4 RESONANCE ç April 2002 SERIES ç ARTICLE In th is se rie s o f a rticles, w e in te n d to h av e a g li m p se o f so m e o f th e la n d m a rk s in a n c i e n t In - d ia n m a th e m a ti c s w ith sp e c ia l e m p h a sis o n n u m - b e r th e o ry . T h is issu e fe a tu re s a b ri e f o v e rv ie w o f so m e o f th e h ig h p ea k s o f m a th e m a ti c s in a n - c ie n t In d ia . In th e n e x t p a rt w e sh a ll d e sc rib e A ry a b h a ta 's g e n e ra l so l u tio n in in te g e rs o f th e e q u a t i o na x¡by=c. In su b se q u e n t i n sta l m e n ts w e sh a l l d i scu ss i n so m e d e ta i l tw o o f th e m a - jo r co n trib u ti o n s b y In d i a n s i n n u m b e r th e o ry . T h e c l i m a x o f th e In d i a n a ch i e v e m e n ts i n a l g e - b ra a n d n u m b e r th e o ry w a s th e i r d ev e l o p m e n t o f th e i n g e n i o u s chakravalam e th o d fo r so l v i n g , i n i n te g e rs , th e e q u a tio n x2¡D y2= 1, e rr o n e o u sl y k n o w n a s th e P e ll e q u a ti o n . W e sh a ll l a te r d e - sc rib e th e p a rtia l so lu ti o n o f B ra h m a g u p ta a n d th en th e c o m p le te so lu tio n d u e to J a y a d e v a a n d B h a sk a ra ch a ry a . V e d i c M a th em a ti cs: T h e S u l b a S u tra s M a th em a ti cs, in its early st a ges,d ev el o p ed m a i n ly a l o n g tw o b ro a d ov erl a p p in g trad i ti o n s: (i) th e geom etric an d(ii) th e a ri th m etical an d algeb ra ic. A m on g th e p re- G reek an ci en t ci v i l i za ti o n s, i t i s i n I n d i a th a t w e see a stro n g em p h asis o n b oth th ese g reat strea m s of m a th - em a ti cs. O th er a n ci en t civ il i za ti on s like th e E gyp ti a n an d th e B a b y lo n i a n h a d p rog ressed ess en ti a ll y a l o n g th e co m p u tati o n al trad i ti o n . A S eid en b erg, a n em i n en t al - geb rai st a n d h i st o ria n o f m a th em a tics, tr a ce d th e o ri g i n o f so p h isti ca ted m a th em atics to th e o ri g in a tors of th e R ig V ed ic ritu als ([1, 2]). Mathematics in Ancient India 1. An Overview Amartya Kumar Dutta Amartya Kumar Dutta is an Associate Professor ofMathematics at the Indian Statistic al Institute, Kolkata. His research interest is in commutative algebra. Keywords. Taittiriya Samhita, Sulba-sutras, Chakravala method, Meru- Prastara, Vedic altars, Yukti- bhasa, Madhava series.
I n t h is s e r ie s o f a r t ic le s , w e in t e n d t o h a v e a
g lim p se o f s o m e o f th e la n d m a rk s in a n c ie n t In -
d ia n m a th e m a tic s w ith sp e c ia l e m p h a sis o n n u m -
b e r th e o ry . T h is issu e fe a tu re s a b rie f o v e rv ie w
o f so m e o f th e h ig h p ea k s o f m a th e m a tic s in a n -
c ie n t In d ia . In th e n e x t p a rt w e sh a ll d e sc rib e
A ry a b h a ta 's g e n e ra l so lu tio n in in te g e rs o f th e
e q u a t io n a x ¡ by = c . In su b se q u e n t in sta lm e n ts w e s h a ll d is c u s s in s o m e d e t a il tw o o f t h e m a -
jo r co n trib u tio n s b y In d ia n s in n u m b e r th e o ry .
T h e c lim a x o f th e In d ia n a ch ie v e m e n ts in a lg e -
b r a a n d n u m b e r t h e o r y w a s t h e ir d e v e lo p m e n t
o f th e in g e n io u s chakravala m e th o d fo r so lv in g , in
in te g e rs , th e e q u a tio n x 2 ¡ D y 2 = 1 , e rr o n e o u sly
k n o w n a s th e P e ll e q u a tio n . W e sh a ll la te r d e -
sc rib e th e p a rtia l so lu tio n o f B ra h m a g u p ta a n d
th en th e c o m p le te so lu tio n d u e to J a y a d e v a a n d
B h a sk a ra ch a ry a .
V e d ic M a t h e m a t ic s : T h e S u lb a S u t r a s
M ath em a tics, in its ea rly sta g es, d ev elo p ed m a in ly a lo n g tw o b road ov erla p p in g tra d itio n s: (i) th e geo m etric a n d (ii) th e a rith m etica l a n d alg eb ra ic. A m on g th e p re-G reek an cien t civ iliza tio n s, it is in In d ia th a t w e see
a stro n g em p h asis o n b o th th ese g rea t strea m s o f m a th -em a tics. O th er a n cien t civ iliza tio n s lik e th e E gy p tian an d th e B a b y lo n ia n h a d p rog ressed essen tia lly a lon g th e co m p u tatio n a l trad ition . A S eid en b erg, a n em in en t a l-geb raist a n d h isto ria n o f m a th em a tics, traced th e o rigin of so p h istica ted m a th em atics to th e origin ators of th e R ig V ed ic ritu als ([1, 2]).
T h e o ld est kn ow n m ath em atics tex ts in ex isten ce a re th e S u lba-sutras of B a u d h ay an a, A p astam b a a n d K a ty ay a n a w h ich form p art o f th e litera tu re o f th e S u tra p erio d ofth e la ter V ed ic a g e. T h e S u lb asu tra s h ad b een estim a ted
to h av e b een com p osed arou n d 80 0 B C (som e recent re-search ers are su gg estin g ea rlier d a tes). B u t th e m a th e-m atica l k n ow led ge record ed in th ese sutras (ap h orism s)are m u ch m ore a n cien t; fo r th e S u lb a a u th o rs em p h a sise th a t th ey w ere m erely statin g fa cts a lrea d y k n ow n to th e co m p osers of th e B rah m an as an d S a m h itas o f th e ea rly V ed ic a g e ([3 ], [1], [2 ]).
