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Home , Indian mathematics, Mathematics in India

Book Review in 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 manuscripts, Did you know that Vedic priests were using the so- a widespread appreciation of the greatest achieve- called 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 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 π/4 1 1 1 was proven using the = − 3 + 5 − ··· this both in India and the West. The prosperity for arctangent and, with ingenious and success of India has created support for a new 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 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 and 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 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 , in- over them; (ii) there are “retroflex” versions of t, d, and n cluding especially , 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 ”, secular ˙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 , 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 (e.g., hand, it is a primer on plane , with many the “” 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 ’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 . 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 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 , reached India along with Greek some rules; and he produced rewrite rules that . 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 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 distinctive features of Indian mathematics. is based on three-dimensional projections, using 9 5 right triangles inside the , 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 λ, then J. Indian Philosophy, pp. 89–141 (esp. sections 11–12), the declination δ of P is given by sin(δ) sin(λ). sin(i), ¯ = 2006; or B. Gillon, “As. .t¯adhy¯ayi and linguistic theory”, J. i the inclination of the ecliptic. If I understand it right, Indian Philosophy, pp. 445–468, 2007. later writings suggest this was proven by considering the 7 “Notes on Yukti-Bh¯as. ¯a: Recursive methods in Indian planar right triangle given by P,P1,P2, where P1 is the mathematics”, forthcoming in a book entitled Studies in orthogonal projection of P onto the plane of the equator the History of . (inside the sphere!) and P2 is its projection onto the line

386 Notices of the AMS 57, 3 which I find much simpler and more natural than the Mathematical Seed, has been published11 in ’s use of Menelaus’s theorem. Above all, as English by Agathe Keller. One hopes she will mentioned above, they found the finite difference follow this with an edition of the astronomical equation satisfied by samples sin(n. θ) of sine chapters. And Glen van Brummelen has written a (see Plofker, section 4.3.3). This seems to have cross-cultural study of the use of trigonometry, set the future development of mathematics∆ and entitled The Mathematics of the Heavens and the astronomy in India on a path totally distinct from Earth,12 which compares in some detail Greek and anything in the West (or in China). Indian work. It is important to recognize two essential dif- Chapters 5 and 6 of Plofker’s book, entitled ferences here between the Indian approach and The Genre of Medieval Mathematics and The Devel- that of the Greeks. First of all, whereas Eudoxus, opment of “Canonical” Mathematics, are devoted Euclid, and many other Greek mathematicians to the sixth through twelfth centuries of In- created pure mathematics, devoid of any actual dian mathematical work, starting with Aryabhat¯ .a numbers and based especially on their invention and ending with Bh¯askara (also called Bh¯askara of indirect reductio ad absurdum arguments, the II or Bh¯askaracharya, distinguishing him from Indians were primarily applied mathematicians the earlier Bh¯askara). This was a period of in- focused on finding algorithms for astronomical tense mathematical-astronomical activity from predictions and philosophically predisposed to which many works have survived, and I want reject indirect arguments. In fact, Buddhists and to touch on some of its high points. We find Jains created what is now called Belnap’s four- already in the seventh century the full arithmetic valued logic claiming that assertions can be true, of negative numbers in ’s Br¯ahma- false, neither, or both. The Indian mathematics sphut.a-siddh¯anta (see Plofker, p. 151). This may tradition consistently looked for constructive argu- sound mundane but, surprisingly, nothing sim- ments and justifications and numerical algorithms. ilar appears in the West until Wallis’s Algebra So whereas Euclid’s Elements was embraced by Is- published in 1685.13 And in the Bakhsh¯al¯i man- lamic mathematicians and by the Chinese when uscript, an incredibly rare birch bark manuscript Matteo Ricci translated it in 1607, it simply didn’t unearthed by a farmer’s plow in 1881, we find fit with the Indian way of viewing math. In fact, algebraic equations more or less in the style of there is no evidence that it reached India before Viète, Fermat, and Descartes. It is incomplete and the eighteenth century. neither title nor author survives, but paleographi- Secondly, this scholarly work was mostly carried cal evidence suggests that it was written between out by Brahmins who had been trained since a very the eighth and twelfth centuries, and