Courtesy:BillCasselman

Courtesy:BillCasselman

S&TScience

UnderstandingancientIndianmathematics

S.G.Dani

AportionofadedicationtabletinarockcutVishnutempleinGwaliorbuiltin876AD.Thenumber270seenintheinscriptionfeaturestheoldestextantzeroinIndia.

ABuddhistcaveinscriptiononatraderoutethroughNaneghat,WesternIndia,fromaround100BC,depictingthenumber17withnonplacevaluenumeralsfor10and7.

Itishightimewestudiedourmathematicalheritagewithdiligenceandobjectivity

QuiteoftenIfindthatconversations,withpeoplefromvariouswalksoflife,onancientIndianmathematicsslidetoVedicmathematicsofthe16sutrasfame,whichissupposedtoendowonewithmagicalpowersofcalculation.Actually,the16sutraswereintroducedbyBharatiKrishnaTirthaji,whowastheSankaracharyaofPurifrom1925untilhepassedawayin1960,associatingwiththemproceduresforcertainarithmeticaloralgebraiccomputations.Thus,thissocalledVedicmathematics(VM)isessentiallya20thcenturyphenomenon.Neitherthesutrasnortheproceduresthattheyaresupposedtoyield,orcorrespondto,haveanythingtodowitheithertheVedas,orevenwithanypostVedicmathematicaltraditionofyoreinIndia.Theimagethatitmayconjureup

