Mathematics in Ancient India 1

SERIES ç ARTICLE Mathematics in Ancient India 1. An Overview

Amartya Kumar Dutta

In th is series of articles, w e inten d to h ave a glimpse ofsom e ofthe landm arks in ancient In- dian m athem aticsw ith specialem phasison num - ber theory. T his issue features a brief overview of som e of the high peaks of m athem atics in ancient India. In the next part w e shalldescribe Amartya Kumar Dutta is A ryabhata's general solution in integers of the an Associate Professor of equation ax ¡ by = c. In subsequent instalments Mathematics at the Indian Statistical w e sh all d iscu ss in som e d etail tw o of th e m a- Institute, Kolkata. His jor contributions by Indians in num ber theory. research interest is in T he climax of the Indian achievem ents in alge- commutative algebra. bra and num ber theory w as theirdevelopm ent ofthe ingenious chakravala m ethod for solving,in integers,the equation x 2 ¡ Dy2 =1, erron eou sly know n as the P ellequation. W e shalllater de- scrib e th e p artial solution of B rah m agu p ta an d then the com plete solution due to Jayadeva and B haskaracharya.

V edicM athem atics:T he SulbaSutras

M ath em atics, in its early stages, d evelop ed m ainly alon g tw o b road overlap p ing trad ition s: (i) th e geom etric an d (ii) th e arithm etical an d algeb raic. A m on g th e p re- G reek ancient civilizations, itisin India that w e see a stron g em p h asis on b oth th ese great stream s of m ath - e m a tic s. O th e r a n c ie n t c iv iliza tio n s lik e th e E g y p tia n an d th e B ab y lon ian h ad p rogressed essen tially alon g th e com putationaltradition. A Seidenberg,an em inent al- Keywords. geb raist an d h istorian of m ath em atics, traced th e origin Taittiriya Samhita, Sulba-sutras, of sop h isticated m ath em atics to th e originators of th e Chakravala method, Meru- Prastara, Vedic altars, Yukti- R ig V ed ic rituals ([1,2]). bhasa, Madhava series.

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T h e oldest k n ow n m ath em atics texts in ex isten ce are th e Sulba-sutrasofB audhayana,A pastam ba and K atyayana E A F w h ich form p art of th e literatu re of th e S u tra p eriod of th e later V ed ic age. T h e S u lbasu tras h ad b een estim ated O P B to have been com posed around 800 B C (som e recent research ers are su ggesting earlier d ates).Butthemathe- D R Q m atical k n ow led ge record ed in th ese sutras (ap h orism s) arem uch m oreancient;forthe Sulbaauthorsem phasise H C G th at th ey w ere m erely stating facts alread y k n ow n to th e com p osers of th e B rah m an as an d S am h itas of th e KATYAYANA early V edic age ([3], [1], [2]). From the sulba. T h e S u lbasu tras give a com p ilation of th e resu lts in m athem atics that had been used for the designing and con stru ction s ofth e variou s elegan t V ed ic ¯ re-altars righ t fro m th e d a w n o f c iv iliza tio n . T h e a lta rs h a d ric h sy m - b olic sign i¯can ce an d h ad to b e con stru cted w ith accuracy. T he designs of several of these brick-altars are q u ite involved { for instan ce, th ere are con stru ction s d e- p icting a falcon in ° ight w ith cu rved w ings, a ch ariot- w h eel com p lete w ith sp okes or a tortoise w ith exten d ed h ead an d legs! C on stru ction s of th e ¯ re-altars are d e- scrib ed in an en orm ou sly d evelop ed form in th e Sata- patha B rahm ana (c. 2000 B C ; vide [3]);someofthem are m en tion ed in th e earlier T a ittiriy a S a m h ita (c. 3000 BC;vide[3]); b u t th e sacri¯cial ¯ re-altars are referred { w ith ou t ex p licit con stru ction { in th e even earlier R ig VedicSam hitas,the oldeststrataoftheextantVediclit- erature. T he descriptions ofthe ¯re-altars from the T ait- Vakrapaksa-syenacit. tiriya S am h ita on w ard s are ex actly th e sam e as th ose First layer of construction found inthe laterSulbasutras. (after Baudhayana)

P lane geom etry stands on tw o im portant pillars having applications throughout history: (i) th e resu lt p op - ularly know n as the `Pythagoras theorem ' and (ii) th e p rop erties of sim ilar ¯ gu res. In th e S u lbasu tras, w e see an ex p licit statem en t of th e P y th agoras th eorem an d its ap p lication s in variou s geom etric con stru ction s su ch as con stru ction of a squ are eq u al (in a re a ) to th e

