Review of Kim Plofker’s review Mathematics in

Plofker offers a coherent historical account of mathematics in India, found in , starting from the , until the seventeenth century. She carefully adheres to documentary historical evidence, and points to controversies on dating some of the literature. The book is very well written and provides a resource that mathematics teachers in India can greatly benefit from.

R. RAMANUJAM

I. Between ignorance and pride As a child growing up in Tiruchirappalli, I used to marvel at the Rājagōpuram, the majestic tower that welcomed us into the Srirangam temple. I recall my teacher in school asking us to think about how we might determine the height of the tower. He had the hidden agenda (which we easily saw through) of discussing trigonometry. While I used to admire the thousands of sculptures, the music played daily, and the beauty of the Tamil and Sanskrit verses we heard, in that and other temples around me, it never occurred to me to ask how mathematics might have been used in the construction of the temples.

Indeed, I saw Sanskrit and Tamil as classical languages that I was trained in, with great literature in them. If someone had

mathematical knowledge, I would not have believed any of it. Nobodytold me thatdid tellthis me literature either. included significant scientific and

82 At Right Angles | Vol. 3, No. 3, Nov 2014 On the other hand, I had been raised on a staple have no idea whether did use much diet of ‘modern’ mathematics that apparently . had Greek origins, but was mostly developed in This is perhaps an appropriate place to also remark Europe in the last six hundred years or so, spiced on another brand of pride in , with mysterious references to a ‘glorious past’ in which tends to be altogether jingoistic. Typically, India. Ancient India had come up with the number this covers ‘ancient’ Indian mathematics with gold zero (presumably because of its philosophical and glory, attributing all kinds of mathematical predilections) and thus the positional number knowledge to wise folk ‘thousands of years ago’. system, and thus was really responsible for all Indeed, ‘Vedic mathematics’ is touted these days as mathematical knowledge in an essential sense. a cure for all mathematical maladies among school Indian mathematicians had been great in algebra students. This brand tends to be nationalistic and (though I had no clue in what way), but seriously ahistorical, often reveling in false pride. speaking, I thought, India contributed to the foundation and little else, Europeans actually most welcome, but also needed in that it serves names like Aryabha a, Bh skara and Brahmagupta, toIn providethis scenario, a much Kim needed Plofker’s cultural book and is not historic only 2 R. RAMANUJAM butbuilt little the edificeof their that actual I came work. to The love. mathematics I knew famous I perspective that is so clearly lacking in many of us. 2 R. RAMANUJAM ṭ ā studied, especiallyand thusat the the higher positional secondary number level system,Reading and thus it, one was embarksreally responsible on a fascinating for alljourneymathemat- and thus the positional number system, and thus was really responsible for allormathemat- at University,ical was knowledge clearly ‘modern’. in an I essentialcould not sense.into Indian India, mathematician over centuries,s and had visits been beautiful greatin algebra conceive of Indian scholars, writing in Indian mathematical landscapes. It is but natural to feel ical knowledge in an essential sense. Indian mathematicians had been great in algebra (though I had no clue in what way), but seriously speaking, I thought, India contributed languages, producing any such mathematics. a sense of pride in the achievements of ancestors (though I had no clue in what way), but seriously speaking, I thought, India contributed to the foundation and little else, Europeansfrom actuallyone’s own built geographicalthe edifice regions; that Ithe came pleasure to love. I Why, you may wonder, this public confession of to the foundation and little else, Europeans actually built the edifice that I came to love. I knew famous names like , Bhisaskara¯ much deeper and Brahmagupta, when we understand but little which of their actual such appalling historical ignorance: only because . ¯ knew famous names like Aryabhat.a, Bhaskara and Brahmagupta, but little ofI their later actualdiscoveredwork. that The scores mathematics of teachers Iof studied, especially at the higher secondary level or at University, them in their own historical context. work. The mathematics I studied, especially at the higher secondary level or atmathematics University, thatwas I clearly interacted ‘modern’. with were I couldsimilarly not conceivespecific of Indian achievements scholar wes, writingare admiring, in Indian placing languages, was clearly ‘modern’. I could not conceive of Indian scholars, writing in Indian languages, producing any such mathematics. Mathematics in India was published producing any such mathematics. in mathematics is usually not considered a in 2008, and has already achieved prominence as afflicted. A history of the development of ideas Kim Plofker’s prerequisite for teachingWhy, you the maysubject, wonder, and cursory this public confessionan authoritative of suchtext on appallin its subject.g historical However, ignorance:it does Why, you may wonder, this public confession of such appalling historicalacquaintance ignorance: onlyprovided because in textbooks I later is discovered of little that scoresnot yet of seem teachers to be as of well ma knownthematics among that Indian I interacted help. University mathematics is studded with X’s teachers of mathematics as it deserves to be. only because I later discovered that scores of teachers of mathematics that I interacted theorem and Y’swith formula, were where similarly X and afflicted. Y are almost A history of the development of ideas in mathematics is with were similarly afflicted. A history of the development of ideas in mathematicsnever Mishra is usuallyor Chung. not Popular considered books like a prerequisite E. T. forII. The teaching Content the subj ect, and cursory acquaintance usually not considered a prerequisite for teaching the subject, and cursory acquaintanceBell’s Men of Mathematicsprovided in reinforce textbooks this is view, of little so it help. University mathematics is studded with X’s theorem provided in textbooks is of little help. University mathematics is studded with X’sis not theorem surprising.and (Comment Y’s formula,. Whether where mathematics X and Y are almostthe entire never Indian Mishra subcontinent, or Chung and. Popularit is important books like Throughout, Plofker uses the term India to denote and Y’s formula, where X and Y are almost never Mishra or Chung. Popularteachers books like needE. a historical T. Bell’s perspectiveMen of Mathematics to teach reinforce this view, so it is not surprising. (Comment. mathematics well is a relevant question, but it will the banks of the river Indus, a region that is in the E. T. Bell’s Men of Mathematics reinforce this view, so it is not surprising. (Comment. Whether mathematics teachers need a historicalto note this, perspective since early Vedic to teach culture mathematics flourished on well is Whether mathematics teachers need a historical perspective to teach mathematics well is a relevant question, but it will take us too far afield to discuss it here.) takeFor people us too offar this afield history-deprived to discuss it here.) ilk, let me just a relevant question, but it will take us too far afield to discuss it here.) descriptionconfluence of of modern the book’s day contents. , For people of this history-deprived ilk,and India. let me I will just first mention outlinetwo a chapter pearls by from chapter Plofker’s For people of this history-deprived ilk, let me just mention two pearls frommention Plofker’s two pearlsaccount. from One Plofker’s is Madhava’s account. One early is 15-th century discovery of infinite series for the sine, series for the sine, cosine and arctangent functions. that any historian of mathematics faces in the Madhava’s early 15th century discovery of infinite π account. One is Madhava’s early 15-th century discovery of infinite series forNote the that sine, this cosinewas 200 and years arctangent before Gregory’s functions. Note thatThe thisbook was opens 200 with year a sdiscussion before Gregory’s of the difficulties series ( 4 = π 1 1 cosine and arctangent functions. Note that this was 200 years before Gregory’sseries ( = 1 + ) andand Newton’s Newton’s .accounts The other and is claims, Brahmagupta’s and intermingling 7-th cen of turylegends theorem 4 − 3 5 −··· 1 1 calculus. The other is Brahmagupta’s 7th century andIndian factual tradition: records. lack The of documentation, author offers a stunning conflicting 1 3 + 5 ) and Newton’s calculus. The other is Brahmagupta’s 7-th century theorem on the area of a quadrilateral inscribed in a circle. This can be proved easily but involves − −··· theorem on the area of a quadrilateral inscribed on the area of a quadrilateral inscribed in a circle. This can be proved easily but involves significant algebraic manipulation. However,instance we (in havethe Preface, no ide pagea whether vii) of two Brahmagupta reputed did in a circle. This can be proved easily but involves authors, appearing side by side in the same significant algebraic manipulation. However, we have no idea whether Brahmagupta did use much algebra. volume, attributing the emergence of quantitative use much algebra. significant algebraicThis manipulation. is perhaps an However, appropriate we place to also remark on another brand of pride in Indian

