Book Review Mathematics in India Reviewed by David Mumford Mathematics in India neither gives much historical context or explains the importance of astronomy as a driving force Kim Plofker for mathematical research in India. While West- Princeton University Press, 2008 ern scholars have been studying traditional Indian US$39.50, 384 pages mathematics since the late eighteenth century and ISBN-13: 978-0691120676 Indian scholars have been working hard to assem- ble and republish surviving Sanskrit manuscripts, Did you know that Vedic priests were using the so- a widespread appreciation of the greatest achieve- called Pythagorean theorem to construct their fire ments and the unique characteristics of the Indian altars in 800 BCE?; that the differential equation approach to mathematics has been lacking in the for the sine function, in finite difference form, was West. Standard surveys of the history of mathe- described by Indian mathematician-astronomers matics hardly scratch the surface in telling this in the fifth century CE?; and that “Gregory’s” story.3 Today, there is a resurgence of activity in series π/4 1 1 1 was proven using the = − 3 + 5 − ··· this area both in India and the West. The prosperity power series for arctangent and, with ingenious and success of India has created support for a new summation methods, used to accurately compute generation of Sanskrit scholars to dig deeper into π in southwest India in the fourteenth century? the huge literature still hidden in Indian libraries. If any of this surprises you, Plofker’s book is for Meanwhile the shift in the West toward a multi- you. cultural perspective has allowed us Westerners to Her book fills a huge gap: a detailed, eminently shake off old biases and look more clearly at other readable, scholarly survey of the full scope of traditions. This book will go a long way to opening 1 Indian mathematics and astronomy (the two were the eyes of all mathematicians and historians of inseparable in India) from their Vedic beginnings mathematics to the rich legacy of mathematics to to roughly 1800. There is only one other survey, which India gave birth. Datta and Singh’s 1938 History of Hindu Mathe- The first episode in the story of Indian mathe- matics, recently reprinted but very hard to obtain matics is that of the Sulba-s¯utras´ , “The rules of the in the West (I found a copy in a small special- cord”, described in section 2.2 of Plofker’s book.4 ized bookstore in Chennai). They describe in some detail the Indian work in arithmetic and algebra Banarsidass, Delhi. An excellent way to trace the litera- and, supplemented by the equally hard to find Ge- ture is through Hayashi’s article “Indian mathematics” in ometry in Ancient and Medieval India by Sarasvati the AMS’s CD History of Mathematics from Antiquity to Amma (1979), one can get an overview of most the Present: A Selective Annotated Bibliography (2000). 3 topics.2 But the drawback for Westerners is that The only survey that comes close is Victor Katz’s A History of Mathematics. David Mumford is professor of applied mathe- 4There are multiple ways to transcribe Sanskrit (and matics at Brown University. His email address is Hindi) characters into Roman letters. We follow the pre- [email protected]. cise scholarly system, as does Plofker (cf. her Appendix A) 1The word “India” is used in Plofker’s book and in my re- which uses diacritical marks: (i) long vowels have a bar view to indicate the whole of the Indian subcontinent, in- over them; (ii) there are “retroflex” versions of t, d, and n cluding especially Pakistan, where many famous centers where the tongue curls back, indicated by a dot beneath of scholarship, e.g., Takshila, were located. the letter; (iii) h, as in th, indicates aspiration, a breathy 2For those who might be in India and want to find sound, not the English “th”; and (iv) the “sh” sound is copies, Datta and Singh’s book is published by Bharatiya written either as ´sor as s. (the two are distinguishable to Kala Prakashan, Delhi, and Amma’s book by Motilal Indians but not native English speakers). March 2010 Notices of the AMS 385 These are part of the “limbs of the Vedas”, secular Pi˙ngala, who came a few centuries later, ana- compositions5 that were orally transmitted, like lyzed the prosody of Sanskrit verses. To do so, the sacred verses of the Vedas themselves. The he introduced what is essentially binary notation earliest, composed by Baudh¯ayana, is thought to for numbers, along with Pascal’s triangle (the bi- date from roughly 800 BCE. On the one hand, this nomial coefficients). His work started a long line work describes rules for laying out with cords the of research on counting patterns, including