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“How great is the su m , or d i® eren ce, of tw o given squ ares, or to a rec- science which tangle, or to the sum of n squ ares. T h ese con stru c- revealed itself in tio n s im p lic itly in v o lv e a p p lic a tio n o f a lg e b ra ic id e n - titites su ch as (a § b)2 = a 2 + b2 § 2ab, a 2 ¡ b2 = the Sulba, and how (a + b)(a ¡ b), ab = ((a + b)=2)2 ¡ ((a ¡ b)= 2)2 and meagre is my na2 = ((n +1)= 2)2 a 2 ¡ ((n ¡ 1)=2)2 a 2 .Theyre°ect intellect! I have a blending of geom etric and subtle algebraic thinking aspired to cross a n d in sig h t w h ich w e a sso c ia te w ith E u c lid . In fa c t, th e the unconquerable S u lba con stru ction of a sq u are eq u al in area to a given ocean in a mere rectan gle is ex actly th e sam e as given b y E u clid several raft.’’ cen tu ries later ! T h ere are geom etric solution s to w h at B Datta alluding to are algeb raic an d n u m b er-th eoretic p rob lem s. Kalidasa P ythagorastheorem w as know n in otherancient civilizations like the B abylonian, but the em phasisthere was on th e nu m erical an d n ot so m u ch on th e p rop er geo- m etric asp ect w h ile in th e S u lbasu tras on e sees d ep th in both aspects { especially the geom etric. T hisisa subtle pointanalysed indetailby Seidenberg. From certaindi- agram s d escrib ed in th e S u lbasu tras, several h istorian s and m athem aticians like B urk, H ankel,Schopenhauer, Seidenberg and Van der W aerden have concluded that th e S u lba au th ors p ossessed p ro ofs of geom etricalresu lts includingthe Pythagoras theorem { som e ofthedetails “But the Vedic Hindu, areanalysed inthepioneering w ork ofD atta ([2]).One in his great quest of of th e p ro ofs of th e P y th agoras th eorem ,easily d ed u cible the Para-vidya, from th e S u lba verses, is later d escrib ed m ore exp licitly Satyasya Satyam, b y B h askara II (1150 A D ). made progress in the A partfrom the knowledge,skilland ingenuity in geom - Apara-Vidya, etry an d geom etric algeb ra, th e V ed ic civilization w as including the various stron g in th e com p u tation al asp ects of m ath em atics as arts and sciences, to w ell { th ey h an d led th e arithm etic of fraction s as w ell a considerable as surds w ith ease,found good rationalapproxim ations extent, and with a to irration al n u m b ers like th e squ are ro ot of 2, an d , of completeness which cou rse, u sed several sign i¯can t resu lts on m en su ration . is unparallelled in antiquity.’’ A n am azing feature of allancient Indian m athem atical B Datta literatu re, b eginn ing w ith th e S u lbasu tras, is th at th ey

6 RESONANCE ç April 2002 SERIES ç ARTICLE th e y a re c o m p o se d e n tire ly in v e rse s { a n in c re d ib le fe a t! “nor did he [Thibaut] T h is trad ition of com p osing terse sutras, w hich could b e formulate the easily m em orised , en su red th at, insp ite of th e p au city obvious conclusion, a n d p e ris h a b ility o f w ritin g m a te ria ls, so m e o f th e c o re namely, that the k n ow led ge got orally tran sm itted to su ccessive gen era- Greeks were not the tion s. inventors of plane Post-VedicM athem atics geometry, rather it was the Indians.’’ D u ring th e p eriod 600 B C -300 A D , th e G reek s m ad e A Seidenberg profound contributionstom athem atics{ they pioneered th e ax iom atic ap p roach th at is ch aracteristic of m o d ern m athem atics, created the m agni¯cent edi¯ce of E uclid- ean geom etry, fou n d ed trigon om etry, m ad e im p ressive beginnings in num ber theory,and brought out the in- trinsic beauty, elegance and grandeur of pure m athem atics. B ased on the solid foundation provided by E u clid, G reek geom etry soared fu rth er into th e h igh er geom etry of conic sections due to A rch im ed es and A p ol- lonius. A rchimedes introduced integration and m ade several oth er m a jor con tribu tion s in m ath em atics an d p h y sics. B u t after th is b rilliant p h ase of th e G reek s, cre- ativemathematicsvirtuallycametoahaltintheWest till th e m o d ern rev ival.