ofancientrishisengagedinsucharithmeticalexercisesasaretaughttothechildreninthenameofVM,andrepresentingthesolutionsthroughwordstringsofafewwordsinmodernstyledSanskrit,withhardlyanysentencestructureorgrammar,isjusttoofarfromtherealmoftheplausible.Itwouldhaveamountedtoajoke,butfortheauraithasacquiredonaccountofvariousfactors,includingthegeneralignoranceabouttheknowledgeinancienttimes.Itisapitythatalongtraditionofover3,000yearsoflearningandpursuitofmathematicalideashascometobeperceivedbyalargesectionofthepopulacethroughtheprismofsomethingsomundaneandsolackinginsubstancefromamathematicalpointofview,apartfromnotbeinggenuine.TallclaimsThecolossalneglectinvolvedisnotforwantofprideabouttheachievementsofourancientsonthecontrary,thereisalotofwritingonthetopic,popularaswellastechnical,thatisfullofunsubstantiatedclaimsconveyinganalmostsupremeknowledgeourforefathersaresupposedtohavepossessed.Butthereisverylittleunderstandingorappreciation,onanintellectualplane,ofthespecicsoftheirknowledgeorachievementsinrealterms.InthecolonialerathisvarietyofdiscourseemergedasanantithesistothebiasthatwasmanifestintheworksofsomeWesternscholars.Duetotheurgencytorespondtotheadversepropagandaontheonehandandthelackofresourcesinaddressingtheissuesatamoreprofoundlevelontheother,recoursewasoftentakentoshortcuts,whichinvolvedmoreassertivenessthansubstance.TherewereindeedsomeIndianscholars,likeSudhakarDvivedi,whoadheredtoamoreintellectualapproach,buttheywereaminority.Unfortunately,theolddiscoursehascontinuedlongafterthecolonialcontextiswellpast,andlongaftertheworldcommunityhasbeguntoviewtheIndianachievementswithconsiderableobjectivecuriosityandinterest.Itishightimethatweswitchtoamodebettingasovereignandintellectuallyselfreliantsociety,focussingonanobjectivestudyandcriticalassessment,withoutthereferenceframeofwhattheysayandhowwemustassertourselves.AncientIndiahasindeedcontributedagreatdealtotheworld'smathematicalheritage.Thecountryalsowitnessedsteadymathematicaldevelopmentsovermostpartofthelast3,000years,throwingupmanyinterestingmathematicalideaswellaheadoftheirappearanceelsewhereintheworld,thoughattimestheylaggedbehind,especiallyintherecentcenturies.Herearesomeepisodesfromthefascinatingstorythatformsarichfabricofthesustainedintellectualendeavour.VedicknowledgeThemathematicaltraditioninIndiagoesbackatleasttotheVedas.Forcompositionswithabroadscopecoveringallaspectsoflife,spiritualaswellassecular,theVedasshowagreatfascinationforlargenumbers.Asthetransmissionoftheknowledgewasoral,thenumberswerenotwritten,butexpressedascombinationsofpowersof10.Itwouldbereasonabletobelievethatwhenthedecimalplacevaluesystemforwrittennumberscameintobeingitowedagreatdealtothewaynumberswerediscussedintheoldercompositions.Thedecimalplacevaluesystemofwritingnumbers,togetherwiththeuseof0,'isknowntohaveblossomedinIndiaintheearlycenturiesAD,andspreadtotheWestthroughtheintermediacyofthePersiansandtheArabs.Therewereactuallyprecursorstothesystem,andvariouscomponentsofitarefoundinotherancientculturessuchastheBabylonian,Chinese,andMayan.Fromthedecimalrepresentationofthenaturalnumbers,thesystemwastoevolvefurtherintotheformthatisnowcommonplaceandcrucialinvariouswalksoflife,withdecimalfractionsbecomingpartofthenumbersystemin16thcenturyEurope,thoughthisagainhassomeintermediatehistoryinvolvingtheArabs.Theevolutionofthenumbersystemrepresentsamajorphaseinthedevelopmentofmathematicalideas,andarguablycontributedgreatlytotheoveralladvanceofscienceandtechnology.Thecumulativehistoryofthenumbersystemholdsalessonthatprogressofideasisaninclusivephenomenon,andwhilecontributingtotheprocessshouldbeamatterofjoyandpridetothosewithallegiancetotherespectivecontributors,theroleofothersalsooughttobeappreciated.ItiswellknownthatGeometrywaspursuedinIndiainthecontextofconstructionofvedisfortheyajnasoftheVedicperiod.TheSulvasutrascontainelaboratedescriptionsofconstructionofvedisandenunciatevariousgeometricprinciples.ThesewerecomposedintherstmillenniumBC,theearliestBaudhayanaSulvasutradatingbacktoabout800BC.SulvasutrageometrydidnotgoveryfarincomparisontotheEuclideangeometrydevelopedbytheGreeks,whoappearedonthescenealittlelater,intheseventhcenturyBC.Itwas,however,animportantstageofdevelopmentinIndiatoo.TheSulvasutrageometerswereaware,amongotherthings,ofwhatisnowcalledthePythagorastheorem,over200yearsbeforePythagoras(allthefourmajorSulvasutrascontainanexplicitstatementofthetheorem),addressed(withintheframeworkoftheirgeometry)issuessuchasndingacirclewiththesameareaasasquareandviceversa,andworkedoutaverygoodapproximationtothesquarerootoftwo,inthecourseoftheirstudies.Thoughitisgenerallynotrecognised,theSulvasutrageometrywasitselfevolving.Thisisseen,inparticular,fromthedifferencesinthecontentsofthefourmajorextantSulvasutras.Certainrevisionsareespeciallystriking.Forinstance,intheearlySulvasutraperiodtheratioofthecircumferencetothediameterwas,asinotherancientcultures,thoughttobethree,asseeninasutraofBaudhayana,butintheManavaSulvasutra,anewvaluewasproposed,asthreeandonefth.Interestingly,thesutradescribingitendswithanexultationnotahairbreadthremains,andthoughweseethatitisstillsubstantiallyoffthemark,itisagratifyinginstanceofanadvancemade.IntheManavaSulvasutraone