This is perhaps an appropriate place to also remark on another brand of pride in Indian Mathematics, which tends to be altogether jingoistic.At Typi Rightcally, Angles this| Vol. covers 3, No. 3, ‘ancient’Nov 2014 In-83 Mathematics, which tends to be altogether jingoistic. Typically, this covers ‘ancient’ In- dian mathematics with gold and glory, attributing all kinds of mathematical knowledge dian mathematics with gold and glory, attributing all kinds of mathematical knowledge to wise folk ‘thousands of years ago’. Indeed, ‘Vedic mathematics’ is touted these days to wise folk ‘thousands of years ago’. Indeed, ‘Vedic mathematics’ is touted these days as a cure for all mathematical maladies among school students. This brand tends to be as a cure for all mathematical maladies among school students. This brand tends to be nationalistic and ahistorical, often reveling in false pride. nationalistic and ahistorical, often reveling in false pride. astronomy in Vedic India to periods that differ by considerable amusement to readers. The use of as much as two thousand years! The author goes linear interpolation techniques is the highlight on to give a brief account of historiography in this of this chapter. In the calculation of sines, Indian context, and the form of Sanskrit literature in which astronomers recorded only 24 values in steps of knowledge was expressed. 3.75 degrees, which made for easy memorization, and the rest were calculated by interpolation. Chapter 2 on Vedic India, discusses mathematical The fascinating method of three dimensional thought in the Vedas, which were canonical texts projections, using right angles inside the sphere, is (Note for readers: ‘BCE’ stands for “Before the Common by the middle of the first millennium BCE. Era” and is used interchangeably with ‘BC’. elucidatedThis chapter nicely illustrates by Plofker. an important difference Similarly, ‘CE’ has the same meaning as ‘AD’.) The between Greek and Indian mathematicians. The latter were primarily applied mathematicians, in rituals and the geometry in different sizes and who were principally interested in providing cosmic significance of numbers andŚ ūarithmeticlba-sūtras , the rules of cords (or ropes), the earliest of which short tables and learning techniques for generating wereshapes composed of fire altars by Baudhayana, are discussed. offered many longeralgorithms ones and shaped their the justifications. way they approached Memorizing constructions in plane geometry, for instance, ways problems. Moreover, they were not committed to of transforming rectangles into squares or circles. any particular geometric model for astronomical The sūtras included a statement of the theorem phenomena, which allowed them wide we usually attribute to Pythagoras, and reasonably experimentation. good approximations to 2 and π The ‘medieval period’, with its connotation of controversies relating to mathematical ideas in ‘dark ages’ is considered one of intellectual Vedic astronomy and astrology.√ What. Plofker is striking discusses stagnation in European accounts. Chapters 5 is the systematization of large numbers, an and 6, ranging through the work of Āryabhata, elaborate system that is not positional but includes Bhāskara I, Mahāvīra, Bhāskara II (also known as factorization. Bh skarach rya) and others, in a period from the The next chapter takes us into the centuries just sixth to the twelfth centuries CE, show fascinating before and just after the turn of the Common Era. mathematicalā ā development. It is no exaggeration Importantly, it discusses the work of Pānini to say that mathematics emerges as a discipline of century BCE) and Pińgala (third century BCE), in some kind in this period. Mah v ra’s ninth century shaping Sanskrit grammar and the mathematical (fifth work Gaṇita-sāra-sańgraha is a text devoted solely ideas contained therein. The chapter contains an to mathematics. The author summarizesā ī verses attractive analysis of metric structure in poetry and and offers an account of trigonometry and also its relation to binary representations. Important arithmetic and (extensive) algebra, in modern ideas like the use of rewrite rules and recursion notation. (central to modern computer science) inform Every single section of these chapters (including P ninian grammar. The chapter also discusses l vati, trigonometry and controversy related to whether B ja-ga ita, the work of N r yana Pandita) is worth theseā were Greek transmissions (since Alexander’s readingexpositions in detail. of the As work an appetizer, of Mahāvīra let above,me mention Lī ā invasion brought such contact). theī seventhṇ century developmentā ā of the arithmetic It is hard to separate mathematics and astronomy in Sanskrit texts and Chapter 4 takes up this would not appear in Europe until a thousand linkage. The structure and content of siddhāntas yearsof negative later. Anothernumbers is by Brahmagupta’s Bhāskarachārya, formula which for is elaborated. Āryabhatīyam, dated to 499 the area of a quadrilateral inscribed in a circle. CE, is a masterly treatise giving an analysis A brilliant account is the work on ‘Pell’ equation of planetary motion based on epicycles. Its by Brahmagupta: x2 Ny2 = c, showing that notation is most intriguing, and should provide solutions for c and c can be ‘multiplied’ to give 1 −2