O n the other hand,the Indian contribution,w hich began from th e earliest tim es, con tinu ed v igorou sly righ t u p to th e sixteen th centu ry A D , esp ecially in arithm etic, algeb ra an d trigon om etry. In fact, for several cen tu ries afterthedecline oftheG reeks,itw asonlyinIndia,and “Anyway, the tosom e extentC hina,thatone could¯nd an abundance damage had been of creative and original m athem atical activity. Indian done and the mathematicsusedtobeheldinhighesteembycontem- Sulvasutras have p orary sch olars w h o w ere ex p osed to it. F or instan ce, never taken the a m anuscriptfound in a Spanish m onastery (976 A D ) position in the record s ([4],[5]): \T he Indians have an extrem ely subtle history of and penetrating intellect, and w hen it com es to arith- mathematics that m etic, geom etry and other such advanced disciplines, they deserve.’’ other ideas m ust m ake w ay fortheirs. T he bestproof of A Seidenberg

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th is is th e n in e sy m bo ls w ith w h ic h th e y re p re se n t ea c h “The cord stretched num ber no m atter how large." Similar tributew as paid in the diagonal of a by the Syrian scholar Severus Sebokht in 662 A D ([5], rectangle produces [6]). both (areas) which the cords forming T h e D ecim al N otation an d A rith m etic the longer and shorter sides of an India gave to the w orld a priceless gift{ the decimal oblong produce system . T hisprofound anonym ous Indian innovation is separately.’’ unsurpassed for sheerbrilliance ofabstractthoughtand (translation from the utility as a practical invention. T he decimal notation d erives its p ow er m ainly from tw o key strokes of gen ius: Sulbasutras) the concept of place-value and the notion of zero as a d igit. G B H alsted ([7]) highlighted the power of the place-value of zero w ith a beautiful im agery: \T he im - portance ofthe creation ofthe zero m ark can never be exaggerated. T hisgiving to airy nothing,not m erely a local habitation and a nam e, a picture, a sym bol,but helpful pow er, isthe characteristic of the H indu race w hence it sprang. Itis like coining the N irvana into dynam os. N o single m athem atical creation has been m ore potent for “A common source the generalon-go of intelligence and pow er." for the Pythagorean T h e d ecim al sy stem h as a d ecep tive sim p licity as a re- and Vedic mathe- su lt o f w h ich c h ild re n a ll o v e r th e w o rld le a rn it e v e n a t a matics is to be tender age. Ithas an econom y in the num ber ofsym bols sought either in the used asw ellasthe space occupied by a w ritten num ber, Vedic mathematics an ab ility to e® ortlessly ex p ress arb itrarily large n u m - or in an older bers and, above all,com putational facility. T hus the mathematics very tw e lv e -d ig it R o m a n n u m b e r (DCCCLXXXVIII) is s im - much like it. ... What p ly 888 in th e d ecim al sy stem ! was this older, common source M ost ofth e stan d ard resu lts in b asic arith m etic are of In - d ian origin. T h is includ es n eat, sy stem atic an d straigh t- like? I think its forward techniquesofthe fundam entalarithm eticoper- mathematics was a tio n s: a d d itio n , su b tra c tio n , m u ltip lic a tio n , d iv isio n , very much like what tak ing sq u ares an d cu b es, an d ex tracting squ are an d we see in the cu b e roots; th e ru les of op eration s w ith fraction s an d Sulvasutras.’’ su rd s; variou s ru les on ratio an d p rop ortion like th e ru le A Seidenberg

8 RESONANCE ç April 2002 SERIES ç ARTICLE of three; and several com m ercialand related problem s “... the basic point is like incom e an d ex p en d iture, p ro¯ t an d loss, sim p le an d that the dominant com p ou n d interest, d iscou nt, p artn ersh ip, com p u tation s aspect of Old ofthe average impurities ofgold,speeds and distances, Babylonian and the m ixture and cistern problem s similar to those mathematics is its found inm odern texts.T he Indian m ethodsofperform - computational in g lo n g m u ltip lic a tio n s a n d d iv isio n s w e re in tro d u c e d in character ... The E u rop e as late as th e 14th cen tu ry A D . W e h ave b ecom e Sulvasutras know so u sed to th e ru les of op eration s w ith fraction s th at w e ten d to overlook th e fact th at th ey con tain ideas w h ich both aspects w ere unfam iliar to the E gyptians, w ho w ere generally (geometric and pro¯cient inarithm etic,and theG reeks,w ho had som e computational) and of th e m ost b rillian t m ind s in th e h istory of m ath em at- so does the ics. T h e ru le of th ree, b rou ght to E u rop e v ia th e A rab s, Satapatha w as very highly regarded by m erchants during and af- Brahmana.’’ tertherenaissance.Itcam e to be know n as theG olden A Seidenberg R u le fo r its g re a t p o p u la rity a n d u tility in c o m m e rc ia l com putations { m uch space used to be devoted to this ru le by th e early E u rop ean w riters on arithm etic.