Hayashi ar- early age to memorize both sacred and secular gues that its rules and examples date from the Sanskrit verses. Thus they put their mathematics seventh century.14 The manuscript puts equations not in extended treatises on parchment as was in boxes, like our displayed formulas. On p. 159 of done in Alexandria but in very compact (and Plofker’s book, she gives the example from bark cryptic) Sanskrit verses meant to be memorized by fragment 59. The full display in the original is their students. What happened when they needed below. to pass on their sine tables to future generations? They composed verses of sine differences, arguably because these were much more compact than the themselves, hence easier to set to verse and memorize.10 Because their tables listed sines every 3.75 degrees, these first order differences did not closely match the sine table read backward; but the second differences were almost exactly a small 0 5 yu mu 0 sa 0 7 + mu 0 negative multiple of the sines themselves, and this 1 1 1 1 1 they noticed. There are several excellent recent books that give more background on these early devel- Here the 0’s (given by solid dots in the manu- opments. The mathematical sections of the script) stand for unknowns, the 1’s (given by the Aryabhat¯ .¯iya with the seventh century commen- sigma-like subscripted symbols in the manuscript) tary on it by Bh¯askara (I) and an extensive modern commentary, all entitled Expounding 11Springer-Verlag, 2008, for an obscene price of $238!! 12Princeton University Press, 2009. through , the intersection of the equator and the ecliptic. 13For example, both Cardano and Harriot were unsure It is immediate to derive the formula using this triangle. whether to make ( 1) ( 1) equal to 1 or 1. − · − − + 10Using RΥ 3438, the number of minutes in a radian 14See the fully edited and commented edition by Takao = to the nearest integer, and θ 3.75 degrees, they Hayashi, The : An Ancient Indian = calculated R sin(n θ) to the nearest integer. Mathematical Treatise, John Benjamin Pub. Co., 1995. · ∆ ∆ March 2010 Notices of the AMS 387 are just denominators, the yu means 5 is added to not? That he gives many quite complex auxiliary the unknown on its left, + sign signifies that 7 is results on cyclic quadrilaterals suggests he played subtracted, m¯u means root, and s¯a is a pro- with such quadrilaterals extensively.15 noun indicating that the 1st and 3rd unknowns are Indian work on Pell’s equation in the general equal. The whole thing has the modern equivalent: form x2 Ny 2 c also goes back to Brahmagupta. − = √x, 5 w √x. 7 z He discovered its multiplicative property— + = − = solutions for c1 and c2 can be “multiplied” to give This is to be solved in integers, giving x 11. = one for c1c2 (Plofker, pp. 154-156). A complete Note, however, that the Bakhsh¯al¯i manuscript algorithm, known as the “cyclic method”, for does not solve its problems using manipulations constructing a solution to the basic equation of its equations. In Brahmagupta’s treatment of x2 Ny 2 1 was discovered by Jayadeva, whose algebra in the Br¯ahma-Sphut. a-Siddh¯anta, distinct work− is dated= indirectly to the eleventh century colors are used to represent distinct unknowns (Plofker, pp. 194-195). Note again the emphasis (see Plofker’s discussion of his Chapter 18, pp. on construction instead of indirect proofs of 149–157). existence, which are the staple of our treatment of The use of negative numbers to represent points the subject.16 Why such a focus on this equation? on a line to the left of a base point appears One idea is that if x, y is a solution, then x/y is a ¯ ¯ in Bh¯askara’s Lil¯avati. This twelfth-century book, good approximation to √N. described in section 6.2.1 of Plofker’s book, is The discovery of the finite difference equation arguably the most famous of all Indian texts on for sine led Indian mathematicians eventually to ¯ ¯ mathematics. Given the fact that Lil¯avati literally the full theory of for and for means “beautiful” or “playful” and that many sine, cosine, arcsine, and arctangent functions, that verses are addressed to “the fawn-eyed one”, is, for everything connected to the and sphere the conjecture made by a Persian translator that that might be motivated by the applications to the book was written to explain mathematics to astronomy. This work matured over the thousand- Bh¯askara’s daughter seems quite reasonable. year period in which the West slumbered, reaching Another basic tool which appears in all Indian its climax in the work of the school in the manuscripts is what they called the pulverizer. fourteenth to sixteenth centuries. I won’t describe This is an extension of the Euclidean algorithm, the full evolution but cannot omit a mention the idea of starting with two positive integers and of the discovery of the formula for the area and repeatedly subtracting the lesser from the greater. volume of the sphere by Bh¯askara II. Essentially, he They go further than Euclid in using this to sys- rediscovered the derivation found in Archimedes’ tematically write down all solutions of first-order On the Sphere and the I. That is, he sliced integer equations ax by c. It seems unlikely + = the surface of the sphere by equally spaced lines that the Greek algorithm, embedded in the Ele- of latitude and, using this, reduced the calculation ments in highly abstract