Algebra in Cuneiform

Jens Høyrup

TY  - BOOK
T1  - Algebra in Cuneiform: Introduction to an Old Babylonian Geometrical Technique
A1  - Jens Høyrup
AB  - This textbook analyzes a number of texts in “conformal translation,” that is, a translation in which the same Babylonian term is always translated in the same way and, more importantly, in which different terms are always translated differently. Appendixes are provided for readers who are familiar with basic Assyriology but otherwise philological details are avoided. All of these texts are from the second half of the Old Babylonian period, that is, 1800–1600 BCE. It is indeed during this period that the “algebraic” discipline, and Babylonian mathematics in general, culminates. Even though a few texts from the late period show some similarities with what comes from the Old Babylonian period, they are but remnants. Beyond analyzing texts, the book gives a general characterization of the kind of mathematics involved, and locates it within the context of the Old Babylonian scribe school and its particular culture. Finally, it describes the origin of the discipline and its impact in later mathematics, not least Euclid’s geometry and genuine algebra as created in medieval Islam and taken over in European medieval and Renaissance mathematics.
PB  - Max-Planck-Gesellschaft zur Förderung der Wissenschaften
PY  - 2017
SN  - 978-3-945561-15-7
L1  - https://mprl-series.mpg.de/media/textbooks/2/textbooks2.pdf
LK  - https://mprl-series.mpg.de/textbooks/2/index.html
C1  - by-nc-sa
DO  - 10.34663/9783945561157-00
LA  - en
ER  -

Textbooks 2

Algebra in Cuneiform

Introduction to an Old Babylonian Geometrical Technique

Jens Høyrup

An introduction to so-called “Babylonian algebra” that explains its geometrical foundation.

READ ONLINE PDF EPUB

This publication is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Germany (CC BY-NC-SA 3.0 DE) Licence.

An introduction to so-called “Babylonian algebra” that explains its geometrical foundation.

ONLINE PDF EPUB

This publication is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Germany (CC BY-NC-SA 3.0 DE) Licence.

This textbook analyzes a number of texts in “conformal translation,” that is, a translation in which the same Babylonian term is always translated in the same way and, more importantly, in which different terms are always translated differently. Appendixes are provided for readers who are familiar with basic Assyriology but otherwise philological details are avoided. All of these texts are from the second half of the Old Babylonian period, that is, 1800–1600 BCE. It is indeed during this period that the “algebraic” discipline, and Babylonian mathematics in general, culminates. Even though a few texts from the late period show some similarities with what comes from the Old Babylonian period, they are but remnants. Beyond analyzing texts, the book gives a general characterization of the kind of mathematics involved, and locates it within the context of the Old Babylonian scribe school and its particular culture. Finally, it describes the origin of the discipline and its impact in later mathematics, not least Euclid’s geometry and genuine algebra as created in medieval Islam and taken over in European medieval and Renaissance mathematics.

Preface

1 Introduction: The Issue – and Some Necessary Tools

2 Techniques for the First Degree

3 The Fundamental Techniques for the Second Degree

4 Complex Second-degree Problems

5 Application of Quasi-algebraic Techniques to Geometry

6 General Characteristics

7 The Background

8 Origin and Heritage

9 A Moral

Appendix A: Problems for the Reader

Appendix B: Transliterated Texts

Bibliographical Note

Show All

A

Abacus

1

Akkadian

Algebra

and equations

1

and quasi-algebra

1
2
3

meaning of word

1
2
3

Algebra, Arabic

1
2

and geometrical riddles

1

origin

1

Algebra, Babylonian

and Greek theoretical arithmetic

1

arithmetical interpretation

1
2

based on tangible and measurable magnitudes

1
2
3
4

blind alley

1
2

cultural function

1

didactical function

1

discovery

1

erroneous arguments

1

flexible instrument

1
2

origin

1

pretendedly practical problems

1
2
3
4
5

principles of interpretation

1
2
3

problems with no practical applications

1
2
3
4
5

product of the Old Babylonian epoch

1

quasi-disappearance

1

resurgence in reduced form

1

school topic

1

shortcomings of arithmetical interpretation

1
2
3
4
5

variation of coefficients

1
2
3
4

Analysis, Greek

1

Analytic method

1
2
3
4

Angle, Babylonian notion of

1

practically right

1

AO 8862

1
2

#2

1
2
3
4
5
6
7
8

B

Babylonia

1

Babylonian dialect

1

Babylonian mathematics

editions of texts

1

similar to and different from ours

1

bán

1

“Base”

