Mathematics in India

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Book Review

Mathematics in IndiaReviewed by David Mumford

Mathematics in India

Kim Plofker Princeton University Press, 2008

US$39.50, 384 pages ISBN-13: 978-0691120676

Did you know that Vedic priests were using the so-called Pythagorean theorem to construct their firealtars in 800 BCE?; that the differential equationfor the sine function, in finite difference form, wasdescribed by Indian mathematician-astronomersin 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 ingenioussummation methods, used to accurately computeπ in southwest India in the fourteenth century?

If any of this surprises you, Plofker’s book is foryou.Her book fills a huge gap: a detailed, eminently

readable, scholarly survey of the full scope of

Indian1 mathematics and astronomy (the two wereinseparable in India) from their Vedic beginningsto roughly 1800. There is only one other survey,Datta and Singh’s 1938 History of Hindu Mathe- matics , recently reprinted but very hard to obtainin the West (I found a copy in a small special-ized bookstore in Chennai). They describe in somedetail the Indian work in arithmetic and algebraand, supplemented by the equally hard to find Ge- ometry in Ancient and Medieval India by Sarasvati

Amma (1979), one can get an overview of mosttopics.2 But the drawback for Westerners is that

David Mumford is professor of applied mathe-

matics at Brown University. His email address is

[email protected] .1The word “India” is used in Plofker’s book and in my re-

view to indicate the whole of the Indian subcontinent, in-

cluding especially Pakistan, where many famous centers

of scholarship, e.g., Takshila, were located.2For those who might be in India and want to find

copies, Datta and Singh’s book is published by Bharatiya

Kala Prakashan, Delhi, and Amma’s book by Motilal

neither gives much historical context or explainsthe importance of astronomy as a driving forcefor mathematical research in India. While West-ern scholars have been studying traditional Indian

mathematics since the late eighteenth century andIndian scholars have been working hard to assem-

ble and republish surviving Sanskrit manuscripts,a widespread appreciation of the greatest achieve-ments and the unique characteristics of the Indianapproach to mathematics has been lacking in theWest. Standard surveys of the history of mathe-matics hardly scratch the surface in telling this

story.3 Today, there is a resurgence of activity inthis area both in India andthe West. The prosperityand success of India has created support for a newgeneration of Sanskrit scholars to dig deeper intothe huge literature still hidden in Indian libraries.

Meanwhile the shift in the West toward a multi-cultural perspective has allowed us Westerners toshake off old biases and look more clearly at othertraditions. This book will go a long way to openingthe eyes of all mathematicians and historians of mathematics to the rich legacy of mathematics towhich India gave birth.

The first episode in the story of Indian mathe-

matics is that of the ´ Sulba-s¯ utras , “The rules of thecord”, described in section 2.2 of Plofker’s book.4

Banarsidass, Delhi. An excellent way to trace the litera-

ture is through Hayashi’s article “Indian mathematics” in

the AMS’s CD History of Mathematics from Antiquity to

the Present: A Selective Annotated Bibliography (2000).3The only survey that comes close is Victor Katz’s A

History of Mathematics.4There are multiple ways to transcribe Sanskrit (and

Hindi) characters into Roman letters. We follow the pre-

cise scholarly system, as does Plofker (cf. her Appendix A)

which uses diacritical marks: (i) long vowels have a bar

over them; (ii) there are “retroflex” versions of t, d, and n

where the tongue curls back, indicated by a dot beneath

the letter; (iii) h, as in th, indicates aspiration, a breathy

sound, not the English “th”; and (iv) the “sh” sound is

written either as s or as s . (the two are distinguishable to

Indians but not native English speakers).