T h e S u lb asu tra s g ive a co m p ila tio n o f th e resu lts in
m ath em a tics th a t h ad b een u sed for th e d esig n in g a n d co n stru ction s o f th e va riou s eleg an t V ed ic ¯ re-a ltars rig h tfrom th e d aw n o f civ iliza tio n . T h e altars h ad rich sym -b o lic sig n i ca n ce an d h a d to b e co n stru cted w ith ac -cu racy. T h e d esig n s of sev eral o f th ese b rick -a ltars are q u ite in vo lv ed { fo r in sta n ce, th ere are co n stru ction s d e-p ictin g a falco n in ° ig h t w ith cu rved w in gs, a ch ario t-w h eel com p lete w ith sp ok es o r a tortoise w ith ex ten d ed h ea d a n d leg s! C o n stru ction s o f th e ¯ re-a ltars are d e-
scrib ed in a n en o rm o u sly d ev elop ed fo rm in th e Sata-patha B rah m an a (c. 2000 B C ; vid e [3]); som e of th em are m ention ed in th e earlier T aittiriya S am hita (c. 300 0 B C ; v id e [3]); b u t th e sa cri cia l ¯ re-a ltars are referred
{ w ith ou t exp licit con stru ction { in th e even earlier R ig V ed ic S a m h itas, th e o ld est strata o f th e ex tan t V ed ic lit-era tu re. T h e d escrip tio n s of th e ¯ re-a lta rs from th e T ait-tiriya S a m h ita on w ard s are ex actly th e sa m e as th ose fou n d in th e la ter S u lb a su tras.
P lan e g eo m etry stan d s o n tw o im p orta n t p illa rs h av -in g a p p lica tio n s th rou gh o u t h istory : (i) th e resu lt p op -u larly k n ow n a s th e P y th a g ora s th eo rem ' an d (ii) th e p rop erties o f sim ila r ¯ gu res. In th e S u lb asu tra s, w e see a n ex p licit statem en t of th e P y th a go ra s th eorem an d its ap p lica tio n s in va rio u s g eo m etric co n stru ctio n ssu ch as co n stru ction of a squ a re eq u al (in a rea ) to th e
su m , o r d i® eren ce, o f tw o g iv en squ ares, o r to a rec-tan gle, or to th e su m of n squ ares. T h ese co n stru c-tio n s im p licitly in v o lv e ap p lica tion of a lg eb raic id en -titites su ch a s (a § b )2 = a 2 + b 2 § 2 a b , a 2 ¡ b 2 =
(a + b )(a ¡ b ), a b = ((a + b )= 2 )2
¡ ((a ¡ b )= 2)2
a n d n a 2 = ((n + 1 )= 2 )2 a 2 ¡ ((n ¡ 1 )= 2 )2 a 2 . T h ey re° ecta b len d in g of g eo m etric an d su b tle alg eb raic th in k in g an d in sight w h ich w e associate w ith E u clid . In fact, th e S u lb a co n stru ctio n o f a sq u are eq u al in a rea to a giv en recta n gle is ex a ctly th e sam e as g iv en b y E u clid severalcen tu ries la ter ! T h ere are g eo m etric solu tio n s to w h a ta re a lg eb raic a n d n u m b er-th eo retic p rob lem s.
P y th a g oras th eo rem w a s k n ow n in o th er an cien t civ iliza -tio n s like th e B a b y lon ia n , b u t th e em p h a sis th ere w a son th e n u m erical an d n o t so m u ch on th e p rop er geo -m etric asp ect w h ile in th e S u lb asu tras o n e sees d ep th in b o th a sp ects { esp ecia lly th e geo m etric. T h is is a su b tle p o in t a n a ly sed in d etail b y S eid en b erg. F rom certa in d i-a g ram s d escrib ed in th e S u lb a su tra s, sev eral h isto ria n sa n d m a th em a ticia n s lik e B u rk , H a n kel, S ch op en h a u er,S eid en b erg an d V an d er W aerd en h av e con clu d ed th at
th e S u lb a a u th ors p ossessed p ro ofs o f geo m etrica l resu ltsin clu d in g th e P y th ag oras th eo rem { som e o f th e d etailsa re an a ly sed in th e p ion eerin g w o rk of D a tta ([2 ]). O n e o f th e p ro o fs o f th e P y th a g oras th eo rem , ea sily d ed u cib le from th e S u lb a verses, is la ter d escrib ed m o re ex p licitly b y B h askara II (11 50 A D ).
A p a rt from th e k n ow led g e, skill a n d in g en u ity in geo m -etry a n d g eo m etric alg eb ra, th e V ed ic civ iliza tio n w as
stro n g in th e co m p u tation al asp ects o f m a th em a tics a sw ell { th ey h a n d led th e a rith m etic o f fractio n s a s w ella s su rd s w ith ea se, fo u n d g o o d ratio n a l ap p rox im a tio n sto irra tio n al n u m b ers like th e squ a re roo t o f 2, an d , o fco u rse, u sed severa l sig n i ca n t resu lts o n m en su ratio n .
A n a m a zin g fea tu re o f all an cien t In d ian m a th em a tica llitera tu re, b eg in n in g w ith th e S u lb asu tras, is th at th ey
th ey a re c o m p osed en tirely in v erses { a n in cred ib le fea t!T h is trad itio n o f co m p o sin g terse sutras, w h ich co u ld b e ea sily m em o rised , en su red th at, in sp ite o f th e p au city a n d p erish ab ility of w ritin g m a teria ls, so m e o f th e co re
k n ow led g e go t o rally tran sm itted to su ccessiv e gen era-tio n s.