T he excellence and skillattained by the Indians in the “The striking thing foundations ofarithm eticw as primarily due to the ad- here is that we have vantage of the early discovery ofthe decim alnotation { a proof. One will look th e key to all p rincipal ideas in m o d ern arithm etic. F or in vain for such instan ce, th e m o d ern m eth od s for extracting sq u are an d things in Old- cu b e ro ots, d escrib ed b y A ryab h ata in th e 5th centu ry Babylonia. The Old- A D , c le v e rly u se th e id e a s o f p la c e v a lu e a n d ze ro a n d Babylonians, or their th e a lg e b ra ic e x p a n sio n s o f (a + b)2 and (a + b)3 .These predecessors, must m eth o d s w ere intro d u ced in E u rop e on ly in th e 16th have had proofs of centuryA D .A partfrom theexactm ethods,Indiansalso their formulae, but inven ted several ingen iou s m eth o d s for d eterm ination of one does not find ap p roxim ate sq u are ro ots of n on -sq u are n u m b ers, som e them in Old- ofw hich w e shallm ention ina subsequent issue. Babylonia.’’ D ue to thegaps inourknow ledge abouttheearlyphase A Seidenberg of p ost-V ed ic In d ian m ath em atics, th e p recise d etails referring to a verse regarding the origin of decim al notation is not know n. in the Apastamba T he conceptofzero existed by the tim e ofP ingala (dated Sulbasutra on an 200 B C ). T h e id e a o f p la c e -v a lu e h a d b e e n im p lic it in isoceles trapezoid)

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an cien t S an sk rit term inology { as a resu lt, In d ian s cou ld “The diagonal of a e® ortlessly h an d le large n u m b ers righ t from th e V ed ic rectangle produces Age.Thereisterminologyforallmultiplesoftenupto both (areas) which 1018 in early V ed ic literatu re, th e R am ayan a h as term s its length and allthe w ay up to 1055, and the Jaina-Buddhist texts breadth produce show frequent use of large num bers (up to 10140!) fo r separately.’’ their m easurem ents of space and tim e. Expressions of (original verse from su ch large n u m b ers are n ot fou n d in th e con tem p orary the Sulbasutras w ork s of oth er n ation s. E ven th e b rillian t G reek s h ad n o along with the term inology for d en om ination s ab ove th e m yriad (104 ) translation are whiletheRomanterminologystoppedwiththemille given in [2], p.104) (103 ). T h e stru ctu re of th e S an sk rit n u m eral sy stem an d the Indian love for large num bers m ust have triggered th e creation of th e d ecim al system .

A s w e sh all see later, even th e sm allest p ositive integral solution ofth e eq u ation x 2 ¡ Dy2 = 1 could be very large; in fa c t, fo r D = 61,itis (1766319049; 226153980).The earlyIndian solution tothisfairlydeep problem couldbe p artly attribu ted to th e In d ian s' trad ition al fascination fo r la rg e n u m b e rs a n d a b ility to p la y w ith th e m .

D u e to th e ab sen ce of go o d n otation s, th e G reek s w ere n ot stron g in th e com p u tation al asp ects of m ath em atics { on e of th e factors resp on sible for th e even tu al d ecline of G reek m athem atics. A rchimedes (287-212 B C ) did realise the im portance of good notation, and m ade no- tab le p rogress to evolve on e, b u t failed to an ticipate th e In d ian d ecim al sy stem . A s th e great F ren ch m ath em ati- “The Indians have an cian L aplace (1749-1827) rem arked : \Theimportanceof extremely subtle and thisinvention ism ore readilyappreciated w hen one con- penetrating intellect, siders thatitw as beyond the two greatestm en of antiq- and when it comes uity: A rchim edes and A pollonius." to arithmetic, geometry and other T h e d ecim al sy stem w as tran sm itted to E u rop e th rou gh such advanced the A rabs. T he Sanskritw ord \sun ya" w as tran slated disciplines, other in to A ra b ic a s \ sifr" w h ich w as intro d u ced into G er- ideas must make m an y in th e 13th cen tu ry as \c ifra " from w hich w e have way for theirs.’’ th e w ord \cipher". T he w ord \zero" probably com es