84 At Right Angles | Vol. 3, No. 3, Nov 2014 6 R. RAMANUJAM

India had extensive contacts with the Islamic world, and Islamic scholars were the prin- cipal intermediaries of transmission between India and Europe for much of the last millen- nium. Chapter 8 discusses this interaction, raising intriguing questions about why certain theories and methods were absorbed into Indian mathematics and not others. However, the discussion is too brief to get a clear understanding of the issue involved. The book then concludes in Chapter 9 with what we may call the colonial encounter. Direct relations with European culture and education brought both mutual interest and mistrust, shaped by one for c c 1 2 powerIII. Some equations examples and imperialist attitudes to bringing India ‘out of the dark ages’. sines) developed led to a full theory of a calculus It would be disappointing to talk about a book that 6 . The method of finite differencesR. (for RAMANUJAM of polynomials, and for pretty much everything presents mathematics and not talk mathematics at neededIndia to solve had problems extensive related contacts to spheres with the and Islamic world, and Islamic scholars were theIII. prin- SOME EXAMPLES circlescipal in astronomy. intermediaries Bh of transmission between India andnot only Europe the forrichness much of of content, the last but millen- also the of thenium. surface Chapter of the 8sphere discusses using this latitudes interaction, to raisingall. intrig Letuing me present questions a few about examples why certain that illustrate āskarachārya’s division It would be disappointing to talk about a book that presents mathematics and not talk calculatetheories area and and methods volume, werecan only absorbed be termed into as Indian mathematicsnotation alongand notwith others. a translation However, of the the original sheer brilliance. mathematicstrouble taken at all.by Plofker Let me to present present a it few in modern examples that illustrate not only the richness of discussion is too brief to get a clear understandingverses. of the issue involved. The book then content, but also the trouble taken by Plofker to present it in modern notation along with a concludes in Chapter 9 with what we may call the colonial encounter. Direct relations translationIII.1. Transformations of the original verses.of squares and calculationwith European and symbolic culture manipulation and education but broughtalso both mutual interest and mistrust, shaped by This is from the Vedic period, and lets theIn these role ofchapters, proof in Plofker the treatises, discusses the notmathematical only rectangles. power equations and imperialist attitudes to bringingus Indicalculatea ‘out the of side the dark of a square ages’. whose area equals culture, and its relation to society. Problem solving III.1. Transformations of squares and rectangles. This is from the Vedic period, and the sum of, or difference between, the areas of two for various commercial applications and puzzle letsgiven us calculate squares. theConsider side of Figure a square 1(a) whose where area we are equals the sum of, or difference between, solving for amusement are illustrated. YetIII. again SOME we EXAMPLES thegiven areas squares of two ABCD given and squares. EFGH Consider, and KA = FigureLD = FE 1(a) where we are given squares ABCD see the primacy of problem posing and solving, as and= GHEFGH. Then, and LA KAis the= sideLD = ofFE the =‘sum’GH square,. Then LAandis the side of the ‘sum’ square, and MD opposed to theory building, but also rootedness in It would be disappointing to talk about a book(obtained thatMD (obtained presents by locating m byathematics locating point pointM andon notMAD on talk suchAD such that thatLM = LK) is the side of the ‘difference’ social practice. mathematics at all. Let me present a few examplessquare.LM that = LK illust) israte the notside only of the the ‘difference’ richness of square. Chaptercontent, 7 presents but also the the most trouble precious taken jewel by Plofker of to present it in modern notation along with a K A Indiantranslation mathematics, of the namely original the verses. work of the Kerala B A B E school, from the 14th to the 17th centuries, which F E includes the discrete fundamental theorem of M III.1. Transformations of squares and rectangles. This is from the Vedic period, and the founderlets us calculate of the school, the side is one of of a squarethe great whose area equals the sum of, or difference between, H G calculus. Mādhava (approximately 1350 to 1425), mathematiciansthe areas of of two that given time. squares. The school Consider provided Figure 1(a) where we are given squares ABCD powerand seriesEFGH expansions, and KA of= sine,LD = cosineFE = andGH . Then LA is theC sideL of the ‘sum’ square,D G and MD H C D F arctangent. N laka ha’s (obtained by locating point M on AD such that LM = LK) is the side of the ‘difference’ of planets moving in eccentric circles is another (a) (b) square. major achievementī ṅ ṫof this(fifteenth school. The century) explanations model and rationale they provide is also methodologically FIGURE 1. Adapted from Figure 2.3, page 22, of Kim Plofker’s book B K A B A E interesting. F E India had extensive contacts withM the Islamic world, and Islamic scholars were the principal intermediaries of transmission between India and H G Europe for much of the last millennium. Chapter 8 discussesC this interaction,L raisingD intriguingG H C D F questions about why certain theories and methods were absorbed into Indian mathematics(a) and not (b) others. However, the discussion is too brief to get Figure 1. Adapted from Figure 2.3, page 22, a clear understandingFIGURE of1. the Adapted issues involved. from Figure The 2.3, page 22, of Kim Plofker’s book of Kim Plofker’s book book then concludes in Chapter 9 with what we may call the colonial encounter. Direct relations with European culture and education brought both To transform a given square into a rectangle with mutual interest and mistrust, shaped by power a given side and having the same area, consider equations and imperialist attitudes to bringing the illustration in Figure 1(b). Given the square India ‘out of the dark ages’. ABCD, expand its side to the given length BE, forming the rectangle BEFC. Its diagonal BF BEGH (with GH passing