alsondsanimprovementoverthemethoddescribedbyBaudhayanaforndingthecirclewiththesameareaasthatofagivensquare.TheJaintraditionhasalsobeenveryimportantinthedevelopmentofmathematicsinthecountry.UnlikefortheVedicpeople,forJainscholarsthemotivationformathematicscamenotfromritualpractices,whichindeedwereanathematothem,butfromthecontemplationofthecosmos.Jainshadanelaboratecosmographyinwhichmathematicsplayedanintegralrole,andevenlargelyphilosophicalJainworksareseentoincorporatemathematicaldiscussions.NotableamongthetopicsintheearlyJainworks,fromaboutthefifthcenturyBCtothesecondcenturyAD,onemaymentiongeometryofthecircle,arithmeticofnumberswithlargepowersof10,permutationsandcombinations,andcategorisationsofinnities(whosepluralityhadbeenrecognised).AsintheSulvasutratradition,theJainsalsorecognised,aroundthemiddleoftherstmillenniumBC,thattheratioofthecircumferenceofthecircletoitsdiameterisnotthree.InSuryaprajnapti,aJaintextbelievedtobefromthefourthcenturyBC,afterrecallingthetraditionalvaluethreeforit,theauthordiscardsthatinfavourthesquarerootof10.Thisvaluefortheratio,whichisreasonablyclosetotheactualvalue,wasprevalentinIndiaoveralongperiodandisoftenreferredastheJainvalue.ItcontinuedtobeusedlongafterAryabhataintroducedthewellknownvalue3.1416fortheratio.TheJaintextsalsocontainratheruniqueformulaeforlengthsofcirculararcsintermsofthelengthofthecorrespondingchordandthebow(height)overthechord,andalsofortheareaofregionssubtendedbycirculararcstogetherwiththeirchords.ThemeansfortheaccuratedeterminationofthesequantitiesbecameavailableonlyaftertheadventofCalculus.HowtheancientJainscholarsarrivedattheseformulae,whicharecloseapproximations,remainstobeunderstood.JaintraditionAfteralullofafewcenturiesintheearlypartoftherstmillennium,pronouncedmathematicalactivityisseenagainintheJaintraditionfromthe8thcenturyuntilthemiddleofthe14thcentury.GanitasarasangrahaofMahavira,writtenin850,isoneofthewellknownandinuentialworks.Virasena(8thcentury),Sridhara(between850and950),Nemicandra(around980CE),ThakkuraPheru(14thcentury)aresomemorenamesthatmaybementioned.Bythe13thand14thcenturies,IslamicarchitecturehadtakenrootinIndiaandinGanitasarakaumudiofThakkuraPheru,whoservedastreasurerinthecourtoftheKhiljiSultansinDelhi,oneseesacombinationofthenativeJaintraditionwithIndoPersianliterature,includingworkonthecalculationofareasandvolumesinvolvedintheconstructionofdomes,arches,andtentsusedforresidentialpurposes.MathematicalastronomyortheSiddhantatraditionhasbeenthedominantandenduringmathematicaltraditioninIndia.Itourishedalmostcontinuouslyforoversevencenturies,startingwithAryabhata(476550)whoisregardedasthefounderofscienticastronomyinIndia,andextendingtoBhaskaraII(11141185)andbeyond.TheessentialcontinuityofthetraditioncanbeseenfromthelonglistofprominentnamesthatfollowAryabhata,spreadovercenturies:Varahamihirainthesixthcentury,BhaskaraIandBrahmaguptaintheseventhcentury,GovindaswamiandSankaranarayanaintheninthcentury,AryabhataIIandVijayanandiinthe10thcentury,Sripatiinthe11thcentury,BrahmadevaandBhaskaraIIinthe12thcentury,andNarayanaPanditandGanesafromthe14thand16thcenturiesrespectively.Aryabhatiya,writtenin499,isbasictothetradition,andeventothelaterworksoftheKeralaschoolofMadhava(moreonthatlater).Itconsistsof121versesdividedintofourchaptersGitikapada,Ganitapada,KalakriyapadaandGolapada.Therst,whichsetsoutthecosmology,containsalsoaversedescribingatableof24sinedifferencesatintervalsof225minutesofarc.Thesecondchapter,asthenamesuggests,isdevotedtomathematicsperse,andincludesinparticularprocedurestondsquarerootsandcuberoots,anapproximateexpressionforpi'(amountingto3.1416andspeciedtobeapproximate),formulaeforareasandvolumesofvariousgeometricgures,andshadows,formulaeforsumsofconsecutiveintegers,sumsofsquares,sumsofcubesandcomputationofinterest.Theothertwochaptersareconcernedwithastronomy,dealingwithdistancesandrelativemotionsofplanets,eclipsesandsoon.InfluentialworkBrahmagupta'sBrahmasphutasiddhantaisavoluminouswork,especiallyforitstime,onSiddhantaastronomy,inwhichtherearetwochapters,Chapter12andChapter18,devotedtogeneralmathematics.Incidentally,Chapter11isacritiqueonearlierworksincludingAryabhatiyaasinotherhealthyscientificcommunitiesthistraditionalsohadmany,andoftenbitter,controversies.Chapter12iswellknownforitssystematictreatmentofarithmeticoperations,includingwithnegativenumbersthenotionofnegativenumbershadeludedEuropeuntilthemiddleofthesecondmillennium.Thechapteralsocontainsgeometry,includinginparticularhisfamousformulafortheareaofaquadrilateral(statedwithouttheconditionofcyclicityofthequadrilateralthatisneededforitsvalidityapointcriticisedbylatermathematiciansinthetradition).Chapter18isdevotedtothekuttakaandothermethods,includingforsolvingseconddegreeindeterminateequations.AnidentitydescribedintheworkfeaturesalsoinsomecurrentstudieswhereitisreferredastheBrahmaguptaidentity.Apartfromthis,Chapter21hasversesdealingwithtrigonometry.BrahmasphutasiddhantaconsiderablyinfluencedmathematicsintheArabworld,andinturnthelaterdevelopmentsinEurope.BhaskaraIIistheauthorofthefamousmathematicaltextsLilavatiandBijaganita.Apartfrombeinganaccomplishedmathematicianhewasagreatteacherandpopulariserofmathematics.Lilavati,whichliterallymeansonewhoisplayful,'presentsmathematicsinaplayfulway,withseveralversesdirectlyaddressinga