10 RESONANCE ç April 2002 SERIES ç ARTICLE from the Latinised form \zephirum "oftheArabicsifr. “I will omit all Leonardo Fibonacciof Pisa (1180-1240), the ¯rst m a- discussion of the jor European m athem atician ofthe second m illennium, science of the p layed a m a jor role in th e sp read of th e In d ian n u m eral Indians, a people not sy stem in E u rop e. T h e In d ian n otation an d arithm etic the same as the eventuallygot standardised inE urope during the16th- Syrians; of their 17th century. subtle discoveries in T h e d ecim al sy stem stim u lated an d accelerated trad e astronomy, an d com m erce as w ell as astron om y an d m ath em atics. discoveries that are Itisno coincidencethatthem athem aticaland scienti¯c more ingenious than renaissance began in Europe only after the Indian no- those of the Greeks tation w as ad op ted . In d eed th e d ecim al n otation is th e and the Babylonians; very pillarof allm odern civilization. and of their valuable methods of Algebra calculation which W h ile sop h isticated geom etry w as inven ted d u ring th e surpass description.’’ origin of th e V ed ic rituals, its ax iom atisation an d fu r- (Severus Sebokht in th er d evelop m en t w as d on e by th e G reek s. T h e h eigh t 662 AD) reach ed by th e G reeks in geom etry b y th e tim e of A p ol- lonius(260-170 B C ) w asnotm atched by any subsequent a n c ie n t o r m e d ie v a l c iv ilisa tio n . B u t p ro g re ss in g e o m e - try p rop er so on reach ed a p oint of stagn ation . B etw een “Indeed, if one the times of Pappus (300 A D ) { the last big nam e in understands by G reek geom etry { an d m o d ern E u rop e, B rah m agu p ta's algebra the b rillia n t th e o re m s (628 A D ) o n c y c lic q u a d rila te ra ls c o n - application of stitute th e solitary gem s in th e h istory of geom etry. arithmetical Further progress needed new techniques, in fact a com - operations to pletely new approach in m athem atics. T his w as pro- complex magnitudes vided by the em ergence and developm ent ofa new dis- of all sorts, whether cipline { algebra. Itisonly after the establishm ent of rational or irrational an algeb ra cu lture in E u rop ean m ath em atics d u ring th e numbers or space- 16th cen tu ry A D th at a resu rgen ce b egan in geom etry magnitudes, then the th rou gh its algeb raisation b y D escartes an d F erm at in learned Brahmins of early 17th cen tu ry. In fact, th e assim ilation an d re¯ n e- Hindostan are the m en t of algeb ra h ad also set th e stage for th e rem arkab le real inventors of strides in n u m b er th eory an d calcu lus in E u rop e from algebra.’’ th e 17th cen tu ry. H Hankel

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A lg e b ra w a s o n ly im p lic it in th e m a th e m a tic s o f se v - “The intellectual e ra l a n c ie n t c iv ilisa tio n s till it c a m e o u t in th e o p e n potentialities of the w ith the introduction of literal or sym bolic algebra in Indian In d ia. B y th e tim e of A ryab h ata (499 A D ) and B rah- nation are unlimited m agupta (628 A D ), sy m b o lic a lg e b ra h a d e v o lv e d in In - and not many years dia into a distinct branch of m athem atics and becam e would perhaps be on e of its cen tral p illars. A fter evolution th rou gh sev - needed eralstages,algebra has now com e to play a key role in before India can take modernmathematicsbothasanindependentareainits a worthy place in o w n rig h t a s w e ll a s a n in d isp e n sa b le to o l in o th e r ¯ e ld s . world Mathematics.’’ In fact,the 20th century w itnessed a vigorous phase of (Andre Weil in 1936) `a lg e b ra isa tio n o f m a th e m a tic s'. A lg e b ra p ro v id e s e le - gance, sim plicity, precision, clarity and technical pow er in th e h an d s of th e m ath em atician s. It is rem arkab le how earlythe Indians had realised the signi¯ance ofal- geb ra an d h ow stron gly th e lead ing In d ian m ath em ati- cianslikeB rahm agupta (628 A D ) and B haskara II(1150 “The importance of AD) asserted an d estab lish ed th e im p ortan ce of th eir the creation of the newly-founded discipline as w e shallsee in subsequent zero mark can never issues. be exaggerated. This giving to airy In d ian s b egan a sy stem atic u se of sy m b ols to d en ote u n - nothing, not merely a know n quantities and arithm eticoperations. T he four local habitation and a arithm eticoperationsw eredenoted by \yu",\ksh",\gu" name, a picture, a an d \b h a" w h ich are th e ¯ rst letters (or a little m o d i- symbol, but helpful ¯cation) ofthe correspondingSanskritw ords yuta (ad- power, is the dition), ksaya (su b traction ), guna (m u ltip lic a tio n ) and characteristic of the bhaga (division); sim ila rly \ k a " w a s u se d fo r karani(root), Hindu race whence it w h ile th e ¯ rst letters of th e n am es of d i® eren t colou rs w ere used to denote di®erent unknow n variables. T his sprang. It is like intro d u ction of sy m b olic rep resen tation w as an im p or- coining the Nirvana tan t step in th e rap id ad van cem en t of m ath em atics. W h i- into dynamos. No le a ru d im en tary u se of sy m b ols can also b e seen in th e single mathematical G reek textsofD iophantus,itisin Indiathat algebraic creation has been form alism achieved fulldevelopm ent. more potent for the general on-go of T he Indians classi¯ed and m ade a detailed study of intelligence and equations (whichwerecalled sam i-karana), introd u ced power.’’ n egative n u m b ers togeth er w ith th e ru les for arithm etic