1
2

Bisection of a trapezium

1

known before 2200 BCE

1

the argument

1

BM 13901

1
2
3

01

1
2
3

02

1
2
3

10

1
2
3
4
5

12

1
2
3

14

1

23

1
2
3
4
5
6
7
8

BM 15285

#24

1
2
3

BM 85200+VAT 6599

1
2

06

1
2

23

1

“Break”

1

“Bring”

1

bùr

1
2
3
4
5

Bureaucracy, Ur III

1

C

Calculation, techniques of

1

Cardano, Gerolamo

1

Change of scale in one direction

1
2
3
4
5
6
7

City states

1

Civilization, the first

1

“Confront each other”

1
2

“Confrontation”

1
2
3
4

Cuneiform writing

1
2

Ideographic writing

1

change of direction

1

development

1

logograms

1

principles of transcription

1

social use

1
2

syllabic

1
2

“Cut off”

1

Cut-and-paste

1
2
3
4
5
6
7

D

Db2–146

1
2
3

Diagrams

drawn in sand

1

drawn on the tablet

1
2

Dust Abacus

1

E

Eighth degree, problem of

1

“Encounter, make”

1

“Equal by”

1
2
3
4
5

“Equal, 1 joined”

1
2

“Equals” that are not equal

1

Equation, Babylonian

1
2

Equations, operation on

1

Euclid

Elements

1
2

and tradition of geometrical riddles

1

Excavation, problems of

1
2

Explanations, pedagogical

1
2
3
4

F

Factorization

1
2
3

False position

1
2
3
4
5
6
7
8

False value of a magnitude

1
2
3
4
5
6

Fibonacci, Leonardo

1

Field plans

1

First degree, techniques for the

1

G

Genres, mathematical

1

Geometrica

1

Geometry, mental

1

Geometry, practical, Arabic

1

“Go away, make”

1
2

“Go beyond”

1
2

“Go”, repetitive operation

1
2

Grammatical person in mathematical texts

1
2

gú

1

gur

1
2

H

Halves

1

“Hand“, a reckoning board

1

“Head“ meaning beginning

1

“Heap”

1
2
3
4
5
6

History of Mesopotamia

1

Hittites

1

“Hold, make”

1
2
3
4
5
6

producing a surface

1
2

I

igi

1
2
3
4
5
6
7

igûm-igibûm

1
2

Indeterminate equations

1

“Inside” of a magnitude

1
2
3

J

“Join”

1
2
3
4
5
6
7
8
9

K

kùš

1

standard height

1
2

L

Latinity

1

“Lay down”

1
2

“Length”

1

M

Mathematical texts

authors

1

dating

1

language

1
2

Mathematicians, Babylonian?

1

Metrology

for area

1

for hollow measures

1

for horizontal distance

1

for vertical distance

1

for volumes

1

for weight

1

Mina

1

“Modification”

1

“Moiety”

1
2

Moral of history writing

1

Multiplicative operations

1

N

Naive approach

1
2

Negative numbers

absence from Babylonian mathematics

1
2
3

“found“ with the Babylonians

1
2

Neo-Sumerian state

1

and place-value system

1

Neugebauer, Otto

1
2
3
4

nindan

1

Non-normalised equation, technique for

1
2
3

Numerical values

known but not given

1
2

used as names

1

O

Old Babylonian epoch

1

Operations

additive

1

multiplicity of

1

of divisions

1

subtractive

1

Orientalism

1

P

Pacioli, Luca

1
2

pi

1

Place value number system

1
2

“Posit”

1
2

Practitioners, mathematical

and mathematical riddles

1
2

taught in apprenticeship

1

Pride, professional, of scribes

1
2

Problems

about rectangles

1
2
3
4
5
6
7

about squares

1
2
3
4
5
6
7
8

constructed backwards

1
2

Progress

1
2

“Projection”

1
2
3
4
5
6

Proof, numerical

1

Proofs of problem solutions

1

Pure mathematics, Babylonian

1

Q

Quadratic completion

1
2
3
4
5
6
7
8
9
10

Quotation from the statement

1
2

R

“Raise”

1
2
3
4
5
6

Recreational problems

1

Rectangles

primacy compared to triangles

1

Reed, broken, problem of

1
2

Reed, metrological unit

1

Reference volume

1
2
3

Regular numbers

1

Remainder, notions of

1

“Repeat” (“until n”)