P o st-V e d ic M a th e m a tic s
D u rin g th e p erio d 60 0 B C -30 0 A D , th e G reek s m ad e p rofo u n d co n trib u tio n s to m a th em a tics { th ey p io n eered th e ax iom a tic a p p roa ch th a t is ch a racteristic o f m o d ern m ath em a tics, crea ted th e m a g n i cen t ed i ce of E u clid -ea n ge o m etry, fo u n d ed trigo n o m etry, m a d e im p ressive b eg in n in gs in n u m b er th eo ry, a n d b rou g h t o u t th e in -trin sic b ea u ty, eleg a n ce a n d g ra n d eu r of p u re m a th -em a tics. B a sed on th e solid fou n d ation p rov id ed b y E u clid , G reek g eo m etry soa red fu rth er in to th e h ig h erg eo m etry of co n ic sectio n s d u e to A rch im ed es a n d A p ol-lo n iu s. A rch im ed es in tro d u ced in tegratio n a n d m a d e several o th er m a jo r co n trib u tio n s in m a th em a tics a n d p h y sics. B u t a fter th is b rillia n t p h a se o f th e G reek s, cre-
a tiv e m a th em a tics v irtu ally ca m e to a h alt in th e W esttill th e m od ern revival.
O n th e o th er h a n d , th e In d ian co n trib u tio n , w h ich b e-g a n fro m th e ea rliest tim es, co n tin u ed v ig oro u sly righ tu p to th e six teen th cen tu ry A D , esp ecia lly in a rith m etic,a lgeb ra an d trig o n o m etry. In fact, fo r several cen tu riesa fter th e d eclin e of th e G reek s, it w a s o n ly in In d ia, a n d to som e ex ten t C h in a , th a t o n e co u ld ¯ n d a n a b u n d a n ce
o f c reativ e a n d o rig in a l m a th em a tica l a ctiv ity. In d ia n m ath em a tics u sed to b e h eld in h ig h esteem b y co n tem -p o ra ry sch o lars w h o w ere ex p o sed to it. F o r in sta n ce,a m a n u scrip t fo u n d in a S p an ish m o n astery (97 6 A D )reco rd s ([4 ],[5 ]): \T he In dian s h ave a n extrem ely su btle
an d pen etratin g in tellect, a n d w hen it com es to arith-
m etic, geom etry an d other su ch ad van ced disciplin es,
other ideas m ust m ake w ay for theirs. T he best proof of
this is the nine sym bols w ith w hich they represent each
n u m ber n o m a tter ho w large." S im ila r trib u te w as p a id b y th e S y rian sch o la r S everu s S eb ok h t in 66 2 A D ([5],[6 ]).
T h e D e c im a l N o ta tio n a n d A rith m e tic
In d ia gav e to th e w orld a p riceless g ift { th e d ecim a lsy stem . T h is p rofou n d a n o n y m ou s In d ia n in n ov a tio n isu n su rp assed for sh eer b rillia n ce o f ab stra ct th o u g h t a n d u tility a s a p ractica l in ven tio n . T h e d ecim al n o tatio n d eriv es its p ow er m a in ly from tw o k ey strok es o f g en iu s:th e co n cep t of p lac e-va lu e a n d th e n o tio n of zero a s a d igit. G B H a lsted ([7 ]) h ig h ligh ted th e p ow er o f th e p lace-va lu e of zero w ith a b ea u tifu l im ag ery : \T he im -
portance ofthe creation ofthe zero m ark can never be ex-
aggerated. T h is givin g to airy n othin g, n ot m erely a local
habitation an d a n am e, a pictu re, a sym bol, bu t helpfu l
pow er, is the characteristic of the H in du race w hen ce it
spran g. It is like coin in g the N irvan a in to dy n am os. N o
sin gle m athem a tical creation has been m ore poten t for
the gen eral on -go of in telligen ce a n d pow er."
T h e d ecim a l sy stem h a s a d ecep tive sim p licity a s a re-su lt of w h ich ch ild ren allover th e w orld learn it even at a ten d er ag e. It h a s a n eco n om y in th e n u m b er of sy m b olsu sed a s w ell a s th e sp a ce o ccu p ied b y a w ritten n u m b er,a n a b ility to e® ortlessly ex p ress arb itrarily la rge n u m -b ers an d , a b ov e a ll, co m p u tatio n al fa cility. T h u s th e tw elve-d igit R o m an n u m b er (D C C C L X X X V III) is sim -p ly 888 in th e d ecim al system !
M ost of th e stan d ard resu lts in b asic a rith m etic a re o f In -d ian orig in . T h is in clu d es n ea t, sy stem atic a n d straig h t-fo rw a rd tech n iq u es o f th e fu n d a m en tal a rith m etic o p er-a tio n s: a d d itio n , su b tractio n , m u ltip lica tion , d iv ision ,tak in g sq u a res an d cu b es, an d ex tra ctin g squ a re a n d cu b e roo ts; th e ru les o f o p eration s w ith fractio n s a n d su rd s; va rio u s ru les on ratio a n d p rop ortion lik e th e ru le
o f th ree; an d several co m m ercial a n d related p ro b lem slik e in co m e an d ex p en d itu re, p ro¯ t a n d lo ss, sim p le a n d co m p ou n d in terest, d isco u nt, p a rtn ersh ip , co m p u tatio n so f th e av erag e im p u rities of g o ld , sp eed s an d d ista n ces,
a n d th e m ix tu re a n d cistern p rob lem s sim ila r to th ose fou n d in m o d ern texts. T h e In d ia n m eth o d s o f p erform -in g lo n g m u ltip lica tio n s a n d d iv ision s w ere in tro d u ced in E u rop e a s late a s th e 1 4 th cen tu ry A D . W e h av e b eco m e so u sed to th e ru les of op eratio n s w ith fraction s th a t w e ten d to ov erlo ok th e fa ct tha t th ey co n ta in id ea s w h ich w ere u n fa m iliar to th e E g y p tia n s, w h o w ere gen erally p ro¯ cien t in a rith m etic, an d th e G reek s, w h o h a d som e o f th e m ost b rillia n t m in d s in th e h isto ry of m ath em a t-
ics. T h e ru le o f th ree, b rou gh t to E u rop e v ia th e A rab s,w as very h igh ly rega rd ed b y m erch an ts d u rin g a n d af-ter th e ren a issa n ce. It ca m e to b e k n ow n a s th e G o ld en R u le fo r its g reat p o p u la rity a n d u tility in co m m ercia lco m p u tatio n s { m u ch sp a ce u sed to b e d ev o ted to th isru le by th e early E u rop ean w riters on arith m etic.
