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Algebra in Cuneiform MAX-PLANCK-INSTITUT FÜR WISSENSCHAFTSGESCHICHTE Max Planck Institute for the History of Science 2013 PREPRINT 452 Jens Høyrup Algebra in Cuneiform Introduction to an Old Babylonian Geometrical Technique ALGEBRA IN CUNEIFORM Introduction to an Old Babylonian Geometrical Technique Book proposal Corrections are still welcome! Jens Høyrup November 2013 In memory of Peter Damerow MAP TABLE OF CONTENTS Preface .............................................. ix Introduction: The issue – and some necessary tools .............. 1 “Useless mathematics” ............................... 1 The first algebra, and the first interpretation ................ 3 A new reading ..................................... 11 Concerning the texts and the translations .................. 20: Chapter 1: Techniques for the first degree .................... 25 TMSXVI#1...................................... 25 TMSVII#2 ...................................... 33 Chapter 2: The fundamental techniques for the second degree ................................... 39 BM 13901 #1 ..................................... 39 BM 13901 #2 ..................................... 43 YBC 6967 ....................................... 46 BM 13901 #10 .................................... 48 BM 13901 #14 .................................... 50 TMSIX#1and#2 ................................. 55 Chapter 3: Complex second-degree problems .................. 59 TMSIX#3....................................... 59 AO 8862 #2 ...................................... 63 VAT 7532 ....................................... 68 TMS XIII ....................................... 74 BM 13901 #12 .................................... 77 BM 13901 #23 .................................... 79 TMS VIII #1 ...................................... 82 YBC 6504 #4 ..................................... 84 Chapter 4: Application of quasi-algebraic techniques to geometry .................................. 87 VAT 8512 ....................................... 87 BM 85200+VAT 6599 #6 ............................. 94 BM 15285 #24 .................................... 99 Chapter 5: General characteristics .......................... 101 Drawings? ........................................ 101 Algebra ? ........................................ 103 Chapter 6: The background ............................... 107 The scribe school ................................... 107 The first purpose: training numerical calculation ............. 108 VI Algebra in cuneiform The second purpose: professional pride ................... 109 Chapter 7: Origin and heritage ............................ 111 The origin: surveyors’ riddles .......................... 111 The heritage ...................................... 117 Chapter 8: A moral .................................... 123 Appendix A: Problems for the reader ........................ 125 TMSXVI#2...................................... 125 TMSVII#1 ...................................... 126 VAT 8389 #1 ..................................... 126 VAT 8390 #1 ..................................... 129 VAT 8520 #1 ..................................... 131 Str368.......................................... 132 YBC 6504 #1 ..................................... 133 YBC 6504 #3 ..................................... 134 BM 85200+VAT 6599 #23 ............................ 135 Db2-146 ......................................... 135 Appendix B: Transliterated texts ........................... 139 Key to vocabulary and standard translations ................ 139 AO 8862 #2 ...................................... 142 BM 13901 #1, #2, #10, #12, #14 and #23 .................. 143 BM 15285 #24 .................................... 145 BM 85200+VAT 6599 #6 and #23 ....................... 145 Db2-146 ......................................... 146 TMSVII#1and#2................................. 147 TMS VIII #1 ...................................... 148 TMSIX#1,#2and#3............................... 149 TMS XIII ........................................ 150 TMSXVI#1...................................... 151 VAT 7532 ....................................... 151 VAT 8389 #1 ..................................... 152 VAT 8390 #1 ..................................... 154 VAT 8512 ....................................... 155 YBC 6504 ....................................... 156 YBC 6967 ....................................... 158 Bibliographic note ..................................... 159 Index .............................................. 161 Boxes Rudiments of general history .............................2 Cuneiform writing ....................................4 The sexagesimal system ................................7 Preface This book presents an important aspect of Babylonian mathematics, namely the technique or discipline usually known as “Babylonian algebra”. This “algebra” is the earliest example of advanced mathem- atics that has come down to us, for which reason it is spoken of in most general expositions of the history of mathematics. However, most of these expositions rely on translations and interpretations going back to the 1930s. The present book, in contrast, builds on recent research. The traditional interpretation made it possible to establish a list of the results obtained by the Babylonians; of the calculations they were able to perform; and, so to speak, of the formulas they knew. But since its starting point was contemporary mathematical thought it was not able to reconstruct the different thinking which hides behind the Babylonian results. The aim of the present book is to highlight that difference, and thus to show that mathematics can be thought in several ways. A first version of the book was written for students of the Danish high school system in 1998; another version – revised and aug- mented – appeared in French in 2010. This, as well as the present further updated version, addresses itself to those who are interested in the history of mathematics without possessing necessarily mathemat- ical competence beyond what is acquired in high school. Teachers may use it together with their students at various levels. A first approach (in teaching as well as private study) may concentrate on the first-degree equation TMS XVI #1, and the basic second-degree equations, that is, BM 13901 #1 and #2, YBC 6967 and TMS IX #1 and #2. The Introduction and Chapters 5 to 7 provide a general overview. In order to get deeper into the matter one may read the other texts from Chapters 1 and 2, and the texts TMS IX #3, AO 8862 #2, BM 13901 #23 and YBC 6504 #4 from Chapter 3. Those who become passionate may read all the texts from Chapters 1 to 4, and then try their teeth on the texts from Appendix A. VIII Algebra in cuneiform In Appendix B, those who know the rudiments (or more) of Babylonian language and grammar will find transliterations of most of the texts from Chapters 1 to 4 and Appendix A. Introduction The issue – and some necessary tools “Useless mathematics” At some moment in the late 1970s, the Danish Union of Mathe- matics Teachers for the pre-high-school level asked its members a delicate question: to find an application of second-degree equations that fell inside the horizon of their students. One member did find such an application: the relation between duration and counter numbers on a compact cassette reader (thus an application that at best the parents of today’s students will remem- ber!). That was the only answer. Many students will certainly be astonished to discover that even their teachers do not know why second-degree equations are solved. Students as well as teachers will be no less surprised that such equations are taught since 1800 BCE without any possible external reference point for the students – actually for the first 2500 years without reference to possible applications at all (only around 700 CE did Persian and Arabic astronomers possibly start using them in trigonometric computation). We shall return to the question why one taught, and still teaches, second-degree equations. But first we shall have a look at how the earliest second-degree equations, a few first-degree equations and a single cubic equation looked, and examine the way they were solved. We shall need to keep in mind that even though some of the problems from which they are derived look practical (they may refer to mercantile questions, to fortification ramps and to the division of fields), then the mathematical substance is always “pure”, that is, deprived of any immediate application outside mathematics itself. 2 Introduction Rudiments of general history Mesopotamia (“Land between the rivers”) designates since Antiquity the region around the two great rivers Euphrates and Tigris – grossly, contemporary Iraq. Around 3500 BCE, the water level in the Persian Gulf had fallen enough to allow large-scale irrigation agriculture in the southern part of the region, and soon the earliest “civilization” arose, that is, a society centred on towns and organized as a state. The core around which this state took shape was constituted by the great temples and their clergy, and for use in their accounting this clergy invented the earliest script (see the box “Cuneiform writing”, page 4). The earliest script was purely ideographic (a bit like modern mathemat- ical symbolism, where an expression like E = mc2 can be explained and even pronounced in any language but does not allow us to decide in which language Einstein thought). During the first half of the third millennium, however, phonetic and grammatical complements were introduced, and around 2700 BCE the language
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