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 an- cient 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. 4 RESONANCE ç April 2002 SERIES ç ARTICLE 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 re- search 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 ac- curacy. 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 hav- ing 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 RESONANCE ç April 2002 5 SERIES ç ARTICLE “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 civiliza- tions 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.