March 2010 Notices of the AMS 385



These are part of the “limbs of the Vedas”, secular

compositions5 that were orally transmitted, likethe sacred verses of the Vedas themselves. The

earliest, composed by Baudhayana, is thought to

date from roughly 800 BCE. On the one hand, thiswork describes rules for laying out with cords thesacrificial fire altars of the Vedas. On the other

hand, it is a primer on plane geometry, with manyof 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 Baudhayana’s theorem).It is completely clear that this result was known tothe Babylonians circa 1800 BCE, but they did not

state it as such—like all their mathematical results,it is only recorded in examples and in problems

using it. And, to be sure, there are no justifica-

tions for it in the ´ Sulba-s¯ utras either—these sutrasare 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 thatof another. There is much more in these sutras:

for example, Euclidean style “geometric algebra”,very good approximations to

√ 2, and reasonable

approximations to π .Another major root of Indian mathematics is the

work of Pan. ini and Pingala (perhaps in the fifth

century BCE and the third century BCE respec-

tively), described in section 3.3 of Plofker’s book.Though Pan. ini is usually described as the greatgrammarian of Sanskrit, codifying the rules of thelanguage that was then being written down for the

first time, his ideashave a much wider significance

than that. Amazingly, he introduced abstract symbols to denote various subsets of letters and words

that would be treated in some common way in

some rules; and he produced rewrite rules thatwere to be applied recursively in a precise or-

der.6 One could say without exaggeration that heanticipated the basic ideas of modern computer

science. One wishes Plofker had described Pan. ini’sideas at more length. As far as I know, there isno exposition of his grammar that would make

it accessible to the non-linguist/Sanskrit scholar.P. P. Divakaran has traced the continuing influence

of the idea of recursion on Indian mathematics, 7

leading to the thesis that this is one of the majordistinctive features of Indian mathematics.

5Technically, they are called smr.ti (“remembered text”)

as opposed to sruti (“heard”, i.e., from divine sources).6

To get a glimpse of this, see Plofker, p. 54; F. Staal,

“Artificial languages across sciences and civilization”,

J. Indian Philosophy, pp. 89–141 (esp. sections 11–12),

2006; or B. Gillon, “As . t . ¯ adhy¯ ay i and linguistic theory”, J.

Indian Philosophy, pp. 445–468, 2007.7“Notes on Yukti-Bh¯ as . ¯ a: Recursive methods in Indian

mathematics”, forthcoming in a book entitled Studies in

the History of Mathematics in India.

Pingala, who came a few centuries later, ana-lyzed the prosody of Sanskrit verses. To do so,he introduced what is essentially binary notationfor numbers, along with Pascal’s triangle (the bi-nomial coefficients). His work started a long lineof research on counting patterns, including manyof the fundamental ideas of combinatorics (e.g.,the “Fibonacci” sequence appears sometime in500-800 CE in the work of Virahanka). There is aninteresting treatment of this early period of Indianmathematics in Frits Staal’s excellent recent bookDiscovering the Vedas ,8 ch.14. For example, Staaltraces recursion back to the elaborate and precisestructure of Vedic rituals.

After this period, unfortunately, one encoun-ters a gap, and very little survives to show whatmathematicianswere thinking about formore than500 years. This was the period of Alexander’s in-vasion, the Indo-Greek Empire that existed side byside with the Mauryan dynasty including Asoka’sreign, and the Indo-Scythian and Kushan empires

that followed. It was a period of extensive trade between India and the West, India and China. Wasthere an exchange of mathematical ideas too? Noone knows, and this has become a rather politicalpoint. Plofker, I believe, does a really good job dis-cussing the contentious issues, stating in section4.6 the “consensus” view but also the other pointsof view. She states carefully the arguments on bothsides and lets the reader take away what he or shewill. She deals similarly with the early influencesfrom the Middle East in section 2.5 and of theexchanges with the Islamic world in Chapter 8.

For my part, I follow my late colleague DavidPingree, whotraineda wholegeneration of scholarsin ancient mathematics and astronomy. He arguesthat the early version of Greek astronomy, dueto Hipparchus, reached India along with Greekastrology. The early Indian division of the eclipticinto twenty-eight Naks . atras , (the moon slept witha different wife every night in each trip around theecliptic)was replaced by the Greek zodiacof twelvesolar constellations and—more to the point—ananalysis of solar, lunar, and planetary motion

based on epicycles appears full-blown in the greattreatise, the ¯ Aryabhat .¯ iya of Aryabhat.a, writtenin 499 CE. But also many things in the Indiantreatment are totally different from the Greek

version. Their treatment of spherical trigonometryis based on three-dimensional projections, using