many sacrificial fire altars of the Vedas. On the other of the fundamental ideas of combinatorics (e.g., hand, it is a primer on plane geometry, with many the “Fibonacci” sequence appears sometime in of the same constructions and assertions as those 500-800 CE in the work of Virah¯anka). There is an found in the first two books of Euclid. In particular, interesting treatment of this early period of Indian as I mentioned above, one finds here the earliest mathematics in Frits Staal’s excellent recent book 8 explicit statement of “Pythagorean” theorem (so it Discovering the Vedas, ch.14. For example, Staal might arguably be called Baudh¯ayana’s theorem). traces recursion back to the elaborate and precise It is completely clear that this result was known to structure of Vedic rituals. the Babylonians circa 1800 BCE, but they did not After this period, unfortunately, one encoun- state it as such—like all their mathematical results, ters a gap, and very little survives to show what it is only recorded in examples and in problems mathematicians were thinking about for more than using it. And, to be sure, there are no justifica- 500 years. This was the period of Alexander’s in- tions for it in the Sulba-s¯utras´ either—these sutras vasion, the Indo-Greek Empire that existed side by are just lists of rules. But Pythagorean theorem side with the Mauryan dynasty including A´soka’s was very important because an altar often had to reign, and the Indo-Scythian and Kushan empires have a specific area, e.g., two or three times that that followed. It was a period of extensive trade of another. There is much more in these sutras: between India and the West, India and China. Was for example, Euclidean style “geometric algebra”, there an exchange of mathematical ideas too? No very good approximations to √2, and reasonable one knows, and this has become a rather political approximations to π. point. Plofker, I believe, does a really good job dis- Another major root of Indian mathematics is the cussing the contentious issues, stating in section 4.6 the “consensus” view but also the other points work of P¯anini and Pi˙ngala (perhaps in the fifth . of view. She states carefully the arguments on both century BCE and the third century BCE respec- sides and lets the reader take away what he or she tively), described in section 3.3 of Plofker’s book. will. She deals similarly with the early influences Though P¯anini is usually described as the great . from the Middle East in section 2.5 and of the grammarian of Sanskrit, codifying the rules of the exchanges with the Islamic world in Chapter 8. language that was then being written down for the For my part, I follow my late colleague David first time, his ideas have a much wider significance Pingree, who trained a whole generation of scholars than that. Amazingly, he introduced abstract sym- in ancient mathematics and astronomy. He argues bols to denote various subsets of letters and words that the early version of Greek astronomy, due that would be treated in some common way in to Hipparchus, reached India along with Greek some rules; and he produced rewrite rules that astrology. The early Indian division of the ecliptic were to be applied recursively in a precise or- 6 into twenty-eight Naks. atras, (the moon slept with der. One could say without exaggeration that he a different wife every night in each trip around the anticipated the basic ideas of modern computer ecliptic) was replaced by the Greek zodiac of twelve science. One wishes Plofker had described P¯an. ini’s solar constellations and—more to the point—an ideas at more length. As far as I know, there is analysis of solar, lunar, and planetary motion no exposition of his grammar that would make based on epicycles appears full-blown in the great it accessible to the non-linguist/Sanskrit scholar. treatise, the Aryabhat¯ .¯iya of Aryabhat¯ .a, written P. P. Divakaran has traced the continuing influence in 499 CE. But also many things in the Indian 7 of the idea of recursion on Indian mathematics, treatment are totally different from the Greek leading to the thesis that this is one of the major version. Their treatment of spherical trigonometry distinctive features of Indian mathematics. is based on three-dimensional projections, using 9 5 right triangles inside the sphere, an approach Technically, they are called smr.ti (“remembered text”) as opposed to ´sruti (“heard”, i.e., from divine sources). 8Penguin Books, 2008. 6To get a glimpse of this, see Plofker, p. 54; F. Staal, 9 A basic formula in the Gola section of the Aryabhat¯ .¯iya is “Artificial languages across sciences and civilization”, that if P is a point on the ecliptic with longitude λ, then J.