12 RESONANCE ç April 2002 SERIES ç ARTICLE op eration s involving zero an d n egative n u m b ers, d is- “The ingenious covered resu lts on su rd s, d escrib ed solution s of linear method of an d q u ad ratic eq u ation s, gave form u lae for arithm etic expressing every an d geom etric p rogression as w ell as iden titites involving sum m ation of¯niteseries,and applied severaluseful possible number resu lts on p erm u tation an d com b ination s includ ing th e using a set of ten n n symbols (each fo rm u la e fo r P r and C r . T h e enlargem ent of the num - ber system to includenegative num bersw as a m om en- symbol having a tous step in the developm ent of m athem atics. T hanks place value and an to th e early recogn ition of th e ex isten ce of n egative n u m - absolute value) b ers, th e In d ian s cou ld give a u n i¯ed treatm en t of th e emerged in India. various form s of quadratic equations (w ith p ositive co- The idea seems so e± cients), i.e ., ax2 + bx = c; ax 2 + c = bx ; bx + c = ax2 . simple nowadays T h e In d ian s w ere th e ¯ rst to recogn ise th at a q u ad ratic that its significance equation has two roots. Sridharacharya (750 A D ) gave is no longer the w ell-know n m ethod ofsolvinga quadraticequation appreciated. Its b y com p leting th e sq u are { an idea w ith far-reach ing simplicity lies in the consequences in m athem atics. T he P ascal's triangle for way it facilitated n q u ick com p u tation of C r isdescribed by H alayudha in calculations and th e 10th centu ry A D as M eru -P ra sta ra 700 years b efore placed arithmetic itw asstated by Pascal;and H alayudha'sM eru-Prastara foremost among w as only a clari¯cation of a rule invented by P ingala useful inventions. m ore th an 1200 years earlier (around 200 B C )! The importance of T h u s, as in arithm etic, m an y top ics in h igh -sch o ol al- this invention is more gebra had been system atically developed in India. T his readily appreciated knowledgewenttoEuropethroughtheArabs.Theword when one considers yava in A ryabhatiyabhasya of B haskara I (6th centu ry that it was beyond AD) m eaning \to m ix" or \to separate" has a± nity w ith the two greatest men th at of al-jabr ofal-K hw arizm i(825 A D ) fro m w h ich th e of antiquity, w ord algebra isderived. In hisw idelyacclaim ed text on Archimedes and h istory of m ath em atics, C a jori ([8]) concludesthechap- Apollonius.’’ te r o n In d ia w ith th e fo llo w in g re m a rk s: \ ...it is re m a rk - able to w hat extent Indian m athem atics enters into the sc ie n ce o f o u r tim e . B o th th e fo rm a n d th e sp irit o f th e arithm etic and algebra of m odern tim es are essentially Indian. T hink of our notation of num bers, brought to perfection by the H indus,think of the Indian arithm eti-

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caloperations nearlyas perfectas our ow n,think oftheir “India has given to elegant algebraicalm ethods,and then judge w hether the antiquity the earliest B rahm ins on the banks ofthe G anges are not entitled to scientific physicians, somecredit." and, according to Sir William Hunter, she B ut ancient Indian algebra w ent far beyond the high has even contributed schoollevel.T he pinnacleofIndian achievem entw asat- to modern medical tained intheirsolutions ofthe hard and subtlenum ber- science by the th eoretic p rob lem s of ¯ n d ing integer solution s to equ a- discovery of various tions of ¯rst and second degree. Such equations are chemicals and by called ind eterm inate or D iop h an tine eq u ation s. B u t alas, teaching you how to th e In d ian w ork s in th is area w ere to o far ah ead of th e reform misshapen tim es to be noticed by contem porary and subsequent ears and noses. civilisations! A s C ajori lam ents, \Unfortunately,some Even more it has of the m ost brilliant results in indeterm inate analysis, done in mathematics, found in theH indu w orks,reached E urope too lateto ex- for algebra, ertthein°uence they w ouldhave exerted,had they com e geometry, astronomy, tw o or three cen turies earlier."Withoutsomeaware- ness of the Indian contributions in this¯eld,itisnot and the triumph of p ossible to get a tru e p ictu re of th e d ep th an d skill at- modern science – ta in e d in p o st-V e d ic In d ia n m a th e m a tic s th e ch a ra c te r mixed mathematics – o f w h ich w a s p rim a rily a lg e b ra ic . W e sh a ll d isc u ss so m e were all invented in of th ese nu m b er-th eoretic con tribu tion s from th e n ex t India, just so much as instalm en t. the ten numerals, the very cornerstone of Trigonom etry and C alculus all present civilization, were discovered in A part from developing the subject of algebra proper, India, and, are in Indians also began a process ofalgebrisation and conse- quent sim pli¯cation ofother areas ofm athem atics. For reality, sanskrit instan ce, th ey d evelop ed trigon om etry in a sy stem atic words.” m an n er, resem b ling its m o d ern form , an d im p arted to Swami Vivekananda it its m o d e rn a lg e b ra ic c h a ra c te r. T h e a lg e b ra isa tio n o f the study ofin¯nitesimalchangesled tothediscovery of key principles ofcalculusby the tim e ofB haskaracharya (1150 A D ) som e of w hich w e shallm ention in a subsequent issue. C alculus in Indialeaped to an am azing h eight in th e an alytic trigon om etry of th e K erala sch o ol in th e 14th cen tu ry.