right triangles inside the sphere,9 an approach

8Penguin Books, 2008.9

A basic formula in the Gola section of the Aryabhat.iya is

that if P is a point on the ecliptic with longitude λ, then

the declination δ of P is given by sin(δ) = sin(λ). sin(i),

i the inclination of the ecliptic. If I understand it right,

later writings suggest this was proven by considering the

planar right triangle given by P, P 1, P 2, where P 1 is the

orthogonal projection of P onto the plane of the equator

(inside the sphere!) and P 2 is its projection onto the line

386 Notices of the AMS Volume 57, Number 3



which I find much simpler and more natural than

Ptolemy’s use of Menelaus’s theorem. Above all, asmentioned above, they found the finite differenceequation satisfied by samples sin(n.∆θ) of sine(see Plofker, section 4.3.3). This seems to haveset the future development of mathematics andastronomy in India on a path totally distinct fromanything in the West (or in China).

It is important to recognize two essential dif-ferences here between the Indian approach andthat of the Greeks. First of all, whereas Eudoxus,Euclid, and many other Greek mathematicianscreated pure mathematics, devoid of any actualnumbers and based especially on their inventionof indirect reductio ad absurdum arguments, theIndians were primarily applied mathematiciansfocused on finding algorithms for astronomicalpredictions and philosophically predisposed toreject indirect arguments. In fact, Buddhists andJains created what is now called Belnap’s four-valued logic claiming that assertions can be true,

false, neither, or both. The Indian mathematicstradition consistently lookedfor constructive arguments and justifications and numerical algorithms .So whereas Euclid’s Elements was embraced by Is-lamic mathematicians and by the Chinese whenMatteo Ricci translated it in 1607, it simply didn’tfit with the Indian way of viewing math. In fact,there is no evidence that it reached India beforethe eighteenth century.

Secondly, this scholarly work was mostly carriedout by Brahmins who had been trainedsince a veryearly age to memorize both sacred and secularSanskrit verses. Thus they put their mathematics

not in extended treatises on parchment as wasdone in Alexandria but in very compact (and

cryptic) Sanskrit verses meant to be memorized bytheir students. What happened when they neededto pass on their sine tables to future generations?They composed verses of sine differences , arguably

because these were much more compact than thesines themselves, hence easier to set to verse andmemorize.10 Because their tables listed sines every3.75 degrees, these first order differences did notclosely match the sine table read backward; butthe second differences were almost exactly a smallnegative multiple of the sines themselves, and thisthey noticed.

There are several excellent recent books thatgive more background on these early devel-opments. The mathematical sections of the¯ Aryabhat .¯ iya with the seventh century commen-

tary on it by Bhaskara (I) and an extensivemodern commentary, all entitled Expounding

through Υ , the intersection of the equator and the ecliptic.

It is immediate to derive the formula using this triangle.10Using R = 3438, the number of minutes in a radian

to the nearest integer, and ∆θ = 3.75 degrees, they

calculated R · sin(n∆θ) to the nearest integer.

the Mathematical Seed , has been published11 inEnglish by Agathe Keller. One hopes she willfollow this with an edition of the astronomicalchapters. And Glen van Brummelen has written across-cultural study of the use of trigonometry,entitled The Mathematics of the Heavens and the

Earth,12 which compares in some detail Greek andIndian work.

Chapters 5 and 6 of Plofker’s book, entitledThe Genre of Medieval Mathematics and The Devel- opment of “Canonical” Mathematics , are devotedto the sixth through twelfth centuries of In-dian mathematical work, starting with Aryabhat.aand ending with Bhaskara (also called BhaskaraII or Bhaskaracharya, distinguishing him fromthe earlier Bhaskara). This was a period of in-tense mathematical-astronomical activity fromwhich many works have survived, and I wantto touch on some of its high points. We findalready in the seventh century the full arithmeticof negative numbers in Brahmagupta’s Br¯ ahma-

sphut . a-siddh¯ anta (see Plofker, p. 151). This maysound mundane but, surprisingly, nothing sim-ilar appears in the West until Wallis’s Algebra

published in 1685.13 And in the Bakhsh¯ al i man-uscript, an incredibly rare birch bark manuscriptunearthed by a farmer’s plow in 1881, we findalgebraic equations more or less in the style of Viète, Fermat, and Descartes. It is incomplete andneither title nor author survives, but paleographi-cal evidence suggests that it was written betweenthe eighth and twelfth centuries, and Hayashi ar-gues that its rules and examples date from the

seventh century.14 The manuscript puts equations

in boxes, like our displayed formulas. On p. 159 of Plofker’s book, she gives the example from barkfragment 59. The full display in the original is

below.

