Book Review:
Mathematics in India. Princeton University Press: Princeton, 2008.
By Kim Plofker

Reviewed by David Mumford, Professor of Applied Mathematics at Brown University and winner of the Fields Medal – the most prestigious prize in mathematics.

Did you know that Vedic priests were using the socalled Pythagorean theorem to construct their fire altars in 800 BCE?; that the differential equation for the sine function, in finite difference form, was described by Indian mathematician-astronomers in the fifth century CE?; and that “Gregory’s” series π/4 = 1−1/3 +1/5 −· · · was proven using the power series for arctangent and, with ingenious summation methods, used to accurately compute π in southwest India in the fourteenth century? If any of this surprises you, Plofker’s book is for you.

Her book fills a huge gap: a detailed, eminently readable, scholarly survey of the full scope of Indian1 mathematics and astronomy (the two were inseparable in India) from their Vedic beginnings to roughly 1800. There is only one other survey, Datta and Singh’s 1938 History of Hindu Mathematics, recently reprinted but very hard to obtain in the West (I found a copy in a small specialized bookstore in Chennai). They describe in some detail the Indian work in arithmetic and algebra and, supplemented by the equally hard to find Geometry in Ancient and Medieval India by Sarasvati Amma (1979), one can get an overview of most topics.2 But the drawback for Westerners is that neither gives much historical context or explains the importance of astronomy as a driving force for mathematical research in India. While Western scholars have been studying traditional Indian mathematics since the late eighteenth century and Indian scholars have been working hard to assemble and republish surviving Sanskrit manuscripts, a widespread appreciation of the greatest achievements and the unique characteristics of the Indian approach to mathematics has been lacking in the West. Standard surveys of the history of mathematics hardly scratch the surface in telling this story.3 Today, there is a resurgence of activity in this area both in India and the West. The prosperity and success of India has created support for a new generation of Sanskrit scholars to dig deeper into the huge literature still hidden in Indian libraries. Meanwhile the shift in the West toward a multicultural perspective has allowed us Westerners to shake off old biases and look more clearly at other traditions. This book will go a long way to opening the eyes of all mathematicians and historians of mathematics to the rich legacy of mathematics to which India gave birth.

The first episode in the story of Indian mathematics is that of the Sulba-sutras, “The rules of the cord”, described in section 2.2 of Plofker’s book.4 These are part of the “limbs of the Vedas”, secular compositions5 that were orally transmitted, like the sacred verses of the Vedas themselves. The earliest, composed by Baudh¯ayana, is thought to date from roughly 800 BCE. On the one hand, this work describes rules for laying out with cords the sacrificial fire altars of the Vedas. On the other hand, it is a primer on plane geometry, with many of the same constructions and assertions as those found in the first two books of Euclid. In particular, as I mentioned above, one finds here the earliest explicit statement of “Pythagorean” theorem (so it might arguably be called Baudh¯ayana’s theorem). It is completely clear that this result was known to the Babylonians circa 1800 BCE, but they did not state it as such—like all theirmathematical results, it is only recorded in examples and in problems using it. And, to be sure, there are no justifications for it in the Sulba-sutras either—these sutras are just lists of rules. But Pythagorean theorem was very important because an altar often had to have a specific area, e.g., two or three times that of another. There is much more in these sutras: for example, Euclidean style “geometric algebra”, very good approximations to p2, and reasonable approximations to π.

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Image:
A page from a manuscript of Lilavati showing the Pythagorean Theorem