defines the rectangle At Right Angles | Vol. 3, No. 3, Nov 2014 85 8 R. RAMANUJAM ¯ III.3. Aryabhat.a. Consider the right triangles in Figure 2(a). It is easy to see by similarity d a s of triangles that −s = a. Now consider the two circles in Figure 2(b) with diameters d1 2 8 and d , andR. RAMANUJAM let a = a + a . We thus have a (d a )=s = a (d a ). Now, with 2 1 2 1 · 1 − 1 2 · 2 − 2 KIM PLOFKER’S MATHEMATICSsome easy IN INDIA manipulation, we get a rule7 given by Aryabhat¯ a: III.3. Aryabhat¯ a. Consider the right triangles in Figure 2(a). It is easy to see by similarity. through the pointTo transform of intersection a given square of AD. into and a BF rectangle), with a given side and having the same of triangles that d a = s . Now consider the two circles in Figure 2(b) with diameters d whose areaarea, equals consider that theof ABCD. illustration in Figure−s 1(b).a Given the square ABCD, expand(d its2 sidea) toa (d11 a) a a1 = −2 · , a2 = − · and d2, and let a = 8a1 + a2. We thus have a1 (d1 (adR.1)= RAMANUJAMa)+(s =da22 (ad)2 a2). Now,(d1 witha)+(d2 a) the given Consider length BE ,values forming for the the rectangle sine BEFC. Its diagonal BF· defines− the− rectangle−· − − − III.2. Sine values. some easy manipulation, we get a rule given by Aryabhat¯ a: BEGH (with GH passing through the point¯ of intersection of AD and BF), whose. area function from Var hamihira’s PańcasiddhāIII.3.ntikāAryabhat, .a. Consider the right triangles in Figure 2(a). It is easy to see by similarity equals that of ABCD. of triangles that d a = s . Now consider the two circles in Figure 2(b) with diameters d around sixth century CE. 8 (d2III.4.a) Brahmagupta.−as a ConsiderR. RAMANUJAM(d1 quadrilaterala) a ABCD inscribed in a circle1 as in Figure 3, ā a1 =and d , and− let a·= a +,a . Wea2 thus= have a −(d · a )=s2 = a (d a ). Now, with (d12 ¯ a)+(d2 1a) 2 (d1 a1)+(1d2 1 a) 2 2 2 R be the radius of a circle. Consider an arcIII.3. of −Aryabhatwith diagonals.a. −ConsiderAC theand rightBD trianglesintersecting− in· Figure¯ − − 2(a). at H Itat is righteasy· to angles. see−by similarity Let GHJ be the line through III.2. Sine values. Consider valuessome for the easy sine manipulation, functiond a froms we getVarahamihira’s¯ a rule givenPa´nca- by Aryabhat.a: the circle which subtends an angle θ at the centreof triangles H, that perpendicular−s = a. Now to considerCD. Then the two Brahmagupta’s circles in Figure theorem 2(b) with states diameters that dJ1is the midpoint of AB. siddhantik¯ a¯, around sixth century CE. 2 KIM PLOFKER’SLet MATHEMATICS IN INDIA and d27, and let III.4.a = a1 Brahmagupta.+ a2(.d We thusa) a have Considera1 (d1 quadrilaterala1)=s(d= a2a)( ABCDda2 a2) . Now, with θ refer to the length of the For proof,a = let BE2 and AF be, the· perpendicularsa−= 1 from· −B and A to CD. Then Brahmagupta III.4. Brahmagupta.someConsider easy manipulation, quadrilateralinscribed1 we in get ABCDa− circle a rule· inscribed givenas in by FigureAryabhat¯ in2 a3, circle witha: diagonals as− in· Figure 3, To transform a given square intochord a rectangle correspondingLet withR be a thegiven to radiusthis side arc ofand a(so circle. having it is Considera the length samean arcargues of the that circle(d1 whicha)+( subtendsd2 a) an angle θ (d. 1 a)+(d2 a) of the circle. Let Sin with diagonals AC and BD intersectingAC and at BDH−at intersecting right angles.− at LetH GHJ be− the line− through area, consider the illustration in Figurerather 1(b). thanat Given a the ratio). centre the sqThen: ofuare theABCD circle., expandLet Sinθ itsrefer side to to the length of the chord(d2 correspondinga) a to this (d1 a) a H, perpendicular to CD. Then Brahmagupta’sGHJa1 = be the line theorem− through· states, H, thataperpendicular2 =(BEJ is theHG− midpoint)+( ·toAF ofHGAB). the given length BE, forming the rectangle BEFCarc (so. Its it is diagonal a lengthBF ratherdefines than thea ratio). rectangle Then: (d1 a)+(d2 a) JH = (d1− a)+(d2 a) − , • Sin 30° = √R2/4; Sin 45° = √R2/2; III.4. Brahmagupta. Consider− quadrilateral− ABCD at rightinscribed− angles.2 in− Let a circle as in Figure 3, BEGH (with GH passing through the point of intersectionFor of AD proof,and letBFBE), whoseand AF areabe the perpendicularsCD. Then Brahmagupta’s from B and theoremA to CD. states Then Brahmaguptathat J 2 2 with diagonals AC and BD intersecting at H at right angles. Let GHJ be the line through Sin 60° = √R −- Sin 30°; 2 2 and fromis thethis midpoint2 we2 see of thatAB. For the proof, ‘height’ let of BEJ andis halfway AF between those of B and C, i.e., equals that of ABCD. Sin30◦argues= R that/4; Sin45◦ = R /2; Sin60◦ = √R Sin 30◦; • HIII.4., perpendicularBrahmagupta. to CDConsider. Then− Brahmagupta’s quadrilateral ABCD theoreminscribed states in that a circleJ is the as midpoint in Figure of 3, AB. 2 2 JG =(beBE the+ perpendicularsAF)/2. from B and A to CD. Then • Sin (90° -−θ) = √SinR(−Sin90 θθ; )=√R2 Sin 2θ; (BE HG)+(AF HG) • ◦ − − Forwith proof, diagonalsJH let=BEACBrahmaguptaandand−AFBDbeintersecting the argues perpendiculars− at thatH at, right from angles.B and LetAGHJto CDbe. Thenthe line Brahmagupta through III.2. Sine values. Consider values• Sin for θthe = √ sine60(R function Sin- Sin(90θ = from° 602( Var R· θahamihira’s¯)).Sin(90 argues2Pa´nca-Hθ,)) perpendicular. that to CD. Then2 Brahmagupta’s theorem states that J is the midpoint of AB. • − ◦ − · siddhantik¯ a¯, around sixth century CE. For proof, let BE and AF be the perpendiculars fromB B and A to CD. Then Brahmagupta Commentand. The from third this rule we issee accurate that only the when‘height’R/2 of= 60;J is the halfway verse(BE itself betweenHG offers)+(AF a thoseHG of)J B and C,A i.e., Comment. The third rule is− accurate only whenargues that JH = − − , roughJG approximation=(BE + AF of D)/=2. (360)2/10. 2 Let R be the radius of a circle.R Consider/2 = 60; anthe arc verse of the itself circle offers which a s ubtendsrough an angle θ (BE HG)+(AF HG) 2 and from this we see that theJH = ‘height’− of J is halfway− between, those of B and C, i.e., at the centre of the circle. Let Sinapproximationθ refer to the length of D of =√ the(360) chord/10 corresponding. to this and from this we see that the2 ‘height’ of J is arc (so it is a length rather than a ratio). Then: JG =(BE + AF)/2. H and from this wehalfwayB see that between theJ ‘height’ those ofA J ofis B halfway and A, between those of B and C, i.e., JG =(BE + AFi.e.,)/2. JG = (BE + AF)/2. 2 2 2 2 Sin30 = R /4; Sin45 = R /2; Sin60 = √R Sin 30 ; B J • ◦ ◦ ◦ − ◦ A s s Sin(90 θ)=√R2 Sin 2θ; B • ◦ − − H J A Sinθ = 60(R Sin(90 2 θ)). a d a a1 a2 D • − ◦ − · − H F G Comment. The third rule is accurate only when R/2 = 60; the verse itself offers a H E rough approximation of D = (360)2/10. C Diameter d2 D DiameterF d1 G D (a) (b)E F G D C FIGURE 3. AdaptedE from FigureF 5.6, page 146, of Kim Plofker’s book FIGURE 2. Adapted from Figure 5.3, page 131, of KimC Plofker’s book G s s E C a d a a1 a2 − IGURE F 3. AdaptedFIGURE from Figure3.Figure Adapted 5.6, 3. fromAdaptedpage Figure 146, from of 5.6, KimFigure page Plofker’s 146,5.6, page of Kim book 146, Plofker’s book FIGURE 3. Adapted fromof Figure Kim Plofker’s 5.6, page book 146, of Kim Plofker’s book