01

51

yu mu 01

sa 01

71

+ mu 0

Here the 0’s (given by solid dots in the manu-script) stand for unknowns, the 1’s (given by thesigma-like subscripted symbols in the manuscript)

11Springer-Verlag, 2008, for an obscene price of $238!! 12

Princeton University Press, 2009.13For example, both Cardano and Harriot were unsure

whether to make (−1) · (−1) equal to −1 or +1.14See the fully edited and commented edition by Takao

Hayashi, The Bakhshali Manuscript: An Ancient Indian

Mathematical Treatise, John Benjamin Pub. Co., 1995.




Chapter 7 of Plofker’s book is devoted to thecrown jewel of Indian mathematics, the work of the Kerala school. Kerala is a narrow fertile strip

between the mountains and the Arabian Sea alongthe southwest coast of India. Here, in a numberof small villages, supported by the Maharaja of

Calicut, an amazing dynasty17 of mathematiciansand astronomers lived and thrived. A large pro-portion of their results were attributed by laterwriters to the founder of this school, Madhava of Sangamagramma, who lived from approximately1350 to 1425. It seems fair to me to compare himwith Newton and Leibniz. The high points of theirmathematical work were the discoveries of thepower series expansions of arctangent, sine, andcosine. By a marvelous and unique happenstance,there survives an informal exposition of theseresults with full derivations, written in Malayalam,the vernacular of Kerala, by Jyes. t.hedeva perhapsabout 1540. This book, the Gan. ita-Yukti-Bh¯ as . ¯ a ,has only very recently been translated into English

with an extensive commentary.18 As a result, this book gives a unique insight into Indian methods.Simply put, these are recursion, induction, andcareful passage to the limit.

I want to give one example in more detail, thederivation of the power series expansion for sine.It seems most transparent to explain the idea of the proof in modern form and then to indicate howJyes. t.hadeva’s actual derivation differed from this.The derivation is based on the integral equationfor sine:

θ − sin(θ) =

θ

0(1 − cos(β))dβ

= θ

0

β0

sin(α)dαdβ = (K ∗ sin)(θ)

where K(x,y) = max(0, x− y).

Jyes. t.hadeva uses a finite difference form of thisequation using discrete samples of sine: he sub-divides the arc [0, θ] into n “arc-bits” of size∆θ = θ/n and, choosing a big radius R (like3438, see above), he works with sampled “Rsines”Bk = R · sin(k∆θ) and also the “full chord” of thearc-bit: 2R sin(∆θ/2). Then, based on the formulafor the second difference of sines which goes backto Aryabhata, he derives:

θ − sin(θ) ≈ bB1 − Bn = (2sin(∆θ/2))2

· ((B1 +· · · + Bn−1)+ (B1 +· · · + Bn−2)

+· · ·+ (B2 + B1)+ B1) .

17The names we know form an essentially linear se-

quence of teacher and student (sometimes son).18

Translated and edited by the late K. V. Sarma, with

notes by K. Ramasubramanian, M. D. Srinivas, and M. S.

Sriram, and published in India in 2008 by the Hindustan

Book Agency at a price of $31 and distributed in the West

by Springer for $199, a 640% markup.

Note that the right-hand side is exactly the finitedifference version of the double integral in the cal-culus version. Here is how Jyes. t.hadeva expressedthis formula (p. 97 of Sarma’s translation; here“repeated summation” stands for the sum of sumson the right):

Here, multiply the repeated sum-

mation of the Rsines by the squareof the full chord and divide bythe square of the radius. . . . Inthis manner we get the result thatwhen the repeated summation of the Rsines up to the tip of a par-ticular arc-bit is done, the resultwill be the difference between thenext higher Rsine and the corre-sponding arc. Here the arc-bit hasto be conceived as being as minuteas possible. Then the first Rsinedifference will be the same as thefirst arc-bit. Hence, if multiplied by

the desired number, the result willcertainly be the desired arc.

