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Algebra in Cuneiform

Introduction to an Old Babylonian Geometrical Technique
An introduction to so-called “Babylonian algebra” that explains its geometrical foundation.

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

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.
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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
logograms
principles of transcription
social use
syllabic

“Cut off”

Cut-and-paste

D

Db2–146

Diagrams

drawn in sand
drawn on the tablet

Dust Abacus

E

Eighth degree, problem of

“Encounter, make”

“Equal by”

“Equal, 1 joined”

“Equals” that are not equal

Equation, Babylonian

Equations, operation on

Euclid

Elements
and tradition of geometrical riddles

Excavation, problems of

Explanations, pedagogical

F

Factorization

False position

False value of a magnitude

Fibonacci, Leonardo

Field plans

First degree, techniques for the

G

Genres, mathematical

Geometrica

Geometry, mental

Geometry, practical, Arabic

“Go away, make”

“Go beyond”

“Go”, repetitive operation

Grammatical person in mathematical texts

gur

H

Halves

“Hand“, a reckoning board

“Head“ meaning beginning

“Heap”

History of Mesopotamia

Hittites

“Hold, make”

producing a surface

I

igi

igûm-igibûm

Indeterminate equations

“Inside” of a magnitude

J

“Join”

K

kùš

standard height

L

Latinity

“Lay down”

“Length”

M

Mathematical texts

authors
dating
language

Mathematicians, Babylonian?

Metrology

for area
for hollow measures
for horizontal distance
for vertical distance
for volumes
for weight

Mina

“Modification”

“Moiety”

Moral of history writing

Multiplicative operations

N

Naive approach

Negative numbers

absence from Babylonian mathematics
“found“ with the Babylonians

Neo-Sumerian state

and place-value system

Neugebauer, Otto

nindan

Non-normalised equation, technique for

Numerical values

known but not given
used as names

O

Old Babylonian epoch

Operations

additive
multiplicity of
of divisions
subtractive

Orientalism

P

Pacioli, Luca

pi

Place value number system

“Posit”

Practitioners, mathematical

and mathematical riddles
taught in apprenticeship

Pride, professional, of scribes

Problems

about rectangles
about squares
constructed backwards

Progress

“Projection”

Proof, numerical

Proofs of problem solutions

Pure mathematics, Babylonian

Q

Quadratic completion

Quotation from the statement

R

“Raise”

Recreational problems

Rectangles

primacy compared to triangles

Reed, broken, problem of

Reed, metrological unit

Reference volume

Regular numbers

Remainder, notions of

“Repeat” (“until n”)

Representation

fundamental
fundamental, Babylonian
geometric
of areas by line segments

Riddle format

Riddles, geometric

adopted and transformed by the school

Riddles, geometric, tradition of

and modern mathematics

Riddles, mathematical

their functions

S

sar

“Scatter”

School dimension of figures

Scribe school

Scribes

profession of
their duties

Second degree

complex problems
fundamental techniques

Second degree equations, practical application of

“Separate”

Sexagimal system

Shekel

sìla

“Sixty”

Square and square roots

Square roots, approximated

Squares

concentric

Standard units

“Steps of”

Str. 368

Structure diagrams

Substractive magnitudes

Sum, notions of

Sumerian

dead language
learned language of scribes
support for professional pride

“Surface”

Surveyors

Akkadian
riddle tradition of

Synonyms in mathematical terminology

T

Tables

equal, 1 joined
learned by heart
metrological
of cubic “equals”
of igi
of multiplication
of squares and ”equals“

Tablets

damaged
for rough work
support for writing

Talent (weight unit)

“Tear out”

Terminology, Babylonian mathematical

Third degree, problems of

Thureau-Dangin, François

TMS IX

#1
#2
#3

TMS VII

#1
#2

TMS VIII

#1

TMS XIII

TMS XVI

#1
#2

Translation

conformal
of numbers
principles

True value of a magnitude

U

Units

Ur, centre of neo-Sumerian state

Ur III

uš, unit

V

Variables

VAT 7532

VAT 8389

#1

VAT 8390

#1

VAT 8512

VAT 8520

#1

W

Width

Y

YBC 6504

01
03
04

YBC 6967

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