Mathematics in African History and Cultures

430 Pages·2007·3.04 MB·English

Paulus Gerdes & Ahmed Djebbar

MATHEMATICS
IN AFRICAN HISTORY
AND CULTURES:

AN ANNOTATED BIBLIOGRAPHY

African Mathematical Union
Commission on the History of Mathematics in Africa
(AMUCHMA)
Mathematics in African History and Cultures

Second edition, 2007

First edition:
African Mathematical Union, Cape Town, South Africa, 2004

ISBN: 978-1-4303-1537-7
Published by Lulu.
Copyright © 2007 by Paulus Gerdes & Ahmed Djebbar

Authors

Paulus Gerdes
Research Centre for Mathematics, Culture and Education,
C.P. 915, Maputo, Mozambique
E-mail: [email protected]

Ahmed Djebbar
Département de mathématiques, Bt. M 2, Université de Lille 1,
59655 Villeneuve D’Asq Cedex, France
E-mail: [email protected],
[email protected]

Cover design inspired by a pattern on a mat woven in the 19th century
by a Yombe woman from the Lower Congo area (Cf. GER-04b, p. 96).

2
Table of contents

page
Preface by the President of the African 7
Mathematical Union (Prof. Jan Persens)
Introduction 9
Introduction to the new edition 14
Bibliography A 15
B 43
C 65
D 77
E 105
F 115
G 121
H 162
I 173
J 179
K 182
L 194
M 207
N 223
O 228
P 234
R 241
S 252
T 274
U 281
V 283
3
Mathematics in African History and Cultures
page
W 290
Y 296
Z 298

Appendices

1 On mathematicians of African descent / 307
Diaspora
2 Publications by Africans on the History of 313
Mathematics outside Africa (including
reviews of these publications)
3 On Time-reckoning and Astronomy in 317
African History and Cultures
4 String figures in Africa 338
5 Examples of other Mathematical Books and 343
Booklets published by African
Mathematicians
6 Board Games in Africa 356
7 Note on Research Inspired by the Historical 370
Reconstruction of Mathematical Ideas in the
‘Sona’ Geometric Tradition Of Southern-
Central Africa

Indices

1 Subject Index 375
2 Country Index 383
3 Regional Index 390
4 Author Index 392
5 Ethnographic and Linguistic Index 408
6 Journal Index 411
7 Index of mathematicians 424
4
page
Members of the African Mathematical Union 428
Commission on the History of Mathematics in Africa
(AMUCHMA)
AMUCHMA website 429

Illustrations

Shape of a plaited nonahedron (Mozambique) 76
Detail of Ibn Muncim’s manuscript with his 83
arithmetic triangle four centuries before
Pascal (1623-1662)
Example of a monolinear (lu)sona 132
Example of a monolinear engraving from 133
Ancient Egypt
Example of a twill woven band on a gipatsi 134
Example of a litema wall decoration (Lesotho) 140
Makhuwa circular tray with woven multiple 142
spiral structure
Example of a Yombe woven plane pattern 148
Example of a symmetric (lu)sona composed 150
of two monolinear halves
Hexagonal woven strip from Benin, Kenya, 206
Mozambique, Nigeria
Example of a Kuba two-colour design 222
(Congo)
A magical square in a manuscript of Al- 260
Kishnâwî
Example of a woven strip design from 306
Zanzibar (Tanzania)
Example of a Lunda-design 391

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Mathematics in African History and Cultures