Diameter d2

Diameter d1 III.5. Bhāskara II. (a) (b) two from the many beautiful results of Bh skara II Figure 2. Adapted from Figure 5.3, page 131, (from the 12th century). It is difficult to pick only one or FIGURE 2. Adapted from Figure 5.3, pageof 131, Kim of Plofker’s Kim Plofker’s book book 8 R. RAMANUJAM ā . Consider the right triangles Here is one from the Līlāvatī: suppose that we III.3. Āryabhaṭa 2 2 III.3.in FigureAryabhat¯ 2(a).a. It isConsider easy to see the by right similarity triangles of in Figure 2(a). It is easy to seex, byy such similarity that x ± y − 1 = . 2 triangles that d a s Now consider the two z for z an integer, or fraction. Assume a number of triangles that −s = a. Now consider the two circleswish in Figureto find (rational) 2(b)4 with diameters3 d1 circles in Figure 2(b) with diameters d and d , and a and let2 x = 8a + 1, y = 8a . Then the right and d2, and let a = a1 + a2. We thus have1 a12 (d1 a1)=s = a2 (d2 a24). Now,4 2 with let a = a + a . We thus have a · (d a ) = s2 = a · · − hand side becomes· − 16a (4a ± 4a + 1), which is 1 2 1 1 1 2 ¯ some easy manipulation, we get a rule given by Aryabhata perfect.a: square. (This is from verses 56–72 of (d2 a2). Now, with some easy manipulation, we get − Līlāvatī, discussed on pages 186–187 a rule given by ryabha a.(d a) a (d a) a − a = 2 − · , a = book.)1 − · 1 (d a)+(d a) 2 (d a)+(d a) of Plofker’s Ā 1ṭ − 2 − 1 − 2 − 86 At Right Angles | Vol. 3, No. 3, Nov 2014 III.4. Brahmagupta. Consider quadrilateral ABCD inscribed in a circle as in Figure 3, with diagonals AC and BD intersecting at H at right angles. Let GHJ be the line through H, perpendicular to CD. Then Brahmagupta’s theorem states that J is the midpoint of AB. For proof, let BE and AF be the perpendiculars from B and A to CD. Then Brahmagupta argues that (BE HG)+(AF HG) JH = − − , 2 and from this we see that the ‘height’ of J is halfway between those of B and C, i.e., JG =(BE + AF)/2.

B J A

H

D F G E C

FIGURE 3. Adapted from Figure 5.6, page 146, of Kim Plofker’s book KIM PLOFKER’S MATHEMATICS10 IN INDIA R. RAMANUJAM9