prettyyoungwoman,andexamplespresentedthroughreferencetovariousanimals,trees,ornaments,andsoon.(Legendhasitthatthebookisnamedafterhisdaughterafterherweddingfailedtomaterialiseonaccountofanaccidentwiththeclock,butthereisnohistoricalevidencetothateffect.)Thebookpresents,apartfromvariousintroductoryaspectsofarithmetic,geometryoftrianglesandquadrilaterals,examplesofapplicationsofthePythagorastheorem,trirasika,kuttakamethods,problemsonpermutationsandcombinations,etc.TheBijaganitaisanadvancedleveltreatiseonAlgebra,thefirstindependentworkofitskindinIndiantradition.Operationswithunknowns,kuttakaandchakravalamethodsforsolutionsofindeterminateequationsaresomeofthetopicsdiscussed,togetherwithexamples.Bhaskara'sworkonastronomy,SiddhantasiromaniandKaranakutuhala,containseveralimportantresultsintrigonometry,andalsosomeideasofCalculus.TheworksintheSiddhantatraditionhavebeeneditedonasubstantialscaleandtherearevariouscommentariesavailable,includingmanyfromtheearliercenturies,andworksbyEuropeanauthorssuchasColebrook,andmanyIndianauthorsincludingSudhakaraDvivedi,KuppannaSastriandK.V.Sarma.ThetwovolumebookofDattaandSinghandthebookofSaraswatiAmmaserveasconvenientreferencesformanyresultsknowninthistradition.Variousdetailshavebeendescribed,withacomprehensivediscussion,intherecentbookbyKimPlofker.TheBakhshalimanuscript,whichconsistsof70foliosofbhurjapatra(birchbark),isanotherworkofsignicanceinthestudyofancientIndianmathematics,withmanyopenissuesaroundit.ThemanuscriptwasfoundburiedinaeldnearPeshawar,byafarmer,in1881.ItwasacquiredbytheIndologistA.F.R.Hoernle,whostudieditandpublishedashortaccountonit.HelaterpresentedthemanuscripttotheBodleianLibraryatOxford,whereithasbeensincethen.FacsimilecopiesofallthefolioswerebroughtoutbyKayein1927,whichhavesincethenbeenthesourcematerialforthesubsequentstudies.Thedateofthemanuscripthasbeenasubjectofmuchcontroversysincetheearlyyears,withtheestimateddatesrangingfromtheearlycenturiesofCEtothe12thcentury.TakaoHayashi,whoproducedwhatisperhapsthemostauthoritativeaccountsofar,concludesthatthemanuscriptmaybeassignedsometimebetweentheeighthcenturyandthe12thcentury,whilethemathematicalworkinitmaymostprobablybefromtheseventhcentury.Carbondatingofthemanuscriptcouldsettletheissue,buteffortstowardsthishavenotmaterialisedsofar.Aformulaforextractionofsquarerootsofnonsquarenumbersfoundinthemanuscripthasattractedmuchattention.AnotherinterestingfeatureoftheBakhshalimanuscriptisthatitinvolvescalculationswithlargenumbers(indecimalrepresentation).KeralaschoolLetmenallycometowhatiscalledtheKeralaSchool.Inthe1830s,CharlesWhish,anEnglishcivilservantintheMadrasestablishmentoftheEastIndiaCompany,broughttolightacollectionofmanuscriptsfromamathematicalschoolthatourishedinthenorthcentralpartofKerala,betweenwhatarenowKozhikodeandKochi.Theschool,withalongteacherstudentlineage,lastedforover200yearsfromthelate14thcenturywellintothe17thcentury.ItisseentohaveoriginatedwithMadhava,whohasbeenattributedbyhissuccessorsmanyresultspresentedintheirtexts.ApartfromMadhava,NilakanthaSomayajiwasanotherleadingpersonalityfromtheschool.TherearenoextantworksofMadhavaonmathematics(thoughsomeworksonastronomyareknown).NilakanthaauthoredabookcalledTantrasangraha(inSanskrit)in1500AD.Therehavealsobeenexpositionsandcommentariesbymanyotherexponentsfromtheschool,notableamongthembeingYuktidipikaandKriyakramakaribySankara,andGanitayuktibhashabyJyeshthadevawhichisinMalayalam.Sincethemiddleofthe20thcentury,variousIndianscholarshaveresearchedonthesemanuscriptsandthecontentsofmostofthemanuscriptshavebeenlookedinto.AneditedtranslationofthelatterwasproducedbyK.V.SarmaandithasrecentlybeenpublishedwithexplanatorynotesbyK.Ramasubramanian,M.D.SrinivasandM.S.Sriram.AneditedtranslationofTantrasangrahahasbeenbroughtoutmorerecentlybyK.RamasubramanianandM.S.Sriram.TheKeralaworkscontainmathematicsataconsiderablyadvancedlevelthanearlierworksfromanywhereintheworld.Theyincludeaseriesexpansionforpi'andthearctangentseries,andtheseriesforsineandcosinefunctionsthatwereobtainedinEuropebyGregory,LeibnitzandNewton,respectively,over200yearslater.Somenumericalvaluesforpi'thatareaccurateto11decimalsareahighlightofthework.Inmanyways,theworkoftheKeralamathematiciansanticipatedcalculusasitdevelopedinEuropelater,andinparticularinvolvesmanipulationswithindenitelysmallquantities(inthedeterminationofcircumferenceofthecircleandsoon)analogoustotheinnitesimalsincalculusithasalsobeenarguedbysomeauthorsthattheworkisindeedcalculusalready.HonouringthetraditionAlotneedstobedonetohonourthisrichmathematicalheritage.Theextantmanuscriptsneedtobecaredfortopreventdeterioration,cataloguedproperlywithdueupdatesand,mostimportant,theyneedtobestudieddiligentlyandthendingsplacedinpropercontextonthebroadcanvassoftheworldofmathematics,fromanobjectivestandpoint.Lettheoccasionofthe125thbirthanniversaryofthegeniusofSrinivasaRamanujan,aglobalmathematiciantothecore,inspireusasanation,toapplyourselvestothistask.(TheauthorisDistinguishedProfessor,SchoolofMathematics,TataInstituteofFundamentalResearch,Mumbai.)Keywords:SrinivasaRamanujan,Ramanujanbirthanniversary,mathematicalheritage,Vedicmathematics,Kerala

schoolofmathematics,Sulvasutrageometry,mathematicsandJaintradition,mathematicsandSiddarthatradition