1

Representation

1
2
3
4
5
6

fundamental

1
2

fundamental, Babylonian

1

geometric

1
2

of areas by line segments

1

Riddle format

1
2
3
4

Riddles, geometric

1

adopted and transformed by the school

1
2

Riddles, geometric, tradition of

1
2

and modern mathematics

1

Riddles, mathematical

1
2
3

their functions

1

S

sar

1
2
3

“Scatter”

1

School dimension of figures

1

Scribe school

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

Scribes

1
2
3

profession of

1

their duties

1

Second degree

complex problems

1

fundamental techniques

1

Second degree equations, practical application of

1

“Separate”

1

Sexagimal system

1
2

Shekel

1

sìla

1
2
3

“Sixty”

1

Square and square roots

1

Square roots, approximated

1
2

Squares

concentric

1

Standard units

1

“Steps of”

1
2
3

Str. 368

1
2

Structure diagrams

1

Substractive magnitudes

1
2

Sum, notions of

1

Sumerian

1

dead language

1

learned language of scribes

1
2

support for professional pride

1

“Surface”

1
2
3

Surveyors

1
2

Akkadian

1
2

riddle tradition of

1
2

Synonyms in mathematical terminology

1
2
3

T

Tables

1

equal, 1 joined

1

learned by heart

1
2
3

metrological

1

of cubic “equals”

1

of igi

1
2
3
4
5
6
7
8

of multiplication

1
2
3

of squares and ”equals“

1

Tablets

damaged

1

for rough work

1
2

support for writing

1

Talent (weight unit)

1

“Tear out”

1
2
3
4
5
6
7

Terminology, Babylonian mathematical

1

Third degree, problems of

1
2

Thureau-Dangin, François

1
2
3
4

TMS IX

1

#1

1

#2

1

#3

1
2
3
4
5
6
7

TMS VII

1
2

#1

1
2

#2

1
2
3
4

TMS VIII

#1

1
2
3

TMS XIII

1
2
3

TMS XVI

1
2

#1

1
2
3
4
5

#2

1

Translation

conformal

1
2

of numbers

1
2

principles

1

True value of a magnitude

1

U

Units

1

Ur, centre of neo-Sumerian state

1

Ur III

1
2

uš, unit

1

V

Variables

1

VAT 7532

1
2
3
4
5
6

VAT 8389

1

#1

1
2

VAT 8390

1

#1

1

VAT 8512

1
2
3

VAT 8520

1

#1

1
2

W

Width

1
2

Y

YBC 6504

1
2

01

1

03

1

04

1

YBC 6967

1
2
3
4
5
6
7
8
9
10

Information

ISBN

978-3-945561-15-7

DOI

10.34663/9783945561157-00

Publication Date

Dec. 8, 2017

Print on Demand

currently unavailable

Suggested Citation

Høyrup, Jens (2017). Algebra in Cuneiform: Introduction to an Old Babylonian Geometrical Technique. Berlin: Max-Planck-Gesellschaft zur Förderung der Wissenschaften.

Submitted by

Robert K. Englund

Algebra in Cuneiform

Introduction to an Old Babylonian Geometrical Technique

A

Abacus

Akkadian

principal language

“Akkadian method”

Al-Khwārizmī

Algebra

and equations

and quasi-algebra

meaning of word

Algebra, Arabic

and geometrical riddles

origin

Algebra, Babylonian

and Greek theoretical arithmetic

arithmetical interpretation

based on tangible and measurable magnitudes

blind alley

cultural function

didactical function

discovery

erroneous arguments

flexible instrument

origin

pretendedly practical problems

principles of interpretation

problems with no practical applications

product of the Old Babylonian epoch

quasi-disappearance

resurgence in reduced form

school topic

shortcomings of arithmetical interpretation

variation of coefficients

Analysis, Greek

Analytic method

Angle, Babylonian notion of

practically right

AO 8862

#2

B

Babylonia

Babylonian dialect

Babylonian mathematics

editions of texts

similar to and different from ours

bán

“Base”

Bisection of a trapezium

known before 2200 BCE

the argument

BM 13901

01

02

10

12

14

23

BM 15285

#24

BM 85200+VAT 6599

06

23

“Break”

“Bring”

bùr

Bureaucracy, Ur III

C

Calculation, techniques of

Cardano, Gerolamo

Change of scale in one direction

City states

Civilization, the first

“Confront each other”

“Confrontation”

Cuneiform writing

Ideographic writing

change of direction

development