The Madhava¯ – Leibniz series for π: III.5. Bhaskara¯ II. It is difficult to pick only one or two• from the many beautiful results KIMKIM PLOFKER’S PLOFKER’SMATHEMATICSMATHEMATICS IN IN INDIA INDIA 4D 4D 4D 99 n 1 4D n 4Dn of Bhaskara¯ II (from the 12-th century). C + ...+( 1) − +( 1) ≈ 1 − 3 5 − 2n 1 − (2n)2 + 1 III.5.III.5. BhBhaskara¯askara¯ II. II. ItIt is is difficult difficultKIM to PLOFKER’Sto pick pick only onlyMATHEMATICS one one or or two two from from IN theINDIA the many many beautiful beautiful r resultsesults 9 − KIM PLOFKER’S MATHEMATICS IN INDIA The9 Madhava¯ – Gregory series for the arctangent: Hereofof Bh Bhaskara¯askara¯ is one II II from (from (from the the the 12-thL 12-thil¯ avat¯ century). century).i¯: suppose that we wish• to find (rational) x,y such that 3 5 2 III.5.2 Bhaskara¯ 2 II. It is difficult to pick only one or two from the manyRSin beautifulθ R4Sin resultsθ RSin θ III.5. Bhaskara¯ II. It is difficultx y to pick1 = onlyz for onez oran two integer, from the or many fraction. beautiful Assume results a number a andθ = let x = 8a + 1, + ... 1Cosθ − 3Cos 3θ 5Cos 5θ − of Bhaskara¯ II (from the 12-thof± century).Here BhHere3−askara¯ is is one one II from from (from the the theLLil¯il¯ 12-thavat¯avat¯ i¯i¯:: century). suppose suppose that that we4 we wish wish4 to to2 find find (rational) (rational) xx,,yy suchsuch that that y = 8a . Then the right hand side becomes 16a (4a 4a(Here+1 we), assumewhichthat is a Sin perfectθ < Cos square.θ.) xx22 yy22 11==zz22 forfor zz anan integer, integer, or or fraction. fraction. Assume Assume± a a number number aa andand let let xx==88aa44++1,1, (This±± is from−− verses 56–72 of Lil¯ avat¯ i¯, discussed on pages 186–187 of Plofker’s book.) Here is one from the Lil¯ avat¯yy==Herei¯8:8aa suppose33..is Then ThenHere one thethe thatis from one right right we from the hand wish hand Lthe sideil¯ tosideavat ¯B findīja-ga becomes becomesi¯: (rational) supposeṇita. 16 16Supposeaa44x((, that44yaa4such4 we we44 aa that22 wish++The11)),, M whichto whichadhava¯ find is is – (rational) a a Newton perfect perfect series square. square.x,y forsuch the Sine: that 2 2 • 2 2 2 2 2 want to2 solve the equation Nx + 1 = y . When±±4 3 4 5 7 x y 1 = z for z an integer,x(This(Thisy oris is from fromfraction.1 = verses versesz Assumefor 56–72 56–72z an ofa of integer, numberLLil¯il¯avat¯avat¯ i¯i¯, ora, discussed discussedand fraction. let x on on= Assumepages8 pagesa + 186–1871, 186–187 a number of of Plofker’s Plofker’sa and book.) book.) letθ2x = 8a θ+2 1, θ ± − Here is one from the Bija-gan¯ ita. Suppose we want to solve the• SinThe equationθ Mādhava= θ Nx - Newton+ 1 = seriesy . + for the Sine:... y = 8a3. Then the right hand side± becomes3−we already 16a4(4a know4 4a a2 solution+.1), which a, b isfor a perfectthe ‘auxiliary’4 square.4 2 − R2 3! − R4 5! R6 7! −  y = 8a . Then the right2 ± hand2 side becomes 16a (4a 4a +1), which2 is a2 perfect· square.· · WhenHere¯Here we isalready is¯ equation one one from from know Na the the a+BB ija-gan¯solutionija-gank¯ = b foritaitaa .some., Suppose Supposeb for k the we we ‘auxiliary’ want want to to solve solve± equation the the equation equationNa +NxNxk =22++b11=for=yy2 some2.. k, (This is from verses 56–72 of(ThisLilavat¯ isi from, discussed versesKIM on 56–72PLOFKER’S pages 186–187of..MATHEMATICSLil¯ avat¯ ofi¯, Plofker’s discussed IN INDIA book.) on pages 186–187 of9 Plofker’s book.) integer m such that am+b is also an integer and 2 22 22 weWhenWhen find an we we integer already alreadym know knowsuch a a solution thatsolutionkaa,,bbisforfor also the the an ‘auxiliary’ ‘auxiliary’ integer equation and equationm NaNaIV.N++ Fillingiskk= minimized.=bb for fora huge some some (Findingk kgap,, 2 , we find an | − | IV. FILLING A HUGE GAP Here is one from theIII.5.Bija-gan¯suchBhaskara¯mitais. Suppose called II.|m It−N is the| we difficultispulverizer minimized. want to to pick solveamam technique(Finding only++ thebb one equation orsuch, twogoing mNx from is2 backcalled+ the1 = many to they2Aryabhat2¯.2 beautifula.) results wewe. find find an an integer integermmsuchsuch that that¯ kk isis also also an an integer integer and and mm NN isisThe minimized.. minimized. story of mathematics (Finding (Finding2 in India,2 dating from of Bhaskara¯ Here II (from ispulverizer one the from 12-th technique the century).Bija-gan, goingita back.2 Suppose to ryabha2 we wanta.)| | to−− solve|| the equation Nx + 1 = y . When we already know a solutionsuchsuchmam,bisisfor called called the ‘auxiliary’ the thepulverizerpulverizer equation technique technique. Na ,+, going goingk = b back backfor some to toAryabhatAryabhat¯¯ k, a.)a.)Vedic times to the 1600s, is one that needed to be The story of.. mathematics2 in India,2 dating from Vedic times to the 1600s, is one that we find an integer m such thatWhenam+b weis also2 already an integer2 2 know and2 a solutionm22 2N ais,b minimized.for the ‘auxiliary’ (Finding equation Na +k = b for some k, HereNow, is onek N from.1Now,+( the mNL.1il¯ avat¯+ N(mi)=¯: supposeNm) = trivially,m that trivially, we so wish so weĀ we to can findcanneeded ‘compose’ ṭ (rational) to bex told, thesey such in equations detail, that and Plofker to get a tells new the tale admirably. Its chief virtue is that we find an integer22 22−m such¯ that| 22 −am+| b is also an integer and m2 N is minimized. (Finding such m is called the pulverizerx2 one:y2 techniqueNow,1Now,= z2NNfor‘compose’..11, goingz+(+(anmm integer, back theseNN)=)= to or equationsAryabhatmm fraction.trivially,trivially,k.a.) to Assume get so so we wea new a can can number one: ‘compose’ ‘compose’a and these letthesex =Its equations equations8 chiefa4 + 1,virtue to to get get is athat a new new scholarly commitment −− − scholarly commitment| told− in| isdetail, never and compromised, Plofker tells and the historical tale admirably.record is carefully adhered ± 3suchone:one:− m is called the pulverizer technique4 4 , going2 back to Aryabhat¯ is nevera.) compromised, and historical record is y = 8a . Then the right hand side becomes2 162 a2 (4a 42a +1),to.2 which What is it a offers perfect2 is square. an. informed understanding of the tremendous achievements of math- 2 2 2 N(am + b) + k(m − N) = (bm + Na) . Now, N.1 +(m N)=m trivially, so we can ‘compose’N(am + theseb)22 + equationsk(±m22 toN)=( get abm new+ Na22 )carefully. adhered to. What it offers is an informed − (This is from verses 56–72 of Lil¯ avat¯NN((amami¯, discussed++bb)) ++kk on((mm pages−NN 186–187)=()=(ematiciansbmbm+ of+Na Plofker’sNa)) in.. India, book.) their methods and style. There is much to be proud of, and there is no We2 then see2 that (bm +2 Na) is also− −divisible by k, understanding of the tremendous achievements of one: We thenNow, seeN. that1 +((bmm + NaN)=) ism alsotrivially, divisible so by wek can,need and ‘compose’ to hence exaggerate we these get claims yet equations either. another to auxiliary get a new WeWe then then see seeand that that hence¯((bmbm we+−+Na Naget)) yetisis also alsoanother divisible divisible auxiliary by bykk, ,equation and and hence hence we we get get2mathematicians yet yet another another2 auxiliary auxiliary in India, their methods and style. Hereone: is one2 from2 the2 B2ija-gan2. ita. Suppose2 we want to solve the equation Nx + 1 = y . Nequation(am + b) Na+ k(+m2 k 2=N )=(b2 , wherebm + Na) . When weequationequation alreadyNaNa knowNa1 22+ + −a1kk2 solution2 == bbb12 2,, ,where wherea where,b for the ‘auxiliary’ equation NaI was2 + ak little= b2 disappointedforThere some is kmuch, at the to lackbe proud of reference of, and to there non-Sa isnskritic no need work in mathematics 11 11 111 2 2 2 am+b 2 We then see that (bm +weNa find) is an also integer divisiblem such by thatk, and henceNis( alsoam we an+ integergetb) yet+ k and another(m2 m N auxiliaryinN)=( India,is minimized.bm but+ thenNato) (Findingexaggerate historical. sources claims on either. these are very few indeed. Another source of amk + b m22 | −N− | bm + Na 2 2 2 amam++bb mm ¯ NN disappointmentbmbm++ wasNaNa the lack of any discussion on links with Chinese mathematics, another equation Na1 + k1 = bsuch1, wherem is called the pulverizeraaa111=== technique,,, goingkk1k back11=== to −Aryabhat−−,, , .a.)bb11=b=1 = I was,, a little, disappointed at the lack of reference We then see that (bm + Nakk ) is also divisiblekk k byhighlyk, and developed hencekk we onek get close yet by, another with many auxiliary Chinese travellers carrying texts between the to non-Sanskritic work in mathematics in India, am + b 2 2 2m2 2N 2 2 bm + Na Now,andequationandandNa.1a,ab1+(1,,bb,1mk1Na,,kk11are1areNare+)= integers.k integers. integers.1m=trivially,b1, where If If kk so11 is weisis equal equal can equal ‘compose’ to to to any any any of of these of thecountries. the the desiredequations desired desired Once values values to again, get values 1a 1, new, one22 1, wonders,, 4,4,2, we we what4, are are we documentary are evidence is available. a1 = , 1 1k11=− − , b1 =1 , but then±± historical±± sources on these are very few one: k k k ± ± done.done.done. Otherwise, Otherwise, Otherwise, we we we have have have come come a a a full full full circle, circle, circle, and and and2 start start start over over over ag again.ain. agindeed.ain. Choose Choose Choose Another an an integer integer an source integermm11 of mdisappointment1 was aa11mm11++bb11 am2 + b2 m 2NBut these are relativelybm + Na trivial complaints. For teachers, the book carries some important and a1,b1,k1 are integers. Ifsuchsuchk1 is that that equala1m1+ tob1 anyN(isisama alsoof also+= theb an) an desired+ integer, integer,k(m , values andN and)=(k so so 1bm=, on. on.+2, WeNa We4,) repeat repeat. we, are this thisb until until= kk11 attainsattains, one one of of the the such that andkk1 1a , isb ,also k1 are an integers. integer, If andk is1 soequal on. to− We any repeat this1 the until lackk ofattains any discussion one of theon links with Chinese k1 1 1 1 k − 1 ± ±k messages, in my opinion.k 1 One is a historical understanding of the development of key done. Otherwise, we have comedesireddesired a full values values circle, and and and we we start get get a aover solution solution again. to to ChooseNxNx22++11 an==y integery22.. This Thism is is1 called called the the ‘cyclic’ ‘cyclic’ method. method. Wedesired then see that values(ofbm the and+ Na desired we) is get also values a divisible solution 1, ±2, by ±4, tok,Nx andwe2 henceare+ 1 done.= weideas.y2 get . This yet This another is is called muchmathematics, auxiliary the needed ‘cyclic’ if another one method. is to highly see mathematics developed asone a close creative process (rather such that a1m1+b1 is also an integer, and so on. We repeat this until k attains one of the k1 and a21,b21Otherwise,,k1 2are integers. we have Ifcomek1 ais full equal1 circle, to and any start of the desiredby, with values many 1, Chinese2, 4, travellers we are carrying texts equation Na1 + k1 = b1, where than as received wisdom). Another± ± is contextualization, often claimed to be lacking in desired values and we get a solutionThereThere to is isNx way way2 + too too1 = much muchy2. This in in the isthe calledLLil¯il¯avati¯avati¯ theand ‘cyclic’and the the jyotpatti method.jyotpattitoto pick pickbetween any any one, one, the but but countries. here here is is a a Once again, one wonders done. Otherwise,over again. we Choose have an come integer a full m1 circle,such that andmathematics start over education. again. ChooseSeeing the an context integer in whichm1 many methods arose can be of great There is way too much in the L2 il¯ avati¯ and the jyotpatti to pick any one, but here is a trigonometrictrigonometrica1m identity,1 identity,+amb1+ b a a real real gem: gem:m let letRRNbebe a a radius radius ofbm of a a+ circle circleNa and andαα,what,ββ bebe documentary given given arcs. arcs. Then Then evidence is available. such that a = is also, also an ank integer,= integer,− and, and so on. sob = on.We helprepeat We in repeat this, regard. this until k1 attains one of the trigonometric¯ 1 identity,k1 a real gem:1 let R be a radius1 of a circle and , be given arcs. Then There is way too much in the Lilavati¯ thisand until thek k jyotpatti attains oneto pickofk the any desired one, butvalues herek and is a α β desired values and we1 get a solutionSinSin toααNxCosCos2 +ββ1 =SinSiny2ββ.CosCos Thisαα isBut called these the are ‘cyclic’ relatively method. trivial complaints. For trigonometric identity,and a reala , gem:b ,k letareRwe integers.be aget radius a solution If k ofSinSin ais circle equal (to(αα Nx and toβ2β +)=)= any1α =,β ofy2be. theThis given desired is called arcs.