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IndianmathematicsFromWikipedia,thefreeencyclopedia

IndianmathematicsemergedintheIndiansubcontinent[1]from1200BCE[2]untiltheendofthe18thcentury.IntheclassicalperiodofIndianmathematics(400CEto1600CE),importantcontributionsweremadebyscholarslikeAryabhata,Brahmagupta,Mahvra,BhaskaraII,MadhavaofSangamagramaandNilakanthaSomayaji.Thedecimalnumbersysteminusetoday[3]wasfirstrecordedinIndianmathematics.[4]Indianmathematiciansmadeearlycontributionstothestudyoftheconceptofzeroasanumber,[5]negativenumbers,[6]

arithmetic,andalgebra.[7]Inaddition,trigonometry[8]wasfurtheradvancedinIndia,and,inparticular,themoderndefinitionsofsineandcosineweredevelopedthere.[9]ThesemathematicalconceptsweretransmittedtotheMiddleEast,China,andEurope[7]andledtofurtherdevelopmentsthatnowformthefoundationsofmanyareasofmathematics.

AncientandmedievalIndianmathematicalworks,allcomposedinSanskrit,usuallyconsistedofasectionofsutrasinwhichasetofrulesorproblemswerestatedwithgreateconomyinverseinordertoaidmemorizationbyastudent.Thiswasfollowedbyasecondsectionconsistingofaprosecommentary(sometimesmultiplecommentariesbydifferentscholars)thatexplainedtheprobleminmoredetailandprovidedjustificationforthesolution.Intheprosesection,theform(andthereforeitsmemorization)wasnotconsideredsoimportantastheideasinvolved.[1][10]Allmathematicalworkswereorallytransmitteduntilapproximately500BCEthereafter,theyweretransmittedbothorallyandinmanuscriptform.TheoldestextantmathematicaldocumentproducedontheIndiansubcontinentisthebirchbarkBakhshaliManuscript,discoveredin1881inthevillageofBakhshali,nearPeshawar(moderndayPakistan)andislikelyfromthe7thcenturyCE.[11][12]

AlaterlandmarkinIndianmathematicswasthedevelopmentoftheseriesexpansionsfortrigonometricfunctions(sine,cosine,andarctangent)bymathematiciansoftheKeralaschoolinthe15thcenturyCE.Theirremarkablework,completedtwocenturiesbeforetheinventionofcalculusinEurope,providedwhatisnowconsideredthefirstexampleofapowerseries(apartfromgeometricseries).[13]However,theydidnotformulateasystematictheoryofdifferentiationandintegration,noristhereanydirectevidenceoftheirresultsbeingtransmittedoutsideKerala.[14][15][16][17]

Contents

1Prehistory2Vedicperiod

2.1SamhitasandBrahmanas2.2ulbaStras

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3Pingala4JainMathematics(400BCE200CE)5OralTradition

5.1Stylesofmemorisation5.2TheSutragenre

6Thewrittentradition:prosecommentary7Numeralsandthedecimalnumbersystem8BakhshaliManuscript9ClassicalPeriod(4001600)

9.1Fifthandsixthcenturies9.2Seventhandeighthcenturies9.3Ninthtotwelfthcenturies

10Keralamathematics(13001600)11ChargesofEurocentrism12Seealso13Notes14SourcebooksinSanskrit15References16Externallinks

Prehistory

ExcavationsatHarappa,MohenjodaroandothersitesoftheIndusValleyCivilisationhaveuncoveredevidenceoftheuseof"practicalmathematics".ThepeopleoftheIVCmanufacturedbrickswhosedimensionswereintheproportion4:2:1,consideredfavourableforthestabilityofabrickstructure.Theyusedastandardisedsystemofweightsbasedontheratios:1/20,1/10,1/5,1/2,1,2,5,10,20,50,100,200,and500,withtheunitweightequalingapproximately28grams(andapproximatelyequaltotheEnglishounceorGreekuncia).Theymassproducedweightsinregulargeometricalshapes,whichincludedhexahedra,barrels,cones,andcylinders,therebydemonstratingknowledgeofbasicgeometry.[18]

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TheinhabitantsofInduscivilisationalsotriedtostandardisemeasurementoflengthtoahighdegreeofaccuracy.TheydesignedarulertheMohenjodarorulerwhoseunitoflength(approximately1.32inchesor3.4centimetres)wasdividedintotenequalparts.BricksmanufacturedinancientMohenjodarooftenhaddimensionsthatwereintegralmultiplesofthisunitoflength.[19][20]

Vedicperiod

SamhitasandBrahmanas

ThereligioustextsoftheVedicPeriodprovideevidencefortheuseoflargenumbers.BythetimeoftheYajurvedasahit(1200900BCE),numbersashighas1012werebeingincludedinthetexts.[2]Forexample,themantra(sacrificialformula)attheendoftheannahoma("foodoblationrite")performedduringtheavamedha,andutteredjustbefore,during,andjustaftersunrise,invokespowersoftenfromahundredtoatrillion:[2]

"Hailtoata("hundred,"102),hailtosahasra("thousand,"103),hailtoayuta("tenthousand,"104),hailtoniyuta("hundredthousand,"105),hailtoprayuta("million,"106),hailtoarbuda("tenmillion,"107),hailtonyarbuda("hundredmillion,"108),hailtosamudra("billion,"109,literally"ocean"),hailtomadhya("tenbillion,"1010,literally"middle"),hailtoanta("hundredbillion,"1011,lit.,"end"),hailtoparrdha("onetrillion,"1012lit.,"beyondparts"),hailtothedawn(us'as),hailtothetwilight(vyui),hailtotheonewhichisgoingtorise(udeyat),hailtotheonewhichisrising(udyat),hailtotheonewhichhasjustrisen(udita),hailtosvarga(theheaven),hailtomartya(theworld),hailtoall."[2]

ThesolutiontopartialfractionwasknowntotheRigvedicPeopleasstatesinthepurushSukta(RV10.90.4)

WiththreefourthsPuruawentup:onefourthofhimagainwashere.