form, was transmitted of the area to the of sine. Now, he knew to India, hence more likely that the idea was that cosine differences were sines but, startlingly, discovered independently in India. In fact, Indian he integrates sine by summing his tables! He astronomers had a very pressing application for seems well aware that this is approximate and this algorithm. Although they had, in fact, aban- that a limiting argument is needed but this is doned almost all of the ancient Vedic astronomy, implicit in his work. My belief is that, given his they were not happy doing this and they retained applied orientation, this was the more convincing one startling idea from that tradition: the vast argument. In any case, the argument using the epochs into which the past was divided, the yugas, discrete fundamental theorem of calculus is given all had to begin with one spectacular conjunction a few centuries later by the Kerala school, where of the sun, the moon, and all the planets. To as- one also finds explicit statements on the need for certain when the present yuga began and thus put a limiting process, like: “The greater the number future predictions on a sound basis, they had to [of subdivisions of an arc], the more accurate the solve such integer equations involving the periods circumference [given by the length of the inscribed of the heavenly bodies. polygon]” and “Here the arc segment has to be There are other high points of the work of this imagined to be as small as one wants... [but] since period. One of them is Brahmagupta’s formula for one has to explain [it] in a certain [definite] way, [I] the area of a quadrilateral inscribed in a circle. have said [so far] that a quadrant has twenty-four How he discovered this is a fascinating question. chords.” No justification has been found in any manuscripts earlier than the Kerala work (see below). It can 15Added in proof: I just received a copy of S. Kichenas- be derived from Pythagorean theorem and sim- samy’s article “Brahmagupta’s derivation of the area of a ple geometry but only with substantial algebraic ”, Historia Mathematica, 2009. computation. Did Brahmagupta use algebra, ma- 16See, e.g., Artin’s Algebra, Prentice-Hall, 1991, pp. 434- nipulations of algebraic equations, to find it or 437.

388 Notices of the AMS Volume 57, Number 3 Chapter 7 of Plofker’s book is devoted to the Note that the right-hand side is exactly the finite crown jewel of Indian mathematics, the work of difference version of the double integral in the cal- the Kerala school. Kerala is a narrow fertile strip culus version. Here is how Jyes.t.hadeva expressed between the mountains and the Arabian Sea along this formula (p. 97 of Sarma’s translation; here the southwest coast of India. Here, in a number “repeated summation” stands for the sum of sums of small villages, supported by the Maharaja of on the right): 17 Calicut, an amazing dynasty of mathematicians Here, multiply the repeated sum- and astronomers lived and thrived. A large pro- mation of the Rsines by the square portion of their results were attributed by later of the full chord and divide by writers to the founder of this school, Madhava of the square of the radius. ... In Sangamagramma, who lived from approximately this manner we get the result that 1350 to 1425. It seems fair to me to compare him when the repeated summation of with Newton and Leibniz. The high points of their the Rsines up to the tip of a par- mathematical work were the discoveries of the ticular arc-bit is done, the result power series expansions of arctangent, sine, and will be the difference between the cosine. By a marvelous and unique happenstance, next higher Rsine and the corre- there survives an informal exposition of these sponding arc. Here the arc-bit has results with full derivations, written in , to be conceived as being as minute the vernacular of Kerala, by Jyes.t.hedeva perhaps as possible. Then the first Rsine about 1540. This book, the Gan. ita-Yukti-Bh¯as. ¯a, difference will be the same as the has only very recently been translated into English first arc-bit. Hence, if multiplied by with an extensive commentary.18 As a result, this the desired number, the result will book gives a unique insight into Indian methods. certainly be the desired arc. Simply put, these are recursion, induction, and He is always clear about which formulas are careful passage to the limit. exact and which formulas are approximations. In I want to give one example in more detail, the the modern approach, one has the usual iterative derivation of the power series expansion for sine. solution to the integral equation θ K θ K K It seems most transparent to explain the idea of θ , and you can work out each− term∗ here+ using∗ ∗ −· · · n 1 the proof in modern form and then to indicate how y n y + the indefinite 0 x dx n 1 resulting Jyes.t.hadeva’s actual derivation differed from this. = + in the power series forR sine. Jyes.t.hadeva does The derivation is based on the integral equation the same thing in finite difference form, starting for sine: with sin(θ) θ, and recursively improving the ≈ θ estimate for sine by resubstitution into the left- θ sin(θ) (1 cos(β))dβ hand side of the above identity. Instead of the − = Z0 − θ β integral of powers, he needs the approximate sum p 1 (α)dα dβ (K )(θ) n p k + p sin sin of powers, i.e., k 1 k p 1 O(k ), and he = Z0 Z0 ! = ∗ = = + + has evaluated theseP earlier (in fact, results like where K(x, y) max(0, x y). = − this go way back in Indian mathematics). Then Jyesthadeva uses a finite difference form of this repeated resubstitution gives the usual power . . 