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 5 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 10 Introduction additional bibliography on string figures in Africa. As various studies referred to in the main body analyse mathematical ideas of players of African board games, Appendix 6 presents an additional bibliography on board games in Africa. To complete the image of what research is done by African scholars in the field of the history of mathematics, Appendix 2 lists publications of African scholars on the history of mathematics outside Africa. This research may be related, for instance, to mathematics in Islamic or Arab cultures or to the application of research methodologies developed in Africa to other cultural contexts, like the analysis of mathematical ideas of basket weavers in the Amazon. As instances of AMUCHMA projects to be continued, Appendix 5 presents examples of books and booklets published by African mathematicians. Appendix 7 lists some examples of African mathematical pioneers in the 20th century. Appendix 8 presents an example of a mathematical research field inspired by the historical study of ‘sona’ ideograms from Angola.