TheSatapathaBrahmana(ca.7thcenturyBCE)containsrulesforritualgeometricconstructionsthataresimilartotheSulbaSutras.[21]

ulbaStras

TheulbaStras(literally,"AphorismsoftheChords"inVedicSanskrit)(c.700400BCE)listrulesfortheconstructionofsacrificialfirealtars.[22]MostmathematicalproblemsconsideredintheulbaStrasspringfrom"asingletheologicalrequirement,"[23]thatofconstructingfirealtarswhichhavedifferentshapesbutoccupythesamearea.Thealtarswererequiredtobeconstructedoffivelayersofburntbrick,with

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thefurtherconditionthateachlayerconsistof200bricksandthatnotwoadjacentlayershavecongruentarrangementsofbricks.[23]

Accordingto(Hayashi2005,p.363),theulbaStrascontain"theearliestextantverbalexpressionofthePythagoreanTheoremintheworld,althoughithadalreadybeenknowntotheOldBabylonians."

Thediagonalrope(akayrajju)ofanoblong(rectangle)producesbothwhichtheflank(prvamni)andthehorizontal(tiryamn)produceseparately."[24]

Sincethestatementisastra,itisnecessarilycompressedandwhattheropesproduceisnotelaboratedon,butthecontextclearlyimpliesthesquareareasconstructedontheirlengths,andwouldhavebeenexplainedsobytheteachertothestudent.[24]

TheycontainlistsofPythagoreantriples,[25]whichareparticularcasesofDiophantineequations.[26]Theyalsocontainstatements(thatwithhindsightweknowtobeapproximate)aboutsquaringthecircleand"circlingthesquare."[27]

Baudhayana(c.8thcenturyBCE)composedtheBaudhayanaSulbaSutra,thebestknownSulbaSutra,whichcontainsexamplesofsimplePythagoreantriples,suchas:(3,4,5),(5,12,13),(8,15,17),(7,24,25),and(12,35,37),[28]aswellasastatementofthePythagoreantheoremforthesidesofasquare:"Theropewhichisstretchedacrossthediagonalofasquareproducesanareadoublethesizeoftheoriginalsquare."[28]ItalsocontainsthegeneralstatementofthePythagoreantheorem(forthesidesofarectangle):"Theropestretchedalongthelengthofthediagonalofarectanglemakesanareawhichtheverticalandhorizontalsidesmaketogether."[28]Baudhayanagivesaformulaforthesquarerootoftwo,[29]

Theformulaisaccurateuptofivedecimalplaces,thetruevaluebeing1.41421356...[30]ThisformulaissimilarinstructuretotheformulafoundonaMesopotamiantablet[31]fromtheOldBabylonianperiod(19001600BCE):[29]

whichexpresses2inthesexagesimalsystem,andwhichisalsoaccurateupto5decimalplaces(afterrounding).

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AccordingtomathematicianS.G.Dani,theBabyloniancuneiformtabletPlimpton322writtenca.1850BCE[32]"containsfifteenPythagoreantripleswithquitelargeentries,including(13500,12709,18541)whichisaprimitivetriple,[33]indicating,inparticular,thattherewassophisticatedunderstandingonthetopic"inMesopotamiain1850BCE."SincethesetabletspredatetheSulbasutrasperiodbyseveralcenturies,takingintoaccountthecontextualappearanceofsomeofthetriples,itisreasonabletoexpectthatsimilarunderstandingwouldhavebeenthereinIndia."[34]Danigoesontosay:

"AsthemainobjectiveoftheSulvasutraswastodescribetheconstructionsofaltarsandthegeometricprinciplesinvolvedinthem,thesubjectofPythagoreantriples,evenifithadbeenwellunderstoodmaystillnothavefeaturedintheSulvasutras.TheoccurrenceofthetriplesintheSulvasutrasiscomparabletomathematicsthatonemayencounterinanintroductorybookonarchitectureoranothersimilarappliedarea,andwouldnotcorresponddirectlytotheoverallknowledgeonthetopicatthattime.Since,unfortunately,noothercontemporaneoussourceshavebeenfounditmayneverbepossibletosettlethisissuesatisfactorily."[34]

Inall,threeSulbaSutraswerecomposed.Theremainingtwo,theManavaSulbaSutracomposedbyManava(fl.750650BCE)andtheApastambaSulbaSutra,composedbyApastamba(c.600BCE),containedresultssimilartotheBaudhayanaSulbaSutra.

Vyakarana

AnimportantlandmarkoftheVedicperiodwastheworkofSanskritgrammarian,Pini(c.520460BCE).HisgrammarincludesearlyuseofBooleanlogic,ofthenulloperator,andofcontextfreegrammars,andincludesaprecursoroftheBackusNaurform(usedinthedescriptionprogramminglanguages).