He is always clear about which formulas areexact and which formulas are approximations. Inthe modern approach, one has the usual iterativesolution to the integral equation θ−K ∗θ+K ∗K ∗θ−··· , and you can work out each term here using

the indefinite integrals y

0 xndx = y n+1

n+1 resulting

in the power series for sine. Jyes. t.hadeva doesthe same thing in finite difference form, startingwith sin(θ) ≈ θ, and recursively improving theestimate for sine by resubstitution into the left-hand side of the above identity. Instead of the

integral of powers, he needs the approximate sumof powers, i.e.,

nk=1 kp = kp+1

p+1 + O(kp), and he

has evaluated these earlier (in fact, results likethis go way back in Indian mathematics). Thenrepeated resubstitution gives the usual power

series θ − θ3

6 + θ5

120 − · · · for sine. I consider this

argument to be completely correct, but I am awarethat itis not a rigorous proof by modern standards.It can, however, be converted into such a proof byanyone with basic familiarity with ǫ, δ-techniques.I hope I have whetted your appetite enough so youwill want to feast on the riches laid out in Plofker’s

book and the Gan. ita-Yukti- Bh¯ as . ¯ a itself.

It is very tempting to read the history of math-ematics as a long evolution toward the presentstate of deep knowledge. I see nothing wrong withunderstanding the older discoveries in the lightof what we know now—like a contemporary met-allurgist analyzing ancient swords. Needham, thegreat scholar of Chinese science, wrote “To writethe History of Science we have to take modernscience as the yardstick—that is the only thingwe can do—but modern science will change andthe end is not yet.” Nevertheless, it is much moresatisfying, when reading ancient works, to know as




much as possible about the society in which thesemathematicians worked, to know what mathemat-ics was used for in their society, and how theythemselves lived.

Chapter 1 in Plofker’s book is an extensiveintroduction that gives vital background on thehistory and traditions from which all the Indianwork sprang. A series of extremely helpful appen-dices provide basic facts about Sanskrit, a glossaryof Sanskrit terms, and a list of the most signifi-cant Indian mathematicians with whatever basicfacts about them are known (often distressinglylittle). In places she gives some literal translationssuch as those of numbers in the colorful concretenumber system that uses a standard list of setswith well-known cardinalities, e.g., “In a kalpa , therevolutions 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 thedigits described backwards starting from the one’splace. It is becoming more and more recognized

that, for a good understanding of ancient writings,one needs both an extremely literal translationand one paraphrased so as to be clear in modernterms. In Sanskrit a literal translation of the com-plex compound words sometimes gives additionalinsight into the author’s understanding (as well asthe ambiguities of the text), so one wishes therewere more places in this book where Plofker gavesuch literal translations without using modernexpressions.

I have not touched on the astronomical side of the story. Suffice it to say that almost all treatisesfrom the sixth century on deal with both astron-omy and mathematics. To follow these, one needsa bit of a primer in geocentric astronomy, a van-ishing specialty these days, and Plofker providesa very handy introduction in section 4.1. Just asin Indian mathematics, there is a steady increasein sophistication over the centuries, culminatingin dramatic advances in Kerala. Most strikingly,Nilakan. t.ha in the fifteenth century proposed amodel in which the planets were moving in ec-centric and inclined circles with respect to themean sun moving in its ecliptic orbit—a “virtu-ally” heliocentric model remarkably better thanPtolemy’s.

It is high time that the full story of Indian math-

ematics from Vedic times through 1600 becamegenerally known. I am not minimizing the geniusof the Greeks and their wonderful invention of pure mathematics, but other peoples have beendoing math in different ways, and they have oftenattained the same goals independently. Rigorousmathematics in the Greek style should not be seenas the only way to gain mathematical knowledge.In India, where concrete applications were neverfar from theory, justifications were more informaland mostly verbal rather than written. One shouldalso recall that the European Enlightenment was

an orgy of correct and important but semirigorousmath in which Greek ideals were forgotten. Therecent episodes with deep mathematics flowingfrom quantum field and string theory teach us thesame lesson: that the muse of mathematics can

be wooed in many different ways and her secretsteased out of her. And so they were in India: readthis book to learn more of this wonderful story!

390 Notices of the AMS Volume 57, Number 3

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