6
Preface

One cannot but welcome this very important annotated
bibliography on Mathematics in African History and Culture. We are
already at the beginning of the third millennium and, yet, one is often
struck by attitudes, largely based on ignorance, towards the
mathematical contributions from Africa and by Africans. I am sure
that it is this phenomenon and the collective experience and
knowledge of Gerdes and Djebbar that have led to the
conceptualisation of this publication. It is, indeed, long overdue. This
publication informs us about both the history of mathematics in Africa
and the mathematics in the history of Africa. It is also appropriate that
the contributions of Africans outside Africa, or as is commonly
referred to, the African Diaspora, are included. For, often the
involvement and impact of Africans on life and developments outside
Africa, especially in developed countries, are knowingly and
unknowingly underplayed or even ignored.
7
Mathematics in African History and Cultures
Mathematics in African History and Culture: An Annotated
Bibliography, is bound to have a major impact on the curricula of
courses on (the history of) mathematics in Africa. The role of African
mathematicians in astronomy, time-reckoning and calendars can now
be researched and appreciated more fully. By including the
mathematics in African culture, the authors have attached meaningful
value to the systematic, analytical and structured nature of African
cultures. Thus string figure and board games emerge as meaningful
mathematical activities in addition to being enjoyed as forms of
relaxation. As far as it is known, this bibliography, of over a thousand
references, is not just the most comprehensive ever produced, but also
covers the whole African continent over many centuries as well as
recognising “the historical links across the Mediterranean and the
oceans”. This latter aspect is important because it puts the
achievements in so-called western mathematics into perspective.
I believe that there is a need for African students and
researchers, especially the younger generation, to realise that Africans
have made meaningful contributions to science and mathematics. This
realisation should serve as inspiration to them. The technological and
economic development of Africa in this modern age depends on
various applications of mathematical sciences. Getting to grips with
what has been produced by our forebears is potentially important for
the generation of new knowledge, particularly in this era of
knowledge-based economies.
On behalf of the African Mathematical Union, I sincerely wish
to thank Professors Paulus Gerdes and Ahmed Djebbar for their
contribution. They are, indeed, two stalwarts in our quest for
unearthing and highlighting contributions by Africans to mathematical
research and teaching.

Jan Persens, Ph.D
President of the African Mathematical Union (2000-2004)
Bellville, South Africa
June 2004

8
Introduction

Introduction

One of the first measures taken by the Executive Committee of
African Mathematical Union (AMU), elected at the 2nd Pan-African
Congress of Mathematicians (April 1986, Jos, Nigeria), under the
chairmanship of Professor Aderemi Kuku, was to create an AMU
Commission on the History of Mathematics in Africa (AMUCHMA).
The two authors were appointed chairman and secretary. At the
subsequent congresses in 1991 (Nairobi, Kenya), in 1995 (Ifran,
Morocco) and 2000 (Cape Town, South Africa), the authors were re-
elected. As co-ordinators of the commission, we have tried to
stimulate research, and to collect and disseminate as much information
as possible about the history of mathematics in Africa. Along with the
many papers delivered at conferences and seminars organised during
the years, we published so far 28 issues of the AMUCHMA
Newsletter. To the delegates of the 6th Pan-African Congress of
Mathematicians (September 2004, Tunis, Tunisia), we would like to
present the following bibliography on mathematics in African history
and cultures.
9
Mathematics in African History and Cultures
The bibliography contains over one thousand five hundred
references. It is the result both of the information we collected in the
context of AMUCHMA and of our personal research. The first author
used also the information he gathered as secretary (1991-1995) of the
Southern Africa Mathematical Sciences Association (SAMSA) for the
Who is Who in Mathematics and Mathematics Education in Southern
Africa (5-GER-92, 93, 95).
Our bibliography attempts to encompass the African continent
as a whole, from immortal times to the present, without forgetting the
historical links across the Mediterranean and the oceans. For instance,
several references included in the bibliography highlight the
circulation of mathematicians and of mathematical ideas between the
Maghreb (Northwest Africa) and Andalusia (Iberian Peninsula) during
the ‘Middle Ages.’
The present bibliography is as embracing, complete and update
as was possible. Mainly references in Arabic, French, English,
Portuguese, official languages of the continent, were selected. For
studies from and about the mathematicians of Alexandria (Egypt) we
tried to include the most relevant references from 1980 onwards.
Partial bibliographies on Africa South of the Sahara and the Maghreb
were published earlier in the AMUCHMA Newsletter (GER-92b, 92d;
DJE-95a, 95b), in the international journal Historia Mathematica
(GER-94f) and in the Spanish journal for the history of science and
technology LLULL (GER-04e).

Organisation of the bibliography

Bibliographic references directly referring to mathematical
ideas in African history and cultures are included in the main body of
the bibliography. Several appendices present complementary
bibliographies on themes related to the main theme. For instance, as
several entries in the main body refer to mathematicians of African
descent, Appendix 1 presents additional bibliographic information on
mathematicians of the Diaspora. As during history African
mathematicians were often involved in astronomy, Appendix 3
presents an additional bibliography on time-reckoning, calendars and
astronomy in African cultures. As several authors discuss the use of
string figure games in mathematics education, Appendix 4 presents an
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