Pingala

AmongthescholarsofthepostVedicperiodwhocontributedtomathematics,themostnotableisPingala(pigal)(fl.300200BCE),amusicaltheoristwhoauthoredtheChhandasShastra(chandastra,alsoChhandasSutrachhandastra),aSanskrittreatiseonprosody.Thereisevidencethatinhisworkontheenumerationofsyllabiccombinations,PingalastumbleduponboththePascaltriangleandBinomialcoefficients,althoughhedidnothaveknowledgeoftheBinomialtheoremitself.[35][36]Pingala'sworkalsocontainsthebasicideasofFibonaccinumbers(calledmaatraameru).AlthoughtheChandahsutrahasn'tsurvivedinitsentirety,a10thcenturycommentaryonitbyHalyudhahas.Halyudha,whoreferstothePascaltriangleasMeruprastra(literally"thestaircasetoMountMeru"),hasthistosay:

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"Drawasquare.Beginningathalfthesquare,drawtwoothersimilarsquaresbelowitbelowthesetwo,threeothersquares,andsoon.Themarkingshouldbestartedbyputting1inthefirstsquare.Put1ineachofthetwosquaresofthesecondline.Inthethirdlineput1inthetwosquaresattheendsand,inthemiddlesquare,thesumofthedigitsinthetwosquareslyingaboveit.Inthefourthlineput1inthetwosquaresattheends.Inthemiddleonesputthesumofthedigitsinthetwosquaresaboveeach.Proceedinthisway.Oftheselines,thesecondgivesthecombinationswithonesyllable,thethirdthecombinationswithtwosyllables,..."[35]

ThetextalsoindicatesthatPingalawasawareofthecombinatorialidentity:[36]

Katyayana

Katyayana(c.3rdcenturyBCE)isnotableforbeingthelastoftheVedicmathematicians.HewrotetheKatyayanaSulbaSutra,whichpresentedmuchgeometry,includingthegeneralPythagoreantheoremandacomputationofthesquarerootof2correcttofivedecimalplaces.

JainMathematics(400BCE200CE)

AlthoughJainismasareligionandphilosophypredatesitsmostfamousexponent,Mahavira(6thcenturyBCE)whowasacontemporaryofGautamaBuddha,mostJaintextsonmathematicaltopicswerecomposedafterthe6thcenturyBCE.JainmathematiciansareimportanthistoricallyascruciallinksbetweenthemathematicsoftheVedicperiodandthatofthe"Classicalperiod."

AsignificanthistoricalcontributionofJainmathematicianslayintheirfreeingIndianmathematicsfromitsreligiousandritualisticconstraints.Inparticular,theirfascinationwiththeenumerationofverylargenumbersandinfinities,ledthemtoclassifynumbersintothreeclasses:enumerable,innumerableandinfinite.Notcontentwithasimplenotionofinfinity,theywentontodefinefivedifferenttypesofinfinity:theinfiniteinonedirection,theinfiniteintwodirections,theinfiniteinarea,theinfiniteeverywhere,andtheinfiniteperpetually.Inaddition,Jainmathematiciansdevisednotationsforsimplepowers(andexponents)ofnumberslikesquaresandcubes,whichenabledthemtodefinesimplealgebraicequations(beejganitasamikaran).Jainmathematicianswereapparentlyalsothefirsttousethewordshunya(literallyvoidinSanskrit)torefertozero.Morethanamillenniumlater,theirappellationbecametheEnglishword"zero"afteratortuousjourneyoftranslationsandtransliterationsfromIndiatoEurope.(SeeZero:Etymology.)

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InadditiontoSuryaPrajnapti,importantJainworksonmathematicsincludedtheVaishaliGanit(c.3rdcenturyBCE)theSthanangaSutra(fl.300BCE200CE)theAnoyogdwarSutra(fl.200BCE100CE)andtheSatkhandagama(c.2ndcenturyCE).ImportantJainmathematiciansincludedBhadrabahu(d.298BCE),theauthoroftwoastronomicalworks,theBhadrabahaviSamhitaandacommentaryontheSuryaPrajinaptiYativrishamAcharya(c.176BCE),whoauthoredamathematicaltextcalledTiloyapannatiandUmasvati(c.150BCE),who,althoughbetterknownforhisinfluentialwritingsonJainphilosophyandmetaphysics,composedamathematicalworkcalledTattwarthadhigamaSutraBhashya.

OralTradition

MathematiciansofancientandearlymedievalIndiawerealmostallSanskritpandits(paita"learnedman"),[37]whoweretrainedinSanskritlanguageandliterature,andpossessed"acommonstockofknowledgeingrammar(vykaraa),exegesis(mms)andlogic(nyya)."[37]Memorisationof"whatisheard"(rutiinSanskrit)throughrecitationplayedamajorroleinthetransmissionofsacredtextsinancientIndia.Memorisationandrecitationwasalsousedtotransmitphilosophicalandliteraryworks,aswellastreatisesonritualandgrammar.ModernscholarsofancientIndiahavenotedthe"trulyremarkableachievementsoftheIndianpanditswhohavepreservedenormouslybulkytextsorallyformillennia."[38]

Stylesofmemorisation

ProdigousenergywasexpendedbyancientIndiancultureinensuringthatthesetextsweretransmittedfromgenerationtogenerationwithinordinatefidelity.[39]Forexample,memorisationofthesacredVedasincludeduptoelevenformsofrecitationofthesametext.Thetextsweresubsequently"proofread"bycomparingthedifferentrecitedversions.Formsofrecitationincludedthejapha(literally"meshrecitation")inwhicheverytwoadjacentwordsinthetextwerefirstrecitedintheiroriginalorder,thenrepeatedinthereverseorder,andfinallyrepeatedagainintheoriginalorder.[40]Therecitationthusproceededas:

word1word2,word2word1,word1word2word2word3,word3word2,word2word3...

Inanotherformofrecitation,dhvajapha[40](literally"flagrecitation")asequenceofNwordswererecited(andmemorised)bypairingthefirsttwoandlasttwowordsandthenproceedingas:

word1word2,wordN1wordNword2word3,wordN3wordN2..wordN1wordN,word1word2

Themostcomplexformofrecitation,ghanapha(literally"denserecitation"),accordingto(Filliozat2004,p.139),tooktheform:

word1word2,word2word1,word1word2word3,word3word2word1,word1word2word3word2word3,word3word2,word2word3word4,word4word3word2,word2word3word4...