T h e ex cellen ce a n d sk ill atta in ed b y th e In d ia n s in th e fou n d atio n s o f a rith m etic w a s p rim a rily d u e to th e a d -
va n ta g e of th e ea rly d iscov ery o f th e d ecim a l n otation { th e k ey to a ll p rin cip al id ea s in m o d ern a rith m etic. F orin stan ce, th e m o d ern m eth o d s for ex tra ctin g sq u a re a n d cu b e roo ts, d escrib ed b y A ry ab h a ta in th e 5th cen tu ry A D , clev erly u se th e id ea s of p la ce va lu e a n d zero a n d th e a lg eb raic ex p an sio n s of (a + b )2 an d (a + b )3 . T h ese m eth o d s w ere in tro d u ced in E u ro p e o n ly in th e 1 6th cen tu ry A D . A p art from th e ex a ct m eth o d s,In d ian s also in v en ted several in g en io u s m eth o d s fo r d eterm in a tio n of
a p p rox im a te squ a re roo ts o f n o n -squ a re n u m b ers, so m e o f w h ich w e sh a ll m en tion in a su b seq u en t issu e.
D u e to th e g ap s in ou r k n ow led g e a b o u t th e ea rly p h ase o f p ost-V ed ic In d ian m a th em a tics, th e p recise d etailsrega rd in g th e origin o f d ecim al n o ta tio n is n o t k n ow n .T h e co n cep t of zero ex isted b y th e tim e o f P in g ala (d a ted 2 0 0 B C ). T h e id ea o f p la ce-va lu e h a d b een im p licit in
a n cien t S an sk rit term in o log y { as a resu lt, In d ian s cou ld e® o rtlessly h a n d le large n u m b ers righ t from th e V ed ic A ge. T h ere is term in o lo g y fo r a ll m u ltip les o f ten u p to 1 0 1 8 in ea rly V ed ic literatu re, th e R a m ay a n a h a s term s
all th e w ay u p to 10 5 5
, a n d th e J a in a -B u d d h ist tex tssh ow freq u en t u se o f large n u m b ers (u p to 10 1 4 0 !) fo rth eir m ea su rem en ts of sp a ce an d tim e. E x p ressio n s o fsu ch la rg e n u m b ers are n o t fo u n d in th e co n tem p orary w o rk s o f oth er n a tio n s. E v en th e b rillia n t G reek s h a d n o term in o log y fo r d en om in atio n s a b ov e th e m y riad (10 4 )w h ile th e R o m an term in o lo g y stop p ed w ith th e m ille (10 3 ). T h e stru ctu re o f th e S an skrit n u m eral sy stem a n d th e In d ia n lov e for larg e n u m b ers m u st h av e trig g ered
th e crea tio n of th e d ecim a l system .
A s w e sh a ll see later, ev en th e sm a llest p o sitiv e in teg ralsolu tion o f th e eq u ation x 2 ¡ D y 2 = 1 co u ld b e very la rg e;in fa ct, for D = 61, it is (17 663 19 049 ; 226 153 980 ). T h e ea rly In d ia n so lu tio n to th is fairly d eep p rob lem co u ld b e p a rtly a ttrib u ted to th e In d ian s' trad ition a l fa scin a tio n for large nu m b ers an d ab ility to p lay w ith th em .
D u e to th e ab sen ce o f go o d n o tation s, th e G reeks w ere n o t stro n g in th e co m p u tatio n al asp ects o f m a th em a tics { o n e of th e facto rs resp o n sib le fo r th e ev en tu a l d eclin e of G reek m a th em a tics. A rch im ed es (28 7-2 12 B C ) d id realise th e im p orta n ce o f g o o d n otation , an d m a d e n o -tab le p rog ress to ev olv e on e, b u t failed to a n ticip ate th e In d ia n d ecim a l system . A s th e g reat F ren ch m a th em a ti-cia n L ap lace (17 49 -18 2 7) rem arked : \ T he im portan ce o f
this in ven tion is m ore readily appreciated w hen on e con -
siders that it w as beyond the tw o greatest m en of antiq-uity: A rchim edes and A pollonius."
T h e d ecim a l sy stem w a s tra n sm itted to E u rop e th rou gh th e A rab s. T h e S an sk rit w ord \ sunya " w as tran slated in to A rab ic as \ sifr " w h ich w as in tro d u ced in to G er-m an y in th e 1 3th centu ry as \ cifra " from w h ich w e h av e th e w ord \ cipher " . T h e w o rd \ zero " p rob ab ly com es
from th e L atin ised form \ zephirum " of th e A rab ic sifr .L eo n ard o F ib on acci of P isa (11 80 -12 4 0), th e ¯ rst m a -
jor E u rop ean m ath em a ticia n o f th e secon d m illen n iu m ,p lay ed a m a jo r role in th e sp read o f th e In d ia n n u m eral
system in E u rop e. T h e In d ia n n o tation an d a rith m etic ev en tu ally g o t sta n d a rd ised in E u rop e d u rin g th e 1 6 th -17 th cen tu ry.
T h e d ecim al system stim u lated a n d ac celerated trad e an d com m erce as w ell as astron om y an d m ath em atics.It is n o co in cid en ce th a t th e m a th em a tica l an d scien ti c ren a issa n ce b eg a n in E u rop e o n ly after th e In d ia n n o -tatio n w a s a d op ted . In d eed th e d ecim a l n o tation is th e
very p illa r o f all m o d ern civ iliza tio n .
