The sequence is familiar: Genius of Greek civilization, Dark Ages, rediscovery of Greek learning and the European Renaissance leading up to the present day. Though one can still find versions of it in popular and influential books, such an understanding of the history of mathematics – what George Gheverghese Joseph calls the “classical Eurocentric trajectory” – is rare nowadays, having been replaced by a view which allows for Egyptian and Mesopotamian influence on Greece, differentiates classical Greek culture from the Hellenistic world that followed it, and has Greek learning kept alive by the Arabs during the Dark Ages. This is better, Joseph maintains, but really only a grudging retrenchment, a “modified European trajectory”, that slights the Babylonians, knows nothing of the African roots of Egyptian arithmetic, underestimates Chinese mathematics, willfully misunderstands the Arab contribution and is scandalously ignorant of a rich and complex Indian tradition of mathematical thought and technical achievement.

Joseph, brought up in India and Africa, and teaching in a British university, is, as he recognizes, ideally suited to combat these Euro-centric wrongs. In The Crest of the Peacock we are offered an almost complete survey of extra-European mathematics, from what is called the Ishango Bone – a notched baboon’s fibula of 35,000 BC found in southern Africa – through Mayan numeration and Inca knotted-cord “abacuses”, to the beginnings of written mathematics in Egypt and Babylonia, the complexities of Chinese number theory and algebra, the enormously fertile Arab mathematical culture during the so-called Dark Ages, and the long curve of Indian mathematics from its earliest Vedic beginnings around 1000 BC, through the classical period, up to the early medieval work on infinite series.

Euro-centrism notwithstanding, much of this material can be found in existing books; but what is valuable here is the unified approach that Joseph brings to it, and the non-technical clarity that the attempt to reorder historical priorities and educate his readers out of their European prejudices requires. Much of this material, but not all: the account of Indian mathematics is definitely not part of standard historical treatments of the subject. Joseph takes his title from a text – “As are the crests on peacocks, as are the gems on the heads of snakes, so is mathematics at the head of all knowledge” – and means, I think, to shock his readers into a recognition of a non-European mathematical Other by placing the quotation and its provenance – Vedanga Jyosita, 500 BC – prominently on the back cover. Certainly, I was jolted out of a certain Euro-complacency by the thought that mathematics was long enough developed in India by this time to have engendered tributes to its superior epistemological status.

In fact, the survey of Indian mathematical thought is by far the most original and valuable contribution the book makes. The material is fascinating and intriguing: for example, the claims made for the mathematical culture situated in Kerala, in the south-west of the subcontinent, and in particular for its star turn, Madhava, to have anticipated, by several centuries, results of Gregory and Newton on infinite series. Or, very differently, two millennia earlier, the Vedic and then the Jaina preoccupation with very large numbers, and the latter’s classificatory scheme which partitioned numbers into enumberable, innumerable, infinite, each of which was in turn divided into three, so that, for the third kind, one had the nearly infinite, truly infinite and the infinitely infinite. On this last, Joseph’s explanation of the material, usually clear and not in the least contentious, was, at least to this reader, unconvincing. It is surely inadequate to the essential strangeness of the ideas here, as well as too hastily Eurocentric, to suggest mapping the nine-fold Jaina classification of the numbers on to the finite/infinite binary opposition of Cantor’s set theory.

The existence of a sophisticated mathematical tradition in Kerala (the fruit of recent Indian historiography and not yet fully understood or assimilated) has implications, Joseph believes, for the way a figure such as Srinivasa Ramanujan – the extraordinary young Indian mathematician brought from Kerala to England by G. H. Hardy early in this century – has to be understood. By placing him in the context of such a tradition – albeit one preserved perhaps in astrology, folklore, ritual and verse rather than conventional forms of learning – it might be possible, if I understand Joseph’s preface correctly, to rescue Ramanujan from the mixture of condescending disappointment about his ignorance of modern Western mathematics and mythic evaluation of his isolated genius that has been his – Europeanized – fate; and in so doing, arrive at a richer and less parochial vision of world mathematics. The Crest of the Peacock, among its many merits, allows one to raise such a possibility.

TLS November 15, 1991