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Thatthesemethodshavebeeneffective,istestifiedtobythepreservationofthemostancientIndianreligioustext,thegveda(ca.1500BCE),asasingletext,withoutanyvariantreadings.[40]Similarmethodswereusedformemorisingmathematicaltexts,whosetransmissionremainedexclusivelyoraluntiltheendoftheVedicperiod(ca.500BCE).

TheSutragenre

MathematicalactivityinancientIndiabeganasapartofa"methodologicalreflexion"onthesacredVedas,whichtooktheformofworkscalledVedgas,or,"AncillariesoftheVeda"(7th4thcenturyBCE).[41]Theneedtoconservethesoundofsacredtextbyuseofik(phonetics)andchhandas(metrics)toconserveitsmeaningbyuseofvykaraa(grammar)andnirukta(etymology)andtocorrectlyperformtheritesatthecorrecttimebytheuseofkalpa(ritual)andjyotia(astrology),gaverisetothesixdisciplinesoftheVedgas.[41]Mathematicsaroseasapartofthelasttwodisciplines,ritualandastronomy(whichalsoincludedastrology).SincetheVedgasimmediatelyprecededtheuseofwritinginancientIndia,theyformedthelastoftheexclusivelyoralliterature.Theywereexpressedinahighlycompressedmnemonicform,thestra(literally,"thread"):

Theknowersofthestraknowitashavingfewphonemes,beingdevoidofambiguity,containingtheessence,facingeverything,beingwithoutpauseandunobjectionable.[41]

Extremebrevitywasachievedthroughmultiplemeans,whichincludedusingellipsis"beyondthetoleranceofnaturallanguage,"[41]usingtechnicalnamesinsteadoflongerdescriptivenames,abridginglistsbyonlymentioningthefirstandlastentries,andusingmarkersandvariables.[41]Thestrascreatetheimpressionthatcommunicationthroughthetextwas"onlyapartofthewholeinstruction.TherestoftheinstructionmusthavebeentransmittedbythesocalledGurushishyaparamparai,'uninterruptedsuccessionfromteacher(guru)tothestudent(isya),'anditwasnotopentothegeneralpublic"andperhapsevenkeptsecret.[42]ThebrevityachievedinastraisdemonstratedinthefollowingexamplefromtheBaudhyanaulbaStra(700BCE).

ThedomesticfirealtarintheVedicperiodwasrequiredbyritualtohaveasquarebaseandbeconstitutedoffivelayersofbrickswith21bricksineachlayer.Onemethodofconstructingthealtarwastodivideonesideofthesquareintothreeequalpartsusingacordorrope,tonextdividethetransverse(orperpendicular)sideintosevenequalparts,andtherebysubdividethesquareinto21congruentrectangles.Thebrickswerethendesignedtobeoftheshapeoftheconstituentrectangleandthelayerwascreated.Toformthenextlayer,thesameformulawasused,butthebrickswerearrangedtransversely.[43]Theprocesswasthenrepeatedthreemoretimes(withalternatingdirections)inordertocompletetheconstruction.IntheBaudhyanaulbaStra,thisprocedureisdescribedinthefollowingwords:

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ThedesignofthedomesticfirealtarintheulbaStra

"II.64.Afterdividingthequadrilateralinseven,onedividesthetransverse[cord]inthree.II.65.Inanotherlayeroneplacesthe[bricks]Northpointing."[43]

Accordingto(Filliozat2004,p.144),theofficiantconstructingthealtarhasonlyafewtoolsandmaterialsathisdisposal:acord(Sanskrit,rajju,f.),twopegs(Sanskrit,anku,m.),andclaytomakethebricks(Sanskrit,iak,f.).Concisionisachievedinthestra,bynotexplicitlymentioningwhattheadjective"transverse"qualifieshowever,fromthefeminineformofthe(Sanskrit)adjectiveused,itiseasilyinferredtoqualify"cord."Similarly,inthesecondstanza,"bricks"arenotexplicitlymentioned,butinferredagainbythefemininepluralformof"Northpointing."Finally,thefirststanza,neverexplicitlysaysthatthefirstlayerofbricksareorientedintheEastWestdirection,butthattooisimpliedbytheexplicitmentionof"Northpointing"inthesecondstanzafor,iftheorientationwasmeanttobethesameinthetwolayers,itwouldeithernotbementionedatallorbeonlymentionedinthefirststanza.Alltheseinferencesaremadebytheofficiantasherecallstheformulafromhismemory.[43]

Thewrittentradition:prosecommentary

Withtheincreasingcomplexityofmathematicsandotherexactsciences,bothwritingandcomputationwererequired.Consequently,manymathematicalworksbegantobewrittendowninmanuscriptsthatwerethencopiedandrecopiedfromgenerationtogeneration.

