Understanding Ancient Indian Mathematics - The Hindu

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  • 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,andshado

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