A lg e b ra
W h ile sop h istica ted g eo m etry w as in ven ted d u rin g th e origin o f th e V ed ic ritu a ls, its ax iom atisation an d fu r-th er d evelo p m en t w as d on e b y th e G reek s. T h e h eigh trea ch ed b y th e G reeks in g eo m etry b y th e tim e o f A p ol-lon iu s (26 0-1 7 0 B C ) w a s n o t m a tch ed b y a n y su b seq u en tan cien t o r m ed ieva l civ ilisatio n . B u t p rog ress in geo m e-try p ro p er so o n rea ch ed a p o in t o f stag n a tio n . B etw een th e tim es of P ap p u s (30 0 A D ) { th e last b ig n am e in G reek g eom etry { an d m o d ern E u rop e, B rah m ag u p ta'sb rillia n t th eo rem s (62 8 A D ) on cy clic q u ad rila tera ls co n -stitu te th e solitary g em s in th e h isto ry of g eo m etry.F u rth er p rog ress n eed ed n ew tech n iq u es, in fa ct a co m -p letely n ew a p p roa ch in m ath em a tics. T h is w as p ro -v id ed b y th e em ergen ce an d d evelop m en t o f a n ew d is-
cip lin e { a lg eb ra. It is o n ly a fter th e estab lish m en t ofan a lg eb ra cu ltu re in E u rop ean m ath em atics d u rin g th e 16 th centu ry A D th at a resu rgen ce b ega n in geo m etry th rou g h its alge b raisation b y D esca rtes an d F erm a t in ea rly 1 7 th cen tu ry. In fact, th e a ssim ila tio n an d re¯ n e-m en t of alg eb ra h a d also set th e stag e for th e rem a rka b le strid es in n u m b er th eo ry a n d ca lcu lu s in E u rop e from th e 17 th cen tu ry.
A lgeb ra w as on ly im p licit in th e m a th em a tics o f sev -eral an cien t civ ilisatio n s till it ca m e o u t in th e o p en w ith th e in tro d u ctio n o f literal or sy m b olic a lgeb ra in In d ia . B y th e tim e of A rya b h ata (49 9 A D ) an d B ra h -
m a gu p ta (62 8 A D ), sym b o lic a lg eb ra h a d ev o lve d in In -d ia in to a d istin ct b ran ch of m a th em atics a n d b eca m e o n e of its cen tra l p illars. A fter ev olu tio n th rou gh sev -eral sta ges, alg eb ra h as n ow com e to p lay a key ro le in m o d ern m ath em atics b oth as a n in d ep en d en t area in itsow n righ t a s w ell a s a n in d isp en sa b le too l in oth er ¯ eld s.In fa ct, th e 20 th cen tu ry w itn essed a v ig o ro u s p h a se o fa lg eb raisation of m a th em a tics'. A lgeb ra p rov id es ele-
g a n ce, sim p licity, p recision , clarity an d tech n ica l p ow er
in th e h a n d s of th e m ath em atician s. It is rem a rka b le h ow ea rly th e In d ian s h a d realised th e sig n i a n ce o f al-g eb ra a n d h ow stron g ly th e lea d in g In d ian m a th em ati-cia n s lik e B rah m ag u p ta (62 8 A D ) an d B h a skara II (11 5 0 A D ) a sserted a n d esta b lish ed th e im p o rta n ce of th eirn ew ly -fo u n d ed d iscip lin e as w e sh a ll see in su b seq u en tissu es.
In d ia n s b ega n a sy stem a tic u se o f sym b ols to d en o te u n -
k n ow n q u a n tities a n d a rith m etic o p eratio n s. T h e fou ra rith m etic o p eratio n s w ere d en o ted b y \ y u " , \ k sh ", \g u "a n d \ b h a " w h ich a re th e ¯ rst letters (or a little m o d i-¯ ca tio n ) o f th e co rresp o n d in g S a n sk rit w o rd s yuta (ad -d itio n ), ksaya (su b tractio n ), guna (m u ltip lica tio n ) a n d bhaga (d iv ision ); sim ila rly \ ka " w a s u sed for karani(root),w h ile th e ¯ rst letters o f th e n a m es o f d i® eren t co lo u rsw ere u sed to d en o te d i® eren t u n k n ow n va ria b les. T h isin tro d u ctio n o f sy m b olic rep resen ta tion w as a n im p o r-
tan t step in th e rap id ad va n cem en t of m ath em a tics. W h i-le a ru d im en tary u se o f sy m b ols ca n a lso b e seen in th e G reek tex ts o f D io p h a n tu s, it is in In d ia th a t a lg eb raic form a lism ach iev ed fu ll d ev elop m en t.
T h e In d ian s classi ed an d m a d e a d etailed stu d y ofeq u a tio n s (w h ich w ere ca lled sam i-karana ), in trod u ced n eg a tive n u m b ers to g eth er w ith th e ru les fo r arith m etic
op era tio n s in v o lv in g zero an d n eg a tiv e n u m b ers, d is-co v ered resu lts o n su rd s, d escrib ed solu tio n s o f lin ea ran d q u ad ratic eq u a tio n s, g av e fo rm u la e for a rith m etic an d g eo m etric p rog ressio n as w ell as id en titites in v o lv -
in g su m m a tio n o f ¯ n ite series, an d a p p lied sev era l u sefu lresu lts o n p erm u tation an d co m b in a tio n s in clu d in g th e form u lae for n P r a n d n C r . T h e en la rgem en t o f th e n u m -b er system to in clu d e n eg a tiv e n u m b ers w a s a m om en -tou s step in th e d evelo p m ent o f m a th em a tics. T h an k sto th e ea rly reco gn itio n o f th e ex isten ce o f n eg a tiv e n u m -b ers, th e In d ian s co u ld g iv e a u n i ed trea tm en t o f th e va rio u s form s o f q u ad ra tic eq u ation s (w ith p ositiv e co -e± cien ts), i.e., a x 2 + bx = c ; a x 2 + c = bx ; bx + c = a x 2 .