"Indiatodayisestimatedtohaveaboutthirtymillionmanuscripts,thelargestbodyofhandwrittenreadingmaterialanywhereintheworld.TheliteratecultureofIndiansciencegoesbacktoatleastthefifthcenturyB.C....asisshownbytheelementsofMesopotamianomenliteratureandastronomythatenteredIndiaatthattimeand(were)definitelynot...preservedorally."[44]

Theearliestmathematicalprosecommentarywasthatonthework,ryabhaya(written499CE),aworkonastronomyandmathematics.Themathematicalportionoftheryabhayawascomposedof33stras(inverseform)consistingofmathematicalstatementsorrules,butwithoutanyproofs.[45]However,accordingto(Hayashi2003,p.123),"thisdoesnotnecessarilymeanthattheirauthorsdidnotprovethem.Itwasprobablyamatterofstyleofexposition."FromthetimeofBhaskaraI(600CEonwards),prosecommentariesincreasinglybegantoincludesomederivations(upapatti).BhaskaraI'scommentaryontheryabhaya,hadthefollowingstructure:[45]

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Rule('stra')inversebyryabhaaCommentarybyBhskaraI,consistingof:

Elucidationofrule(derivationswerestillrarethen,butbecamemorecommonlater)Example(uddeaka)usuallyinverse.Setting(nysa/sthpan)ofthenumericaldata.Working(karana)ofthesolution.Verification(pratyayakaraa,literally"tomakeconviction")oftheanswer.Thesebecamerarebythe13thcentury,derivations

orproofsbeingfavouredbythen.[45]

Typically,foranymathematicaltopic,studentsinancientIndiafirstmemorisedthestras,which,asexplainedearlier,were"deliberatelyinadequate"[44]inexplanatorydetails(inordertopithilyconveythebarebonemathematicalrules).Thestudentsthenworkedthroughthetopicsoftheprosecommentarybywriting(anddrawingdiagrams)onchalkanddustboards(i.e.boardscoveredwithdust).Thelatteractivity,astapleofmathematicalwork,wastolaterpromptmathematicianastronomer,Brahmagupta(fl.7thcenturyCE),tocharacteriseastronomicalcomputationsas"dustwork"(Sanskrit:dhulikarman).[46]

Numeralsandthedecimalnumbersystem

ItiswellknownthatthedecimalplacevaluesysteminusetodaywasfirstrecordedinIndia,thentransmittedtotheIslamicworld,andeventuallytoEurope.[47]TheSyrianbishopSeverusSebokhtwroteinthemid7thcenturyCEaboutthe"ninesigns"oftheIndiansforexpressingnumbers.[47]However,how,when,andwherethefirstdecimalplacevaluesystemwasinventedisnotsoclear.[48]

TheearliestextantscriptusedinIndiawastheKharohscriptusedintheGandharacultureofthenorthwest.ItisthoughttobeofAramaicoriginanditwasinusefromthe4thcenturyBCEtothe4thcenturyCE.Almostcontemporaneously,anotherscript,theBrhmscript,appearedonmuchofthesubcontinent,andwouldlaterbecomethefoundationofmanyscriptsofSouthAsiaandSoutheastAsia.Bothscriptshadnumeralsymbolsandnumeralsystems,whichwereinitiallynotbasedonaplacevaluesystem.[49]

TheearliestsurvivingevidenceofdecimalplacevaluenumeralsinIndiaandsoutheastAsiaisfromthemiddleofthefirstmillenniumCE.[50]AcopperplatefromGujarat,Indiamentionsthedate595CE,writteninadecimalplacevaluenotation,althoughthereissomedoubtastotheauthenticityoftheplate.[50]Decimalnumeralsrecordingtheyears683CEhavealsobeenfoundinstoneinscriptionsinIndonesiaandCambodia,whereIndianculturalinfluencewassubstantial.[50]

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Thereareoldertextualsources,althoughtheextantmanuscriptcopiesofthesetextsarefrommuchlaterdates.[51]ProbablytheearliestsuchsourceistheworkoftheBuddhistphilosopherVasumitradatedlikelytothe1stcenturyCE.[51]Discussingthecountingpitsofmerchants,Vasumitraremarks,"When[thesame]claycountingpieceisintheplaceofunits,itisdenotedasone,wheninhundreds,onehundred."[51]Althoughsuchreferencesseemtoimplythathisreadershadknowledgeofadecimalplacevaluerepresentation,the"brevityoftheirallusionsandtheambiguityoftheirdates,however,donotsolidlyestablishthechronologyofthedevelopmentofthisconcept."[51]

Athirddecimalrepresentationwasemployedinaversecompositiontechnique,laterlabelledBhutasankhya(literally,"objectnumbers")usedbyearlySanskritauthorsoftechnicalbooks.[52]Sincemanyearlytechnicalworkswerecomposedinverse,numberswereoftenrepresentedbyobjectsinthenaturalorreligiousworldthatcorrespondencetothemthisallowedamanytoonecorrespondenceforeachnumberandmadeversecompositioneasier.[52]AccordingtoPlofker2009,thenumber4,forexample,couldberepresentedbytheword"Veda"(sincetherewerefourofthesereligioustexts),thenumber32bytheword"teeth"(sinceafullsetconsistsof32),andthenumber1by"moon"(sincethereisonlyonemoon).[52]So,Veda/teeth/moonwouldcorrespondtothedecimalnumeral1324,astheconventionfornumberswastoenumeratetheirdigitsfromrighttoleft.[52]Theearliestreferenceemployingobjectnumbersisaca.269CESanskrittext,Yavanajtaka(literally"Greekhoroscopy")ofSphujidhvaja,aversificationofanearlier(ca.150CE)IndianproseadaptationofalostworkofHellenisticastrology.[53]Suchuseseemstomakethecasethatbythemid3rdcenturyCE,thedecimalplacevaluesystemwasfamiliar,atleasttoreadersofastronomicalandastrologicaltextsinIndia.[52]

IthasbeenhypothesizedthattheIndiandecimalplacevaluesystemwasbasedonthesymbolsusedonChinesecountingboardsfromasearlyasthemiddleofthefirstmillenniumBCE.[54]AccordingtoPlofker2009,

Thesecountingboards,liketheIndiancountingpits,...,hadadecimalplacevaluestructure...Indiansmaywellhavelearnedofthesedecimalplacevalue"rodnumerals"fromChineseBuddhistpilgrimsorothertravelers,ortheymayhavedevelopedtheconceptindependentlyfromtheirearliernonplacevaluesystemnodocumentaryevidencesurvivestoconfirmeitherconclusion."[54]

BakhshaliManuscript

TheoldestextantmathematicalmanuscriptinSouthAsiaistheBakhshaliManuscript,abirchbarkmanuscriptwrittenin"BuddhisthybridSanskrit"[12]intheradscript,whichwasusedinthenorthwesternregionoftheIndiansubcontinentbetweenthe8thand12thcenturiesCE.[55]Themanuscriptwasdiscoveredin1881byafarmerwhiledigginginastoneenclosureinthevillageofBakhshali,nearPeshawar(theninBritishIndiaandnowinPakistan).OfunknownauthorshipandnowpreservedintheBodleianLibraryinOxfordUniversity,themanuscript

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hasbeenvariouslydatedasearlyasthe"earlycenturiesoftheChristianera"[56]andaslateasbetweenthe9thand12thcenturyCE.[57]The7thcenturyCEisnowconsideredaplausibledate,[58]albeitwiththelikelihoodthatthe"manuscriptinitspresentdayformconstitutesacommentaryoracopyofananteriormathematicalwork."[59]

Thesurvivingmanuscripthasseventyleaves,someofwhichareinfragments.Itsmathematicalcontentconsistsofrulesandexamples,writteninverse,togetherwithprosecommentaries,whichincludesolutionstotheexamples.[55]Thetopicstreatedincludearithmetic(fractions,squareroots,profitandloss,simpleinterest,theruleofthree,andregulafalsi)andalgebra(simultaneouslinearequationsandquadraticequations),andarithmeticprogressions.Inaddition,thereisahandfulofgeometricproblems(includingproblemsaboutvolumesofirregularsolids).TheBakhshalimanuscriptalso"employsadecimalplacevaluesystemwithadotforzero."[55]Manyofitsproblemsareofacategoryknownas'equalisationproblems'thatleadtosystemsoflinearequations.OneexamplefromFragmentIII53visthefollowing:

"Onemerchanthassevenasavahorses,asecondhasninehayahorses,andathirdhastencamels.Theyareequallywelloffinthevalueoftheiranimalsifeachgivestwoanimals,onetoeachoftheothers.Findthepriceofeachanimalandthetotalvaluefortheanimalspossessedbyeachmerchant."[60]

Theprosecommentaryaccompanyingtheexamplesolvestheproblembyconvertingittothree(underdetermined)equationsinfourunknownsandassumingthatthepricesareallintegers.[60]

ClassicalPeriod(4001600)

ThisperiodisoftenknownasthegoldenageofIndianMathematics.ThisperiodsawmathematicianssuchasAryabhata,Varahamihira,Brahmagupta,BhaskaraI,Mahavira,BhaskaraII,MadhavaofSangamagramaandNilakanthaSomayajigivebroaderandclearershapetomanybranchesofmathematics.TheircontributionswouldspreadtoAsia,theMiddleEast,andeventuallytoEurope.UnlikeVedicmathematics,theirworksincludedbothastronomicalandmathematicalcontributions.Infact,mathematicsofthatperiodwasincludedinthe'astralscience'(jyotistra)andconsistedofthreesubdisciplines:mathematicalsciences(gaitaortantra),horoscopeastrology(hororjtaka)anddivination(sahit).[46]ThistripartitedivisionisseeninVarhamihira's6thcenturycompilationPancasiddhantika[61](literallypanca,"five,"siddhnta,"conclusionofdeliberation",dated575CE)offiveearlierworks,SuryaSiddhanta,RomakaSiddhanta,PaulisaSiddhanta,VasishthaSiddhantaandPaitamahaSiddhanta,whichwereadaptationsofstillearlierworksofMesopotamian,Greek,Egyptian,RomanandIndianastronomy.Asexplainedearlier,themaintextswerecomposedinSanskritverse,andwerefollowedbyprosecommentaries.[46]

Fifthandsixthcenturies

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SuryaSiddhanta

Thoughitsauthorshipisunknown,theSuryaSiddhanta(c.400)containstherootsofmoderntrigonometry.Becauseitcontainsmanywordsofforeignorigin,someauthorsconsiderthatitwaswrittenundertheinfluenceofMesopotamiaandGreece.[62]

Thisancienttextusesthefollowingastrigonometricfunctionsforthefirsttime:

Sine(Jya).Cosine(Kojya).Inversesine(Otkramjya).

Italsocontainstheearliestusesof:

Tangent.Secant.

LaterIndianmathematicianssuchasAryabhatamadereferencestothistext,whilelaterArabicandLatintranslationswereveryinfluentialinEuropeandtheMiddleEast.

Chhedicalendar

ThisChhedicalendar(594)containsanearlyuseofthemodernplacevalueHinduArabicnumeralsystemnowuseduniversally(seealsoHinduArabicnumerals).

AryabhataI

Aryabhata(476550)wrotetheAryabhatiya.Hedescribedtheimportantfundamentalprinciplesofmathematicsin332shlokas.Thetreatisecontained:

QuadraticequationsTrigonometryThevalueof,correctto4decimalplaces.

AryabhataalsowrotetheAryaSiddhanta,whichisnowlost.Aryabhata'scontributionsinclude:

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Trigonometry:

(Seealso:Aryabhata'ssinetable)

Introducedthetrigonometricfunctions.Definedthesine(jya)asthemodernrelationshipbetweenhalfanangleandhalfachord.Definedthecosine(kojya).Definedtheversine(utkramajya).Definedtheinversesine(otkramjya).Gavemethodsofcalculatingtheirapproximatenumericalvalues.Containstheearliesttablesofsine,cosineandversinevalues,in3.75intervalsfrom0to90,to4decimalplacesofaccuracy.Containsthetrigonometricformulasin(n+1)xsinnx=sinnxsin(n1)x(1/225)sinnx.Sphericaltrigonometry.

