Originally published in the International Study Group on Ethnomathematics (ISGEm) Newsletter, Volume 7, Number 1, January 1991. Located at: http://web.nmsu.edu/~pscott/isgem71.htm. Article reproduced 2003 with permission of the ISGEm Newsletter editor for use in the Ethnomathematics Digital Library (www.ethnomath.org) developed by Pacific Resources for Education and Learning (www.prel.org).

George Gheverghese Joseph (1991). Crest of the Peacock: Non-European Roots of Mathematics. I.B. Taurus, London, 367 pages.

At last, the book many of us have been waiting for is here. Crest of the Peacock is a survey, in some depth, of the non-European mathematics that built the foundation of the modern mathematics we enjoy today. The book is readable, and the mathematics clearly stated in a form accessible to anyone who remembers her high school mathematics. The poetic title comes from the Indian Vedanga Jyotisa, which proclaims, "As are the crests on peacocks, as are the gems on the heads of snakes, so is mathematics at the head of all knowledge."

The quotation suggests a reverent, almost elitist concept of mathematics in early India. However, Joseph is far from elitist in his approach to the history of mathematics. On the contrary, he finds many opportunities to show that the development of mathematics is a response to the needs of people and is made possible by the accumulated productive skills of people. Above all, the weight of the evidence that he presents is a powerful
argument against what he calls the "Eurocentric model" of the history of mathematics, "with Greece as the source and Europe the inheritor and guardian of the Greek heritage." Motivation for the "Eurocentric model" is not a mystery to Joseph who offers this explanation: "The contributions of the colonized people were ignored or devalued as part of the rationale for subjugation and domination."

Main topics discussed in Crest of the Peacock include tally and string records, Maya numerals, the mathematics of ancient Egypt, Babylonia, India, China, and the "Arab" contribution. Given the author's birthplace in Kerala, India, and his Syrian ancestry, it is understandable that he claims "By the second half of the first millennium, the most important contacts for the future of the development of mathematics were those between India and the Arab world. Here he is speaking of more than the development and transmission of Hindu-Arabic numerals whose importance, he says, cannot be overestimated.

Joseph outlines in some detail the early Indian trigonometry, and indeterminate analysis, and the later, little-known (1400-1600) infinite series expansions for trigonometric functions. The rich Chinese contribution is treated with equal respect and similar length (81 pages):
matrix methods for solution of systems of equations, indeterminate analysis, the Chinese anticipation by four centuries of the Horner-Rusini Method and the Pascal triangle for higher-order equations, the use of negative numbers, and double false position solution of equations were also part of Chinese mathematics.

The final chapter is "Prelude to Modern Mathematics - The Arab Contribution." In just 47 pages, Joseph manages to compress the extensive contributions of Islamic math: decimal fractions, inheritance problems, number theory, figurate numbers, algebra, number theory, real numbers, conic sections, the introduction of six basic trigonometric flinctions and identities, and exploration of non-Euclidean geometry.

I'll confess I would have liked to have found more on Abu Hamil, the "Egyptian calculator" whose work extended the algebra of al-Khowarizmi to use several variables, powers up to eight, and irrational solutions. I also missed references to the Cairo "House of Wisdom," a Science Academy where in Yunus, al-Haytham, and other scholars worked. I wondered about the use of the term "Arab" instead of the more general "Islamic". As the author points out, the mathematicians of this tradition came from Persia, Central Asia, Egypt, North Africa and Spain, as well as Arabic countries.

As is the case with some "standard" histories of mathematics, there is a chapter on ancient Egypt and one on Babylonia early in the book. Unlike the standard histories, the chapter on Egypt emphasizes the African origins of the Egyptian civilization. In general, the chapter follows the analysis made by Gillings' in Mathematics in the Time of the Pharaohs. Babylonian mathematics is also discussed at greater length than in the usual text.

Joseph asks, "Is the overly critical attitude to Egyp- tian mathematics found in many textbooks an attempt to counteract the Greeks' (and others') generous acknow- ledgements of the great debt they owed these earlier civilizations. Were this debt acknowledged today, "This would undermine one of the central planks of the Euro- centric view of history and progress." The chapter on Egypt ends with two sentences pointing out that Alexan - dria, Egypt, became the center of Hellenistic mathematics, combining the traditions of Egypt, Babylonia and classical Greece. He says no more about this period because it has been "extensively explored" in other books.

Unfortunately, this also leaves out the work of Hypatia, an opportunity to discuss a woman algebraist. In addition, this understandable omission may create the appearance of an abrupt end to Egyptian mathematics c 300, in contrast to other mathematical traditions whose development is shown as a continuum in India, China and Mesopotamia. The presentation of the African contribution could also have been strengthened by including the Egyptian and North African contributions to medieval Islamic mathematics. These omissions do not take away from the usefulness of this work but indicate possible areas of expansion in the future editions of Crest which I am sure will appear.

(Beatrice Lumpkin)