T h e In d ian s w ere th e ¯ rst to reco g n ise th at a q u ad ratic eq u a tio n h as tw o ro ots. S rid h arach arya (7 50 A D ) ga ve th e w ell-k n ow n m eth o d o f so lv in g a q u a d ratic eq u ation b y co m p letin g th e sq u are { an id ea w ith far-rea ch in g co n seq u en ces in m a th em a tics. T h e P a sca l's tria n g le forq u ick co m p u tation o f n C r is d escrib ed b y H alay u d h a in th e 10 th centu ry A D as M eru -P rastara 7 00 y ears b efore it w a s sta ted b y P a sca l; a n d H a lay u d h a 's M eru -P rastara w as on ly a clari catio n o f a ru le in v en ted b y P in g ala
m ore th an 12 0 0 yea rs ea rlier (a rou n d 20 0 B C )!
T h u s, a s in a rith m etic, m an y top ics in h igh -sch o o l a l-geb ra h ad b een system atically d evelop ed in In d ia. T h isk n ow led g e w en t to E u rop e th rou g h th e A rab s. T h e w o rd yava in A ryabhatiyabhasya of B h a ska ra I (6th cen tu ry A D ) m ea n in g \ to m ix " or \to sep arate" h a s a± n ity w ith th a t o f al-jabr of a l-K h w a rizm i (82 5 A D ) from w h ich th e w ord algeb ra is d erived . In his w id ely acclaim ed text on
h istory of m a th em atics, C a jori ([8 ]) co n clu d es th e ch a p -ter o n In d ia w ith th e follow in g rem a rks: \ ...it is rem ark-
able to w ha t exten t In dian m athem atics e n ters in to the
scien ce o f ou r tim e. B o th the form an d the spirit of the
arithm etic an d algebra of m od ern tim es are essen tially
In dian . T hin k of ou r n otation of n um bers, brou ght to
perfection by the H in du s, thin k of the In dian arithm eti-
cal operation s n early as perfect as ou r o w n , thin k of their
elegant algebraical m ethods, and then judge w hether the
B rahm in s o n the ban ks of the G an ges a re n ot en titled to
som e credit."
B u t an cien t In d ia n algeb ra w en t far b eyo n d th e h igh sch o o l lev el. T h e p in n a cle o f In d ia n a ch iev em en t w a s a t-tain ed in th eir solu tio n s of th e h ard a n d su b tle n u m b er-th eo retic p rob lem s o f ¯ n d in g in teger solu tio n s to eq u a -tio n s of ¯ rst an d secon d d egree. S u ch equ ation s are ca lled in d eterm in a te o r D iop h an tin e eq u a tio n s. B u t alas,th e In d ia n w o rks in th is a rea w ere to o fa r a h ea d of th e tim es to b e n oticed b y con tem p orary an d su b seq u ent
civ ilisation s! A s C a jori lam en ts, \ U n fortu n ately, som e of the m ost brillian t results in in determ in ate an alysis,
found in the H indu w orks, reached E urope too late to ex-
ert the in ° u en ce they w ou ld ha ve exerted, had they com e
tw o or three cen tu ries earlier." W ith ou t som e aw are-n ess o f th e In d ian co n trib u tio n s in th is ¯ eld , it is n o tp o ssib le to get a tru e p ictu re of th e d ep th a n d sk ill at-tain ed in p ost-V ed ic In d ia n m ath em atics th e ch a racterof w h ich w a s p rim a rily alg eb raic. W e sh a ll d iscu ss so m e
of th ese n u m b er-th eo retic co n trib u tio n s from th e n ex tin stalm ent.
T rig o n o m e try a n d C a lc u lu s
A p art from d ev elop in g th e su b ject of a lg eb ra p rop er,In d ia n s a lso b eg a n a p ro cess o f alg eb risation a n d co n se-q u en t sim p li ca tio n o f oth er areas o f m a th em a tics. F o rin stan ce, th ey d ev elop ed trig on o m etry in a system a tic
m an n er, resem b lin g its m o d ern fo rm , a n d im p arted to it its m od ern algeb raic ch aracter. T h e algeb raisation ofth e stu d y o f in n itesim a l ch a n g es led to th e d iscov ery ofkey p rin cip les of calcu lu s b y th e tim e o f B h a ska rach a rya (11 50 A D ) so m e of w h ich w e sh all m ention in a su b -seq u en t issu e. C a lcu lu s in In d ia lea p ed to a n a m azin g h eig h t in th e a n a ly tic trig on o m etry o f th e K erala sch o o lin th e 14 th cen tu ry.
A lth o u g h th e G reek s fo u n d ed trigo n o m etry, th eir p ro-gress w a s h alted d u e to th e a b sen ce of ad eq u ate alge-b raic m a ch in ery an d n o ta tio n s. In d ia n s in ve n ted th e sin e a n d co sin e fu n ctio n s, d iscov ered m o st o f th e sta n -
d ard fo rm u lae a n d id en titites, in clu d in g th e b a sic fo r-m u la fo r sin (A § B ), a n d co n stru cted fa irly a ccu rate sin e ta b les. B rah m a gu p ta (628 A D ) a n d G ov in d asw a m i(88 0 A D ) g av e in terp o latio n fo rm u la e fo r ca lcu la tin g th e sin es o f in term ed ia te a n g les from sin e tab les { th ese a re sp ecial ca ses o f th e N ew ton { S tirlin g a n d N ew ton { G au ssfo rm u la e fo r secon d -o rd er d i® eren ce. R em arka b le a p -p rox im a tio n s fo r ¼ a re g iven in In d ian tex ts in clu d in g 3 :14 16 of A rya b h ata (49 9 A D ), 3 :1 4 1 5 9 2 6 5 3 5 9 o f M a d -
h ava (14 th centu ry A D ) an d 35 5 = 113 of N ilakanta (1500 A D ). A n an on ym ou s w ork K a ran apadd hati (b eliev ed to h av e b een w ritten b y P u tu m a n a S o m ay a jin in th e 1 5th cen tu ry A D ) giv es th e v alu e 3 :141 592 653 589 793 24 w h ich is co rrect u p to sev en teen d ecim a l p lac es.
