted lines) and calculate its area:

which we can add to the original figure, whose area is still 45':

You append 15' to 45'.

We now have a composite square of known area and thus length:

1º is the equalside of 1º.

And all we need to do now is to remove the remains of the broad lines to find out the length of the original square:

You tear out 30' which you have made hold from the inside of 1º: the confrontation is 30'.

We can see that the problem is about completing the square in a very real sense: elements of the figure are created, broken, rearranged, torn out along the way. Every step in the procedure is accounted for, and every instruction makes sense. Further it is now clear that the diagram, whether real or imagined, is intrinsic to its very conceptualisation

As I said earlier, this is no more than a generic example: JH’s method works for literally dozens of Old Babylonian algebraic problems. In Chapter 3 he works through fifteen of them in great detail, after briefly setting out the standard interpretation and detailing his methodology (including a full reference table of his ‘conformal’ translations) in Chapters 1-2. After pausing in Chapter 4 for a discussion of the methods used in the problems themselves, he takes us through around a hundred more examples of Old Babylonian problems in algebra (Chapter 5) and fourteen in ‘quasi-algebraic geometry’ (Chapter 6). The whole is summarised in Chapter 7, where JH tackles the difficult questions of whether we can truly talk about ‘algebra’, ‘equations’, or even ‘mathematics’ in this context. Just as film reviewers never reveal the denouements of the movies they are evaluating, I shall not give the answers away here; but perhaps you can guess them anyway.

So far so good; but we have reached page 309 and by JH’s own admission we have yet to see much history in the sense of contextualisation or diachronic investigation. The discourse analysis comprising the first 308 pages would be more than enough, in anyone else’s book (literal or metaphorical) to establish this work as a ground-breaking classic, but JH’s work is far from done. He devotes a further 100 pages to summarising their historical setting (Chapters 8-9) and to locating them within the long traditions of ‘surveyors’ mathematics’, mathematical riddles, and ‘scholasticized’ versions in the written records of Babylonia, Egypt, the classical Mediterranean, India, and the Islamic Middle East (Chapters 10-11). Like the earlier chapters, these too are revised from articles written by JH over the last decade or so. If anything, they are more ambitious than the first section of the book, and because of their extraordinarily wide scope they are perhaps less consistently successful.

When JH’s subject is the algebraic tradition, he is utterly convincing; when he broadens the discussion to the social and historical implications of his discoveries the weaknesses of his approach are exposed. For instance, in his analysis JH virtually ignores the two other vast sub-corpora of Old Babylonian mathematics: the metrological and arithmetical lists, tables and calculations which belonged to the elementary scribal curriculum on the one hand, and the ‘utilitarian’ problems about quantity surveying which comprise the other half of the advanced corpus on the other. Nor does he use museological information or deal with the physical attributes of the tablets on which his texts are written; both are vital sources of evidence when trying to reconstruct the date and origin of archaeologically unprovenanced material like this. This is not the place to give an exhaustive critique, but let me give one example from Chapter 9, in which JH attempts to untangle the ‘finer [chronological] structure’ of the Old Babylonian mathematical corpus.

One of JH’s key pieces of evidence is a compilation of algebraic problems which ‘has always been regarded as one of the earliest Old Babylonian problem texts’ (p. 338) and which he characterises further on mathematical and linguistic grounds as ‘a witness of the experimental phase when the scribal school first took over a set of surveyors’ riddles and made it a starting point for the creation of a discipline’ (p. 162). The problems are recorded on a four-sided prism, AO 8862, from the ancient city of Larsa and now housed in the Louvre. Now, dates recorded on multiplication tables and other elementary exercises show that there are two different school assemblages from the city of Larsa, one dated to the time of king Rim-Sin (c.1815-1800 BCE) and the other to early in the reign of Hammurabi’s successor Samsu-iluna (c.1750-40 BCE). One dated example from the latter group is a six-sided prism bearing tables of squares, inverse squares, and inverse cubes, dated to 1749 BCE and now in the Louvre, museum number AO 8865. It and AO 8862 belong to a larger group of Old Babylonian school tablets from Larsa. For instance AO 8863 and AO 8864, both hexagonal prisms carrying Sumerian literary compositions from the scribal curriculum, are dated to 1739 BCE. In this light the status of AO 8862 as ‘one of the earliest Old Babylonian problem texts’ immediately looks less secure. [4]

But shortcomings of this type do not deter me from hailing this difficult but exciting book as a major turning point in the study of ancient mathematics. JH is at the forefront of the movement to transform our field from a catalogue of ancient methods expressed in modern terminology to a vital episode in the history of ideas. This book is by no means a gentle introduction to Babylonian mathematics — not something to set your undergraduates to read from cover to cover! — but it certainly repays close and repeated reading. It is an absolute must-read for anyone who wishes to move beyond the folklore of Recorde or the received wisdom of Neugebauer towards a real understanding of the thought processes and the people behind the mathematics of ancient Iraq.


1. JH has given an excellent potted history of the field in Jens Høyrup, ‘Changing trends in the historiography of Mesopotamian mathematics: an insider’s view’, History of Science 34 (1996), 1-32. The seminal article was ‘Algebra and naive geometry. An investigation of some basic aspects of Old Babylonian mathematical thought’, Altorientalische Forschungen 17 (1990), 27-69, 262-354.

2. Marinus Taisbak’s review of Reviel Netz, The shaping of deduction in Greek mathematics (Cambridge 1999), on MAA Reviews, 1999.

3. BM 13901, an Old Babylonian tablet of unknown provenance now in the British Museum, first published by F. Thureau-Dangin in Revue d’Assyriologie 33 (1936), 27-48, and the first example in JH’s book (pp. 11-14). The English translation is mine, a hybrid of Neugebauer’s German and Thureau-Dangin’s French. Two different transliterations are used for the sexagesimal place value system, depending entirely on personal preference. Following Neugebauer, I use a semicolon to represent the boundary between integers and fractions; following Thureau-Dangin, JH uses the notation of degrees, seconds, and minutes.

4. An example of the early group is YBC 11924, a nine times multiplication table dating to 1815 BCE (O. Neugebauer and A. Sachs, Mathematical cuneiform texts (American Oriental Society, 1945): p. 23 no. 99,13b. AO 8865 and AO 8862 are both published by O. Neugebauer, Mathematische Keilschrifttexte (Springer, 1935-37): I 71-75 and I 108-123; II pls. 35-38. Copies of AO 8863 and AO 8864 appear in Henri de Genouillac, Textes Cunéiformes du Louvre, vol. 16 (1930) nos. 87 and 88; editions of their contents (Lipit-Eshtar Hymn B and Iddin-Dagan Hymn B) are published by the Electronic Text Corpus of Sumerian Literature.

Eleanor Robson ( is a cuneiformist whose interests centre on numeracy, literacy and the intellectual history of ancient Iraq. Her publications include Mesopotamian mathematics, 2100-1600 BC (Oxford, 1999) and ‘Words and pictures: new light on Plimpton 322’, American

I. Introduction

II. A New Reading

III. Select Textual Examples

IV. Methods

V. Further "Algebraic" Texts

VI. Quasi-Algebraic Geometry

VII. Old Babylonian "Algebra": A Global Characterization

VIII. The Historical Framework

IX. The "Finer Structure" of the Old Babylonian Corpus

X. The Origin and Transformations of Old Babylonian Algebra

XI. Repercussions and Influences

XII. Index of Tablets

XIII. Index of Akkadian and Sumerian Terms and Key Phrases

XIV. Name Index

XV. Subject Index

XVI. Abbreviations and Bibliography