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A lthou gh th e G reek s fou n d ed trigon om etry, th eir p ro- “The Hindus gress w as halted due to the absence of adequate alge- solved problems in braic m achinery and notations. Indians invented the interest, discount, sine an d cosine fu n ction s, d iscovered m ost of th e stan - partnership, dard form ulae and identitites,including the basicfor- alligation, m u la fo r sin (A § B ), an d con stru cted fairly accu rate summation of sinetables.B rahm agupta (628 A D ) and G ovindasw am i arithmetical and (880 A D ) gave interpolation form ulae forcalculating the geometric series, sines of interm ed iate an gles from sine tab les { th ese are sp ecial cases of th e N ew ton {S tirling an d N ew ton {G au ss and devised rules form ulae for second-order di®erence. R em arkable ap- for determining the proximations for ¼ are given in Indian texts including numbers of 3:1416 of A ryabhata (499 A D ),3:14159265359 of M ad- combinations and hava (14th centu ry A D ) and 355=113 ofN ilakanta (1500 permutations. It AD).A n anonym ouswork K aranapaddhati(b elieved to may here be have been w ritten by Putum ana Som ayajininthe 15th added that chess, century A D ) gives the value 3:14159265358979324 w hich the profoundest of is correct up to seventeen decim al places. all games, had its origin in India.’’ T h e G reeks h ad investigated th e relation sh ip b etw een a F Cajori chord of a circleand the angleitsubtends at the cen- tre { b u t th eir system is qu ite cu m b ersom e in p ractice. T he Indians realised the signi¯cance of the connection between a half-chord and halfof the anglesubtended by the fullchord. In the case of a unitcircle, thisis precisely the sinefunction. T he Indian half-chord was intro d u ced in th e A rab w orld d u ring th e 8th centu ry A D .Europe w as introduced to thisfundam entalnotion th rou gh th e w ork of th e A rab sch olar al-B attan i (858- 929 A D ). T h e A rab s p referred th e In d ian h alf-ch ord to P tolem y 's sy stem of ch ord s an d th e algeb raic ap p roach of the Indians to the geom etricapproach of the G reeks.

T he Sanskritw ord for half-chord \ardha-jya", later ab- breviated as \jy a ", w as w ritten b y th e A rab s as \jy b ". C uriously,thereisa similar-sounding A rab word \ja ib " w h ich m ean s \h eart, b osom , fold, b ay or cu rve". W h en th e A rab w orks w ere b eing tran slated into L atin, th e apparentlym eaninglessw ord \jy b " w as m istaken for th e

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word\ja ib " an d tran slated as \sinus" w hich has several “... it is remarkable to m eanings inLatinincluding\heart,bosom ,fold,bay or what extent Indian curve"! T hisw ord becam e \sine" in the E nglish version. mathematics enters A ryabhata's\k o tijy a "becamecosine. into the science of our time. Both the T hetradition ofexcellenceand originalityinIndian trigo- form and the spirit of n om etry reach ed a h igh p eak in th e ou tstan d ing resu lts the arithmetic of M adhavacharya (1340-1425) on th e p ow er series ex - and algebra of p an sion s of trigon om etric fu n ction s. T h ree cen tu ries b e- modern times are foreGregory(1667), M adhava had described the series essentially Indian. 3 5 Think of our notation µ =tanµ ¡ (1=3)(tan µ) + (1= 5)(tan µ) ¡ of numbers, brought (1= 7)(tan µ)7 + ¢¢¢(jtan µj· 1): to perfection by the Hindus, think of the H is p ro of, as p resen ted in Y uktibhasa,involves the idea Indian arithmetical of integration as the lim it of a sum m ation and corre- operations nearly as sponds to the m odern m ethod ofexpansion and term - perfect as our own, by-term integration. A crucial step isthe use of the think of their elegant resu lt algebraical methods, lim (1 p +2p + ¢¢¢+ (n ¡ 1)p )=n p +1 =1=(p +1): and then judge n !1 whether the T h e exp licit statem en t th at (jtan µj· 1) revealsthe Brahmins levelof sophistication in the understan d ing of in¯nite on the banks of the series including an aw areness of convergence. M adhava Ganges are not a lso d isc o v e re d th e b e a u tifu l fo rm u la entitled to some credit.’’ ¼ = 4= 1¡ 1= 3+ 1= 5 ¡ 1=7+ ¢¢¢; F Cajori ob tained b y p u tting µ = ¼ =4intheMadhava{Gregory series. T h is series w as red iscovered th ree cen tu ries later by Leibniz (1674). A s one of the ¯rst applications of hisnewlyinvented calculus,Leibnizwas thrilled at the d iscovery of th is series w h ich w as th e ¯ rst of th e resu lts givinga connection between ¼ and unitfractions.M ad- h ava also d escrib ed th e series p ¼ = 12 = 1 ¡ 1= 3:3+ 1=5:3 2 ¡ 1= 7:3 3 + ¢¢¢