T h e G reek s h ad in vestig a ted th e relation sh ip b etw ee n a ch o rd o f a circle a n d th e a n g le it su b ten d s a t th e cen -tre { b u t th eir sy stem is q u ite cu m b erso m e in p ractice.
T h e In d ian s realised th e sign i ca n ce o f th e co n n ectio n b etw een a h a lf-ch o rd an d h alf of th e an g le su b ten d ed b y th e fu ll ch o rd . In th e ca se o f a u n it circle, th is isp recisely th e sin e fu n ction . T h e In d ia n h a lf-ch ord w a sin trod u ced in th e A rab w orld d u rin g th e 8th cen tu ry A D . E u rop e w a s in tro d u ced to th is fu n d a m en tal n o tio n th ro u g h th e w ork o f th e A rab sch o la r al-B a ttan i (85 8 -9 2 9 A D ). T h e A rab s p referred th e In d ia n h a lf-ch o rd to P tolem y 's system o f ch o rd s a n d th e alg eb ra ic a p p roa ch
o f th e In d ian s to th e g eo m etric a p p roa ch o f th e G reek s.
T h e S a n sk rit w o rd fo r h a lf-ch o rd \ ardha-jya " , la ter a b -b rev iated as \ jya ", w as w ritten b y th e A rab s a s \ jyb " .C u riou sly, th ere is a sim ila r-sou n d in g A rab w o rd \ jaib "w h ich m ea n s \ h ea rt, b o som , fo ld , b ay o r cu rve" . W h en th e A rab w o rk s w ere b ein g tra n sla ted in to L atin , th e a p p a rently m ea n in g less w ord \ jyb " w as m istaken for th e
w ord \ jaib " an d tra n sla ted a s \ sinus" w h ich h a s severa lm ea n in g s in L atin in clu d in g \ h ea rt, b oso m , fo ld , b ay orcu rve" ! T h is w ord b eca m e \ sin e " in th e E n g lish v ersion .A rya b h a ta's \ kotijya " b ecam e cosine .
T h e trad itio n of ex cellen ce a n d origin ality in In d ia n trig o-n o m etry reach ed a h ig h p ea k in th e o u tstan d in g resu ltsof M a d h ava ch arya (1 34 0-1 42 5 ) on th e p ow er series ex -p a n sio n s o f trigon om etric fu n ctio n s. T h ree cen tu ries b e-fo re G rego ry (16 67 ), M a d h av a h a d d escrib ed th e series
µ = t a n µ ¡ (1 = 3)(tan µ )3 + (1 = 5)(tan µ )5 ¡
(1 = 7)(tan µ )7 + ¢¢¢
(jtan µ
j · 1 ):
H is p ro o f, as p resen ted in Y uktibhasa , in v o lv es th e id ea of in tegratio n as th e lim it of a su m m ation an d corre-sp on d s to th e m o d ern m eth o d o f ex p an sio n a n d term -b y -term in tegratio n . A cru cial step is th e u se of th e resu lt
lim n
(1 p + 2 p + ¢¢¢+ (n ¡ 1 ) p )= n p + 1 = 1 = ( p + 1):
T h e exp licit statem ent th at (jta n µ j · 1 ) rev ea ls th e lev el of sop h istica tion in th e u n d ersta n d in g o f in n ite series in clu d in g an aw aren ess o f co n v ergen ce. M ad h av a a lso d iscov ered th e b ea u tifu l fo rm u la
¼ = 4 = 1 ¡ 1 = 3 + 1 = 5 ¡ 1 = 7 + ¢¢¢;
o b tain ed b y p u ttin g µ = ¼ = 4 in th e M ad h ava {G rego ry series. T h is series w as red iscovered th ree centu ries later
b y L eib n iz (16 74 ). A s o n e o f th e ¯ rst a p p lica tio n s ofh is n ew ly in v en ted ca lcu lu s, L eib n iz w a s th rilled a t th e d iscov ery o f th is series w h ich w a s th e ¯ rst o f th e resu ltsg iv in g a co n n ectio n b etw een ¼ a n d u n it fractio n s. M a d -h av a also d escrib ed th e series
¯ rst g iven in E u rop e b y A S h arp (17 17 ). A g ain , th ree h u n d ered yea rs b efore N ew ton (16 7 6 A D ), M ad h ava h ad d escrib ed th e w ell-k n ow n p o w er series ex p a n sio n s
sin x = x ¡ x
3
= 3 ! + x
5
= 5! ¡ ¢¢¢a n d co s x = 1 ¡ x 2 = 2 ! + x 4 = 4! ¡ ¢¢¢:
T h ese series w ere u sed to co n stru ct a ccu rate sin e a n d co -sin e ta b les for ca lcu lation s in a stro n om y. M a d h ava 's val-u es a re co rrect, in a lm o st all ca ses, to th e eig h th o r n in th d ecim al p lac e { su ch a n a ccu racy w a s n ot to b e a ch iev ed in E u rop e w ith in th ree cen tu ries. M a d h av a's resu ltssh ow th a t ca lcu lu s a n d a n aly sis h ad reach ed rem arka b le
d ep th an d m atu rity in In d ia cen tu ries b efo re N ew ton (16 42 -17 27 ) an d L eib n iz (1 64 6-1 71 6). M ad h ava ch arya m ig h t b e rega rd ed a s th e ¯ rst m a th em a ticia n w h o w o rked in an a ly sis!