Arithmetic:

Continuedfractions.

Algebra:

Solutionsofsimultaneousquadraticequations.Wholenumbersolutionsoflinearequationsbyamethodequivalenttothemodernmethod.Generalsolutionoftheindeterminatelinearequation.

Mathematicalastronomy:

Accuratecalculationsforastronomicalconstants,suchasthe:Solareclipse.Lunareclipse.

Theformulaforthesumofthecubes,whichwasanimportantstepinthedevelopmentofintegralcalculus.[63]

Varahamihira

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Brahmagupta'stheoremstatesthatAF=FD.

Varahamihira(505587)producedthePanchaSiddhanta(TheFiveAstronomicalCanons).Hemadeimportantcontributionstotrigonometry,includingsineandcosinetablesto4decimalplacesofaccuracyandthefollowingformulasrelatingsineandcosinefunctions:

Seventhandeighthcenturies

Inthe7thcentury,twoseparatefields,arithmetic(whichincludedmensuration)andalgebra,begantoemergeinIndianmathematics.Thetwofieldswouldlaterbecalledpgaita(literally"mathematicsofalgorithms")andbjagaita(lit."mathematicsofseeds,"with"seeds"liketheseedsofplantsrepresentingunknownswiththepotentialtogenerate,inthiscase,thesolutionsofequations).[64]Brahmagupta,inhisastronomicalworkBrhmaSphuaSiddhnta(628CE),includedtwochapters(12and18)devotedtothesefields.Chapter12,containing66Sanskritverses,wasdividedintotwosections:"basicoperations"(includingcuberoots,fractions,ratioandproportion,andbarter)and"practicalmathematics"(includingmixture,mathematicalseries,planefigures,stackingbricks,sawingoftimber,andpilingofgrain).[65]Inthelattersection,hestatedhisfamoustheoremonthediagonalsofacyclicquadrilateral:[65]

Brahmagupta'stheorem:Ifacyclicquadrilateralhasdiagonalsthatareperpendiculartoeachother,thentheperpendicularlinedrawnfromthepointofintersectionofthediagonalstoanysideofthequadrilateralalwaysbisectstheoppositeside.

Chapter12alsoincludedaformulafortheareaofacyclicquadrilateral(ageneralisationofHeron'sformula),aswellasacompletedescriptionofrationaltriangles(i.e.triangleswithrationalsidesandrationalareas).

Brahmagupta'sformula:Thearea,A,ofacyclicquadrilateralwithsidesoflengthsa,b,c,d,respectively,isgivenby

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wheres,thesemiperimeter,givenby

Brahmagupta'sTheoremonrationaltriangles:Atrianglewithrationalsides andrationalareaisoftheform:

forsomerationalnumbers and .[66]

Chapter18contained103Sanskritverseswhichbeganwithrulesforarithmeticaloperationsinvolvingzeroandnegativenumbers[65]andisconsideredthefirstsystematictreatmentofthesubject.Therules(whichincluded and )wereallcorrect,withone

exception: .[65]Laterinthechapter,hegavethefirstexplicit(althoughstillnotcompletelygeneral)solutionofthequadraticequation:

Totheabsolutenumbermultipliedbyfourtimesthe[coefficientofthe]square,addthesquareofthe[coefficientofthe]middletermthesquarerootofthesame,lessthe[coefficientofthe]middleterm,beingdividedbytwicethe[coefficientofthe]squareisthevalue.

Thisisequivalentto:

Alsoinchapter18,Brahmaguptawasabletomakeprogressinfinding(integral)solutionsofPell'sequation,[68]

where isanonsquareinteger.Hedidthisbydiscoveringthefollowingidentity:[68]

Brahmagupta'sIdentity: whichwasageneralisationofanearlier

identityofDiophantus:[68]Brahmaguptausedhisidentitytoprovethefollowinglemma:[68]

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Lemma(Brahmagupta):If isasolutionof and, isasolutionof,then:

isasolutionof

Hethenusedthislemmatobothgenerateinfinitelymany(integral)solutionsofPell'sequation,givenonesolution,andstatethefollowingtheorem:

Theorem(Brahmagupta):Iftheequation hasanintegersolutionforanyoneof thenPell'sequation:

alsohasanintegersolution.[69]

Brahmaguptadidnotactuallyprovethetheorem,butratherworkedoutexamplesusinghismethod.Thefirstexamplehepresentedwas:[68]

Example(Brahmagupta):Findintegers suchthat:

Inhiscommentary,Brahmaguptaadded,"apersonsolvingthisproblemwithinayearisamathematician."[68]Thesolutionheprovidedwas:

BhaskaraI

BhaskaraI(c.600680)expandedtheworkofAryabhatainhisbookstitledMahabhaskariya,AryabhatiyabhashyaandLaghubhaskariya.Heproduced:

Solutionsofindeterminateequations.Arationalapproximationofthesinefunction.Aformulaforcalculatingthesineofanacuteanglewithouttheuseofatable,correcttotwodecimalplaces.

Ninthtotwelfthcenturies

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Virasena

Virasena(8thcentury)wasaJainmathematicianinthecourtofRashtrakutaKingAmoghavarshaofManyakheta,Karnataka.HewrotetheDhavala,acommentaryonJainmathematics,which:

Dealswiththeconceptofardhaccheda,thenumberoftimesanumbercouldbehalvedeffectivelylogarithmstobase2,andlists

variousrulesinvolvingthisoperation.[70][71]

Firstuseslogarithmstobase3(trakacheda)andbase4(caturthacheda).