±± values Then the 1, 2.., 4, we are 1 1 1 1 ±± Sinα CosRβR Sinβ Cos± ±αteachers, the book carries some important done. Otherwise,‘cyclic’ we have method. comeSin a full(α circle,β)= and start over again.± Choose an integer. m Sinα Cosβ Sinβ Cosα ¯ messages,1 in my opinion. One is a historical Therea1m1+b1 is way too much in± the Lilavati¯ and theRjyotpatti to pick any one, but here is a suchSin that(α β)= is also an integer,± and so on.. We repeat this until k1 attains one of the ± k1 There is wayR too much in the Līlāvatī and understanding of the development of key ideas. desiredtrigonometricIII.6.III.6. valuesMM andadhava.¯adhava.¯ we get identity, aBelow,Below, solution a let let real toCCNxdenote gem:denote2 + 1 let= the theyR2 circumference. circumference Thisbe a is radius called of the of of aa ‘cyclic’ a circle circle, circle, and method.DDitsitsα diameter,, diameter,β be given and and arcs.RR Then the jyotpatti to pick any one, but here is a This is much needed if one is to see mathematics itsits radius. radius. III.6. Madhava.¯ trigonometricBelow, identity, let C denote a real thegem:Sin circumference letα RCos be aβ radiusSin of β aCos circle,as αa creativeD its diameter, process (rather and R than as received III.6. Madhava.¯ Below,There let C isdenote way too the much circumference in the Lil¯Sinavati¯ of( aαand circle,β the)=Djyotpattiits diameter,to pick and± anyR one, but here. is a its radius. of a circle and α, β be given arcs. Then wisdom). Another is contextualization, often its radius. trigonometric identity, a real gem: let R be a± radius of a circle and α,Rβ be given arcs. Then claimed to be lacking in mathematics education. Sinα Cosβ Sinβ Cosα Seeing the context in which many methods arose Sin(α β)= ± . III.6. Madhava.¯ Below,± let C denoteR the circumference of acan circle, be ofD greatits diameter, help in this and regard.R its radius. III.6. Mādhava. Below, let C denote the Yet another appeal of this story for teachers is that III.6. Madhava.¯ Below, let C denote the circumference of a circle, D its diameter, and R circumference of a circle, D its diameter, and R its it provides an appreciation of how mathematics its radius. radius. can develop in many different ways. A largely 10 R. RAMANUJAM applied body of mathematical knowledge, with π The Madhava¯ – Leibniz series for π: 10 • R. RAMANUJAM lead to amazing discoveries of great mathematical • The4D Mādhava4D 4 D- Leibniz seriesn 1 4 Dfor n 4Dn C + ...+( 1) − +( 1) 2 mostly informal and verbal justifications, can still The Madhava¯ ≈ 1 –− Leibniz3 series5 for−π: 2n 1 − (2n) + 1 richness, as evidenced in Indian mathematics. • − The Madhava¯ – Gregory series for the arctangent: Perhaps this can help in removing the straitjacket • 4D 4D 4D n 1 4D n 4Dn C + ...+( 1) − +( 1) that chokes many students of mathematics in ≈ 1 − 3 RSin5θ RSin−3θ R2nSin 51θ − (2n)2 + 1 θ = + − ... (Here we1Cos assumeθ − 3Cos that3 θSin θ5Cos < Cos5θ −θ.) schools. The Madhava¯• The –Mādhava Gregory series- Gregory for the series arctangent: for the arctangent • (Here we assume that Sin θ < Cos θ.) 3 5 RSinθ RSin θ RSin θ CHENNAI, INDIA The Madhava¯ – Newtonθ = series for the Sine:3 + 5 ... • 1Cosθ − 3Cos θ 5Cos θ − THE INSTITUTE OF MATHEMATICAL SCIENCES, θ 3 θ 5 θ 7 (Here we assumeSinθ = thatθ Sin θ < Cos θ.) + ... E-mail address: [email protected] − R2 3! − R4 5! R6 7! −  · · · The Madhava¯ – Newton series for the Sine: 87 • At Right Angles | Vol. 3, No. 3, Nov 2014 θ 3 θ 5 θ 7 Sinθ = θ IV. FILLING A HUGE+ GAP ... − R2 3! − R4 5! R6 7! −  · · · The story of mathematics in India, dating from Vedic times to the 1600s, is one that needed to be told in detail, andIV. Plofker FILLING tells the A HUGE tale admira GAP bly. Its chief virtue is that scholarly commitment is never compromised, and historical record is carefully adhered to. What it offers is an informed understanding of the tremendous achievements of math- ematiciansThe story in of India, mathematics their methods in India, and style. dating There from is much Vedic to times be proud to the of, and 1600s, there is is one no that neededneed to to exaggerate be told in claims detail, either. and Plofker tells the tale admirably. Its chief virtue is that scholarly commitment is never compromised, and historical record is carefully adhered to. WhatI was it a littleoffers disappointed is an informed at the understanding lack of reference of to the non-Sa tremennskriticdous work achievements in mathematics of math- ematiciansin India, inbut India, then historical their methods sources and on style. these There are very is much few in tdeed.o be proud Another of, and source there of is no disappointment was the lack of any discussion on links with Chinese mathematics, another need to exaggerate claims either. highly developed one close by, with many Chinese travellers carrying texts between the countries. Once again, one wonders what documentary evidence is available. I was a little disappointed at the lack of reference to non-Sanskritic work in mathematics in India,But these butare then relatively historical trivial sources complaints. on these For teachers, are verythe few book in carriesdeed. some Another important source of disappointmentmessages, in my was opinion. the lack One of any is a discussion historical understanding on links with o Cfhinese the development mathematics, of key another highlyideas. developed This is much one closeneeded by, if onewith is many to see Chinese mathematics travellers as a crecarryingative process texts between (rather the countries.than as received Once again, wisdom). one wonders Another whatis contextualization, documentaryeviden often claimedce is available. to be lacking in mathematics education. Seeing the context in which many methods arose can be of great Buthelp these in this are regard. relatively trivial complaints. For teachers, the book carries some important messages, in my opinion. One is a historical understanding of the development of key ideas. This is much needed if one is to see mathematics as a creative process (rather than as received wisdom). Another is contextualization, often claimed to be lacking in mathematics education. Seeing the context in which many methods arose can be of great help in this regard.