U n fo rtu n ately, th e o rig in al tex ts of sev era l o u tsta n d in g m ath em a tician s lik e S rid h a ra , P a d m a n a b h a , J ay a d eva an d M ad h av a h av e n ot b een fou n d y et { it is on ly th rou gh th e o ccasion a l referen ce to som e o f th eir resu lts in su b -
seq u en t com m en ta ries th at w e g et a glim p se of th eirw ork. M ad h ava 's con trib u tio n s are m en tio n ed in sev -era l later tex ts in clu d in g th e T an tra S am grah a (150 0) ofth e great astron om er N ilaka nta S om ay a ji (14 45 -15 4 5)w h o gav e th e h elio cen tric m o d el b efo re C op ern icu s, th e Y uktibhasa (15 40) of Jy esth ad eva (15 00-16 10 ) an d th e an on ym ou s K aran apadd hati. A ll th ese texts th em selvesw ere d iscov ered b y C h a rles W h ish an d p u b lish ed o n ly in 1835.
A m o n g a n cien t m a th em a ticia n s w h o se tex ts h av e b een fou n d , sp ecia lm en tio n m ay b e m a d e o f A rya b h a ta, B rah -m ag u p ta an d B h a ska ra ch a ry a. A ll o f th em w ere em in en tastron om ers a s w ell. W e sh a ll m a k e a b rief m en tion ofsom e o f th eir m a th em a tica l w o rks in su b seq u en t issu es.
T h e In d ia n co n trib u tio n s in a rith m etic, a lg eb ra a n d trigo -n om etry w ere tran sm itted b y th e A rab s an d P ersia n sto E u ro p e. T h e A ra b s also p reserv ed a n d tran sm itted th e G reek h eritag e. A fter m o re th an a th o u sa n d y earso f slu m b er, E u rop e red iscov ered its rich G reek h eritag e a n d ac q u ired som e o f th e fru its o f th e p h en om en al In -d ian p ro gress. It is on th e fou n d a tion form ed b y th e b len d in g o f th e tw o g reat m a th em atica l cu ltu res { th e g eo m etric a n d a x io m a tic trad itio n o f th e G reek s an d th e a lg eb raic an d co m p u tatio n a l trad ition of th e In d ia n s { th at th e m ath em a tica l ren a issa n ce too k p la ce in E u rop e.
In d ia n s m a d e sig n i ca n t co n trib u tio n s in sev eral fron t-lin e a reas o f m a th em a tics d u rin g th e 20 th cen tu ry, esp e-cia lly d u rin g th e seco n d h alf, alth ou g h th is fact is n ot so w e ll-k n ow n am on g stu d en ts p artly b eca u se th e fron tiersof m a th em atics h av e ex p a n d ed far b eyo n d th e scop e ofth e u n iversity cu rricu la . H ow ev er, In d ia n s v irtu a lly to o k n o p art in th e rap id d ev elo p m en t o f m ath em a tics th a ttoo k p la ce d u rin g th e 17 th -19 th cen tu ry { th is p erio d
co in cid ed w ith th e g en eral sta gn a tio n in th e n ation a llife. T h u s, w h ile h ig h -sch o o l m ath em a tics, esp ecia lly in a rith m etic an d a lg eb ra , is m ostly of In d ian o rig in , o n e rarely co m es across In d ia n n a m es in co lleg e an d u n iver-sity cou rses as m ost o f th a t m ath em atics w as created d u rin g th e p erio d ran g in g fro m la te 17 th to ea rly 2 0th cen tu ry. B u t sh ou ld w e fo rg et th e cu ltu re an d g rea t-n ess o f In d ia 's m illen n iu m s b eca u se o f th e ig n ora n ce a n d w ea k n ess of a few cen tu ries?
“Unfortunately,
some of the most
brilliant
results in
indeterminate
analysis, found in
the Hindu works,
reached
Europe too late to
exert the influence
they would have
exerted, had they
come two or threecenturies earlier.’’
F Cajori
Suggested Reading
[1] A Seidenberg, The Origin of Mathematics in Archive for History of Ex-
act Sciences , 1978.
[2] A Seidenberg, The Geometry of Vedic Rituals in Agni, The Vedic Ritual
of the Fire Altar , Vol II, ed F Staal, Asian Humanities Press, Berkeley,
[3] Bibhutibhusan Datta: Ancient Hindu Geometry: The Science of the
Sulbas, Calcutta Univ. Press, 1932, reprinted Cosmo Pub., New Delhi,
1993.
[4] Georges Ifrah, The Universal History of Numbers, John Wiley and Sons,
2000.
[5] G G Joseph, The Crest of The Peacock: Non-European Roots of Math- ematics, Penguin, 1990.
[6] S N Sen, Mathematics Chap 3 of A Concise History of Science in India ,
ed. D M Bose, S N Sen and B V Subbarayappa, INSA, New Delhi , 1971.
[7] G B Halsted, On the foundations and techniques of Arithmetic, Chicago,
1912.
[8] F Cajori, History of Mathematics, Mac Millan, 1931.
[9] Bibhutibhusan Datta, Vedic Mathematics, Chap.3 of The Cultural
Heritage of India Vol VI (Science and Technology) ed. P Ray and S N
Sen, The Ramakrishna Mission Institute of Culture, Calcutta.
[10] B Datta and A N Singh, History of Hindu Mathematics, Asia Publishing
House, Bombay, 1962.
[11] John F Price, Applied Geometry of the Sulba Sutras in Geometry at
Work, ed. C. Gorini, MAA, Washington DC, 2000.
[12] T A Sarasvati Amma, Geometry in Ancient and Medieval India, Motilal
Banarasidass, Delhi , 1999.
[13] S N Sen and A K Bag, Post-Vedic Mathematics, Chap. 4 of The
Cultural Heritage of India Vol. VI ed.PRay and S N Sen, The
Ramakrishna Mission Institute of Culture, Calcutta.
[14] S N Sen and A K Bag, The Sulbasutras, INSA, New Delhi , 1983.
[15] C N Srinivasiengar, The History of Ancient Indian Mathematics, The
World Press, Calcutta, 1967.
“As I look back upon the history of my country,I do not find in the whole world another country which has done quite so much for the improve- ment of the human mind. Therefore I have no words of condemnation for my nation. I tell them, ‘You have done well; only try to do better.’ Great things have been done in the
past in this land, and there is both time and room for greater things to be done yet ... Our ancestors did great things in the past, but we have to grow into a fuller life and march beyond even their great achievement.’’