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¯rst given in E urope by A Sharp (1717).Again,three “Incomparably hundered yearsbeforeN ewton (1676 A D ),M adhava had greater progress described thew ell-know n powerseriesexpansions than in the solution sin x = x ¡ x 3 =3!+ x 5 =5!¡ ¢¢¢ of determinate equations was and cosx =1¡ x 2 =2!+ x 4 =4!¡ ¢¢¢: made by the T h ese series w ere u sed to con stru ct accu rate sine an d co- Hindus in the sine tab les for calcu lation s in astron om y. M ad h ava's val- treatment of u es are correct, in alm ost allcases, to th e eigh th or n inth indeterminate decim alplace { such an accuracy w as not to b e achieved equations. in E u rop e w ithin th ree cen tu ries. M ad h ava's resu lts Indeterminate show that calculus and analysishad reached rem arkable analysis was a d ep th an d m atu rity in In d ia cen tu ries b efore N ew ton subject to which (1642-1727) and Leibniz (1646-1716). M adhavacharya the Hindu m ightberegarded asthe¯rstm athem atician w ho worked mind showed a in a n a ly sis! happy adaptation.’’ F Cajori U n fortu n ately, th e original texts of several ou tstan d ing m athem aticians like Sridhara, Padm anabha, Jayadeva and M adhava have notbeen found yet{ itisonlythrough th e occasion al referen ce to som e of th eir resu lts in su b - sequent com m entaries that w e get a glim pse of their w ork . M ad h ava's con tribu tion s are m en tion ed in sev - erallater texts including the T a n tra S a m gra h a (1500) of th e great astron om er N ilakanta S om aya ji (1445-1545) w ho gave theheliocentricm odelbeforeC opernicus,the Yuktibhasa (1540) of Jyesthadeva (1500-1610) and the anonym ous K aranapaddhati.Allthesetextsthemselves w ere discovered by C harles W hish and published only in 1835.

A m ong ancient m athem aticians w hose texts have been found,specialm ention m ay be m ade ofA ryabhata,B rah- m agupta and B haskaracharya. A llofthem w ereem inent astron om ers as w ell. W e sh all m ake a b rief m ention of som e oftheirm athem aticalw orks in subsequent issues.

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Later D evelopm ents “Unfortunately, some of the most T heIndian contributionsinarithm etic,algebraand trigo- brilliant n om etry w ere tran sm itted b y th e A rab s an d P ersian s results in to E u rop e. T h e A rab s also p reserved an d tran sm itted indeterminate th e G reek h eritage. A fter m ore th an a th ou san d years analysis, found in of slum b er, E u rop e red iscovered its rich G reek h eritage the Hindu works, and acquired som e of the fruitsof the phenom enalIn- reached dian progress. It ison the foundation form ed by the Europe too late to blending of the two great m athem aticalcultures { the exert the influence geom etricand axiom atictradition ofthe G reeks and the they would have algebraic and com putational tradition of the Indians { exerted, had they th at th e m ath em atical ren aissan ce to ok p lace in E u rop e. come two or three Indians m ade signi¯cant contributions in severalfront- centuries earlier.’’ lineareasofm athem aticsduring the20th century,espe- F Cajori c ia lly d u rin g th e se c o n d h a lf, a lth o u g h th is fa c t is n o t so w ell-k n ow n am on g stu d en ts p artly b ecau se th e fron tiers of m athem atics have expanded far beyond the scope of th e u n iversity cu rricu la. H ow ever, In d ian s v irtually to ok no partin the rapid developm ent of m athem atics that took place during the 17th-19th century { thisperiod coincided w ith the general stagnation in the national life . T h u s, w h ile h ig h -sch o o l m a th e m a tic s, e sp e c ia lly in arithm etic and algebra, is m ostly of Indian origin, one rarelycom esacrossIndian nam esincollege and univer- sity cou rses as m ost of th at m ath em atics w as created d u ring th e p eriod ran ging from late 17th to early 20th century. B ut should w e forget the culture and great- nessofIndia'sm illennium s becauseofthe ignorance and w eakness of a few centuries?