algebra in cuneiform

178 Pages·2013·1.34 MB·English

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
TMS XVI #1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
TMS VII #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
TMS IX #1 and #2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Chapter 3: Complex second-degree problems . . . . . . . . . . . . . . . . . . 59
TMS IX #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 Algebraincuneiform
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
TMS XVI #2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
TMS VII #1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
VAT 8389 #1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
VAT 8390 #1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
VAT 8520 #1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Str 368 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
YBC 6504 #1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
YBC 6504 #3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
BM 85200+VAT 6599 #23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Db-146 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
2
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
Db-146 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
2
TMS VII #1 and #2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
TMS VIII #1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
TMS IX #1, #2 and #3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
TMS XIII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
TMS XVI #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,
mostoftheseexpositionsrelyontranslationsandinterpretationsgoing
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-degreeequations,thatis,BM 13901#1and#2,YBC 6967and
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
Chapters1to4,andthentrytheirteethonthetextsfromAppendixA.
VIII Algebraincuneiform
In Appendix B, those who know the rudiments (or more) of
Babylonianlanguageandgrammarwillfindtransliterationsofmostof
the texts from Chapters 1 to 4 and Appendix A.