An Indian history of numbers

Pervez Hoodbhoy¹

BOOK REVIEWED-Mathematics In India

by Kim Plofker

Princeton University Press: 2009. 394 pp. £28.95

Nature 459, 646-647 (4 June 2009) | doi:10.1038/459646a; Published online 3 June 2009

 

F. SOLTAN/SYGMA/CORBIS

Buddha is said to have wooed his future wife by reeling off a huge number series.

In a world divided by culture, politics, religion and race, it is a relief to know one thing that stands above them — mathematics. The diversity among today's mathematicians shows that it scarcely matters who invents concepts or proves theorems; cold logic is immune to prejudice, whim and historical accident. And yet, throughout history, different families of humans have distilled the essence of the cosmos to capture the magic of numbers in many ways.

Mathematics in India shows just how different one of these ways was, and how culture and mathematical development are intimately connected. This carefully researched chronicle of the principal contributions made by a great civilization covers the earliest days of Indian history through to the beginning of the modern period. Regrettably, it stops short of the legendary mathematician Srinivasa Ramanujan (born 1887), whose name is still seen in today's research papers.

Kim Plofker's book fulfils an important need in a world where mathematical historiography has been shaped by the dominance of the Greco-Christian view and the Enlightenment period. Too little has been written on the mathematical contributions of other cultures. One reason for the neglect of Indian mathematics was Eurocentrism — British colonial historians paid it little attention, assuming that Indians had been too preoccupied with spiritual matters to make significant contributions to the exact sciences. Another reason is that many ancient Indian mathematical texts have long been extinct; often, the only indication that they existed comes from scholars who refer to the work of their predecessors. As Plofker wryly notes, two historians of Indian maths recently published articles in the same edited volume, wherein the estimates of their subject's origins differed by about 2,000 years.

Still, surviving Sanskrit texts reveal a rich tradition of Indian mathematical discoveries lasting more than 2,500 years. In the Early Vedic period (1200–600 BC), a decimal system of numbers was already established in India, together with rules for arithmetical operations (ganita) and geometry (rekha-ganita). These were encoded in a complex system of chants, prayers, hymns, curses, charms and other religious rituals. Cryptic phrases called sutras contained arithmetical rules for activities such as laying out a temple or arranging a sequence of sacrificial fires.

Large numbers held immense fascination. Acclamations of praise to the air, sky, times of day or heavenly bodies were expressed in powers of ten that went to a trillion or more. Reputedly, the young Prince Buddha successfully competed for the hand of Princess Gopa by reciting a number table that included names for the powers of ten beyond the twentieth decimal place.

As in other early agricultural civilizations, Indian mathematics probably emerged in response to the need to measure land areas and keep track of financial transactions, incomes and taxation. A rigid caste and class hierarchy reserved the mystery of numbers for elite Brahmins. To maintain personal power, mathematical knowledge was jealously guarded. Its communication was deliberately made difficult, such as in the perplexing rhythmic chant of mathematician Aryabhatta in the fifth century AD: "makhi-bhakhi-phakhi- dhaki-nakhi-nakhi-nakhi-hasjha-skaki-kisga-sghaki-kighva-ghaki..." This recital of values of sine differences in arc minutes would be memorized by aspiring mathematicians in much the same way as verses of the sacred text Bhagavadgita.

The book details the impressive achievements of Indian mathematicians, from Aryabhatta through Brahmagupta, Mahavira, Bhaskara and Madhava, until the Sanskrit tradition became irrelevant with the invasion of modern mathematics from Europe in the nineteenth century. Major discoveries include finding the solution to indeterminate equations and the development of infinite series for trigonometric quantities. Discovered in the fourteenth century by the Kerala school founded by Madhava, these series built on the work of Bhaskara II and grew from the ingenious computation of a circle's circumference. By breaking up the circle into polygons, Madhava was able to calculate the value of pi correct to 11 decimal places. Some developments preceded those in Europe. For example, Reuben Burrow — a British mathematician posted to Bengal as an instructor in the engineers corps — was intrigued by rules he discovered in an unnamed Sanskrit text, and wrote a paper in 1790 entitled 'A Proof that the Hindoos had the Binomial Theorem'.

But how peculiarly Indian was early Indian mathematics? Did it evolve in isolation or did it absorb ideas and knowledge from elsewhere? Cultural pride in their recently reinvigorated country causes some Indians to claim that all worthwhile mathematics originated in ancient India. But this book will not please them. Plofker is not ready to certify that the concept of zero was an Indian invention; it could well have been conveyed by Chinese Buddhist pilgrims. Nor is she willing to believe that differential and integral calculus were anticipated in India ahead of the work of Gottfried Leibniz and Isaac Newton.

The chapter entitled 'Exchanges with the Islamic World' is of particular significance. The Muslim conquest of India brought with it the Islamic mathematical tradition, which was founded on Greek mathematics. Muslims made important advances in maths between the ninth and thirteenth centuries. Greco-Islamic and Indian mathematics were structured quite differently, with the former emphasizing proof and the latter, result. Probably because of Islamic influence, Indian ideas on the nature of mathematical proof moved in the direction of greater rigour.

The book carefully separates fact from hyperbole, copiously quoting formulae. This makes for heavy reading in places, and one wishes that it had been interspersed with vignettes and light anecdotes. It is more of a research monograph than a popular book. But that is the price that scholarship exacts.

Mathematics in India explains how the early development of Indian Maths was influenced by religion, by the need to build temples of specific proportions and to meet astrological imperatives. Similarly, it could be argued that Islamic mathematics was religiously motivated — for example, by the need to know the precise times of daily prayers, and to determine the direction of the holy Kaaba (the Qibla). But a quadratic equation solved by whoever, by whatever means and for whatever purpose must give exactly the same solutions. Ultimately, mathematics is mathematics.