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Algebra in Cuneiform ALGEBRA IN CUNEIFORM Jens Høyrup Roskilde University Roskilde University Algebra in Cuneiform Jens Høyrup This text is disseminated via the Open Education Resource (OER) LibreTexts Project (https://LibreTexts.org) and like the hundreds of other texts available within this powerful platform, it freely available for reading, printing and "consuming." Most, but not all, pages in the library have licenses that may allow individuals to make changes, save, and print this book. Carefully consult the applicable license(s) before pursuing such effects. Instructors can adopt existing LibreTexts texts or Remix them to quickly build course-specific resources to meet the needs of their students. Unlike traditional textbooks, LibreTexts’ web based origins allow powerful integration of advanced features and new technologies to support learning. 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The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. This material is based upon work supported by the National Science Foundation under Grant No. 1246120, 1525057, and 1413739. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation nor the US Department of Education. Have questions or comments? For information about adoptions or adaptions contact [email protected]. More information on our activities can be found via Facebook (https://facebook.com/Libretexts), Twitter (https://twitter.com/libretexts), or our blog (http://Blog.Libretexts.org). This text was compiled on 09/22/2021 TABLE OF CONTENTS An introduction to so-called “Babylonian algebra” that explains its geometrical foundation. ABOUT THE AUTHOR 1: INTRODUCTION - THE ISSUE AND SOME NECESSARY TOOLS 1.1: “USELESS MATHEMATICS” 1.2: THE FIRST ALGEBRA AND THE FIRST INTERPRETATION 1.3: A NEW READING 1.4: CONCERNING THE TEXTS AND THE TRANSLATIONS 1.5: FOOTNOTES 2: TECHNIQUES FOR THE FIRST DEGREE 2.1: TMS XVI #1 2.2: TMS VII #2 2.3: FOOTNOTES 3: THE FUNDAMENTAL TECHNIQUES FOR THE SECOND DEGREE 3.1: BM 13901 #1 3.2: BM 13901 #2 3.3: YBC 6967 3.4: BM 13901 #10 3.5: BM 13901 #14 3.6: TMS IX #1 AND #2 3.7: FOOTNOTES 4: COMPLEX SECOND-DEGREE PROBLEMS 4.1: TMS IX #3 4.2: AO 8862 #2 4.3: VAT 7532 4.4: TMS XIII 4.5: BM 13901 #12 4.6: BM 13901 #23 4.7: TMS VIII #1 4.8: YBC 6504 #4 4.9: FOOTNOTES 5: APPLICATION OF QUASI-ALGEBRAIC TECHNIQUES TO GEOMETRY 5.1: VAT 8512 5.2: BM85200 + VAT 6599 #6 5.3: BM15285 #24 5.4: FOOTNOTES 6: GENERAL CHARACTERISTICS 6.1: DRAWINGS? 6.2: ALGEBRA? 6.3: FOOTNOTES 7: THE BACKGROUND 7.1: THE SCRIBE SCHOOL 7.2: THE FIRST PURPOSE- TRAINING NUMERICAL CALCULATION 7.3: THE SECOND PURPOSE- PROFESSIONAL PRIDE 7.4: FOOTNOTES 1 9/22/2021 8: ORIGIN AND HERITAGE 8.1: THE ORIGIN- SURVEYORS’ RIDDLES 8.2: THE HERITAGE 8.3: FOOTNOTES 9: A MORAL 10: APPEDICES 10.1: APPENDIX A- PROBLEMS FOR THE READER 10.2: APPENDIX B- TRANSLITERATED TEXTS 11: BIBLIOGRAPHICAL NOTE 11.1: BIBLIOGRAPHICAL NOTE 11.2: FOOTNOTES BACK MATTER INDEX GLOSSARY 2 9/22/2021 About the Author Jens Høyrup studied mathematics and physics, with thesis in particle physics (1969). In 1973 he joined the newly founded Roskilde University, first the Faculty of Social Sciences, moving in 1978 to that of Humanities. He has taught broadly on history of ideas and of philosophy and on philosophy of science. His research, though sometimes moving within this wider field, has concentrated on conceptual and social history of premodern and Early Modern mathematics. Jens Høyrup 1 9/8/2021 https://math.libretexts.org/@go/page/48327 CHAPTER OVERVIEW 1: INTRODUCTION - THE ISSUE AND SOME NECESSARY TOOLS 1.1: “USELESS MATHEMATICS” 1.2: THE FIRST ALGEBRA AND THE FIRST INTERPRETATION 1.3: A NEW READING 1.4: CONCERNING THE TEXTS AND THE TRANSLATIONS 1.5: FOOTNOTES 1 9/22/2021 1.1: “Useless mathematics” At some moment in the late 1970s, the Danish Union of Mathematics 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 remember!). 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 have been 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 to use them in trigonometric computation). We shall return to the question why one taught, and still teaches, second-degree equations. But first we shall 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 will 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 siege ramps and to the division of fields), the mathematical substance is always “pure,” that is, deprived of any immediate application outside of mathematics itself. Rudiments of General History https://mprl-series.mpg.de/textbooks/2/2/index.html#7 " data-title="Permanent Link" rel="popover" title="title"> 1 Cuneiform writing: 1 2 " data-title="Keywords, Persons and Locations" rel="popover" title="title"> Mesopotamia (“Land between the rivers”) has designated 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 10). The earliest script was purely ideographic (a bit like modern mathematical symbolism, where an expression like 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 was unmistakably Sumerian. From then on, and until c. 2350, the area was divided into a dozen city- states, often at war with each other for water resources. For this reason, the structure of the state was transformed, and the war leader (“king”) displaced the temples as the centre of power. From around 2600, a professional specialization emerged, due to wider application of writing. Accounting was no longer the task of the high officials of temple and king: the scribe, a new profession, taught in schools and took care of this task. Around 2340, an Akkadian conqueror subdued the whole of Mesopotamia (Akkadian is a Semitic language, from the same language family as Arabic and Hebrew, and it had been amply present in the region at least since 2600). The Akkadian regional state lasted until c. 2200, after which followed a century of competing city states. Around 2100, the city-state of Ur made itself the centre of a new centralized regional state, whose official language was still Sumerian (even though most of the population, including the kings, probably spoke Akkadian). This “neo-Sumerian” state (known as Ur III) was highly bureaucratized (perhaps more than any other state in history before the arrival of electronic computers), and it seems that the place-value number notation was created in response to the demand of the bureaucracy for convenient calculational instruments (see the box “The Sexagesimal Place-Value System,” page 14). Jens Høyrup 1.1.1 9/8/2021 https://math.libretexts.org/@go/page/47591 In the long run, the bureaucracy was too costly, and around 2000 a new phase of smaller states begins. After another two centuries another phase of centralization centred around the city of Babylon sets in—from which moment it is meaningful to speak of southern and central Mesopotamia as “Babylonia.” By now (but possibly since centuries), Sumerian was definitively dead, and Akkadian had become the principal language—in the south and centre the Babylonian and in the north the Assyrian dialect.
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