3 5 equation using discrete samples of sine: he sub- θ θ series θ 6 120 for sine. I consider this divides the arc [0, θ] into n “arc-bits” of size argument− to be+ completely− ··· correct, but I am aware θ θ/n and, choosing a big radius R (like that it is not a rigorous proof by modern standards. = 3438, see above), he works with sampled “Rsines” It can, however, be converted into such a proof by B∆k R sin(k θ) and also the “full chord” of the anyone with basic familiarity with ǫ, δ-techniques. = · arc-bit: 2R sin( θ/2). Then, based on the formula I hope I have whetted your appetite enough so you for the second∆ difference of sines which goes back will want to feast on the riches laid out in Plofker’s to , he∆ derives: book and the Gan. ita-Yukti- Bh¯as. ¯a itself. 2 It is very tempting to read the history of math- θ sin(θ) bB1 Bn (2 sin( θ/2)) − ≈ − = ematics as a long evolution toward the present ((B1 Bn 1) (B1 Bn 2) state of deep knowledge. I see nothing wrong with · + · · · + − + +∆ · · · + − understanding the older discoveries in the light (B2 B1) B1). + · · · + + + of what we know now—like a contemporary met- allurgist analyzing ancient swords. Needham, the 17The names we know form an essentially linear se- quence of teacher and student (sometimes son). great scholar of Chinese science, wrote “To write 18Translated and edited by the late K. V. Sarma, with the we have to take modern notes by K. Ramasubramanian, M. D. Srinivas, and M. S. science as the yardstick—that is the only thing Sriram, and published in India in 2008 by the Hindustan we can do—but modern science will change and Book Agency at a price of $31 and distributed in the West the end is not yet.” Nevertheless, it is much more by Springer for $199, a 640% markup. satisfying, when reading ancient works, to know as

March 2010 Notices of the AMS 389 much as possible about the society in which these an orgy of correct and important but semirigorous mathematicians worked, to know what mathemat- math in which Greek ideals were forgotten. The ics was used for in their society, and how they recent episodes with deep mathematics flowing themselves lived. from quantum field and string theory teach us the Chapter 1 in Plofker’s book is an extensive same lesson: that the muse of mathematics can introduction that gives vital background on the be wooed in many different ways and her secrets history and traditions from which all the Indian teased out of her. And so they were in India: read work sprang. A series of extremely helpful appen- this book to learn more of this wonderful story! dices provide basic facts about Sanskrit, a glossary of Sanskrit terms, and a list of the most signifi- cant Indian mathematicians with whatever basic facts about them are known (often distressingly little). In places she gives some literal translations such as those of numbers in the colorful concrete number system that uses a standard list of sets with well-known cardinalities, e.g., “In a kalpa, the revolutions of the moon are equal to five skies [0], qualities [3], qualities, five, sages [7], arrows [5]”, which means 5,753,300,000 lunar months with the digits described backwards starting from the one’s place. It is becoming more and more recognized that, for a good understanding of ancient writings, one needs both an extremely literal translation and one paraphrased so as to be clear in modern terms. In Sanskrit a literal translation of the com- plex compound words sometimes gives additional insight into the author’s understanding (as well as the ambiguities of the text), so one wishes there were more places in this book where Plofker gave such literal translations without using modern expressions. I have not touched on the astronomical side of the story. Suffice it to say that almost all treatises from the sixth century on deal with both astron- omy and mathematics. To follow these, one needs a bit of a primer in geocentric astronomy, a van- ishing specialty these days, and Plofker provides a very handy introduction in section 4.1. Just as in Indian mathematics, there is a steady increase in sophistication over the centuries, culminating in dramatic advances in Kerala. Most strikingly, N¯ilakan. t.ha in the fifteenth century proposed a model in which the planets were moving in ec- centric and inclined with respect to the mean sun moving in its ecliptic orbit—a “virtu- ally” heliocentric model remarkably better than Ptolemy’s. It is high time that the full story of Indian math- ematics from Vedic times through 1600 became generally known. I am not minimizing the genius of the Greeks and their wonderful invention of pure mathematics, but other peoples have been doing math in different ways, and they have often attained the same goals independently. Rigorous mathematics in the Greek style should not be seen as the only way to gain mathematical knowledge. In India, where concrete applications were never far from theory, justifications were more informal and mostly verbal rather than written. One should also recall that the European Enlightenment was

390 Notices of the AMS Volume 57, Number 3