Virasenaalsogave:

Thederivationofthevolumeofafrustumbyasortofinfiniteprocedure.

ItisthoughtthatmuchofthemathematicalmaterialintheDhavalacanattributedtopreviouswriters,especiallyKundakunda,Shamakunda,Tumbulura,SamantabhadraandBappadevaanddatewhowrotebetween200and600CE.[71]

Mahavira

MahaviraAcharya(c.800870)fromKarnataka,thelastofthenotableJainmathematicians,livedinthe9thcenturyandwaspatronisedbytheRashtrakutakingAmoghavarsha.HewroteabooktitledGanitSaarSangrahaonnumericalmathematics,andalsowrotetreatisesaboutawiderangeofmathematicaltopics.Theseincludethemathematicsof:

ZeroSquaresCubessquareroots,cuberoots,andtheseriesextendingbeyondthesePlanegeometrySolidgeometryProblemsrelatingtothecastingofshadowsFormulaederivedtocalculatetheareaofanellipseandquadrilateralinsideacircle.

Mahaviraalso:

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AssertedthatthesquarerootofanegativenumberdidnotexistGavethesumofaserieswhosetermsaresquaresofanarithmeticalprogression,andgaveempiricalrulesforareaandperimeterofanellipse.Solvedcubicequations.Solvedquarticequations.Solvedsomequinticequationsandhigherorderpolynomials.Gavethegeneralsolutionsofthehigherorderpolynomialequations:

Solvedindeterminatequadraticequations.Solvedindeterminatecubicequations.Solvedindeterminatehigherorderequations.

Shridhara

Shridhara(c.870930),wholivedinBengal,wrotethebookstitledNavShatika,TriShatikaandPatiGanita.Hegave:

Agoodruleforfindingthevolumeofasphere.Theformulaforsolvingquadraticequations.

ThePatiGanitaisaworkonarithmeticandmensuration.Itdealswithvariousoperations,including:

ElementaryoperationsExtractingsquareandcuberoots.Fractions.Eightrulesgivenforoperationsinvolvingzero.Methodsofsummationofdifferentarithmeticandgeometricseries,whichweretobecomestandardreferencesinlaterworks.

Manjula

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Aryabhata'sdifferentialequationswereelaboratedinthe10thcenturybyManjula(alsoMunjala),whorealisedthattheexpression[72]

couldbeapproximatelyexpressedas

HeunderstoodtheconceptofdifferentiationaftersolvingthedifferentialequationthatresultedfromsubstitutingthisexpressionintoAryabhata'sdifferentialequation.[72]

AryabhataII

AryabhataII(c.9201000)wroteacommentaryonShridhara,andanastronomicaltreatiseMahaSiddhanta.TheMahaSiddhantahas18chapters,anddiscusses:

Numericalmathematics(AnkGanit).Algebra.Solutionsofindeterminateequations(kuttaka).

Shripati

ShripatiMishra(10191066)wrotethebooksSiddhantaShekhara,amajorworkonastronomyin19chapters,andGanitTilaka,anincompletearithmeticaltreatisein125versesbasedonaworkbyShridhara.Heworkedmainlyon:

Permutationsandcombinations.Generalsolutionofthesimultaneousindeterminatelinearequation.

HewasalsotheauthorofDhikotidakarana,aworkoftwentyverseson:

Solareclipse.Lunareclipse.

TheDhruvamanasaisaworkof105verseson:

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Calculatingplanetarylongitudeseclipses.planetarytransits.

NemichandraSiddhantaChakravati

NemichandraSiddhantaChakravati(c.1100)authoredamathematicaltreatisetitledGomematSaar.

BhaskaraII

BhskaraII(11141185)wasamathematicianastronomerwhowroteanumberofimportanttreatises,namelytheSiddhantaShiromani,Lilavati,Bijaganita,GolaAddhaya,GrihaGanitamandKaranKautoohal.AnumberofhiscontributionswerelatertransmittedtotheMiddleEastandEurope.Hiscontributionsinclude:

Arithmetic:

InterestcomputationArithmeticalandgeometricalprogressionsPlanegeometrySolidgeometryTheshadowofthegnomonSolutionsofcombinationsGaveaprooffordivisionbyzerobeinginfinity.

Algebra:

Therecognitionofapositivenumberhavingtwosquareroots.Surds.Operationswithproductsofseveralunknowns.Thesolutionsof:

Quadraticequations.Cubicequations.

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Quarticequations.Equationswithmorethanoneunknown.Quadraticequationswithmorethanoneunknown.ThegeneralformofPell'sequationusingthechakravalamethod.Thegeneralindeterminatequadraticequationusingthechakravalamethod.Indeterminatecubicequations.Indeterminatequarticequations.Indeterminatehigherorderpolynomialequations.

Geometry:

GaveaproofofthePythagoreantheorem.

Calculus:

Conceivedofdifferentialcalculus.Discoveredthederivative.Discoveredthedifferentialcoefficient.Developeddifferentiation.StatedRolle'stheorem,aspecialcaseofthemeanvaluetheorem(oneofthemostimportanttheoremsofcalculusandanalysis).Derivedthedifferentialofthesinefunction.Computed,correcttofivedecimalplaces.CalculatedthelengthoftheEarth'srevolutionaroundtheSunto9decimalplaces.

Trigonometry:

DevelopmentsofsphericaltrigonometryThetrigonometricformulas:

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Keralamathematics(13001600)

TheKeralaschoolofastronomyandmathematicswasfoundedbyMadhavaofSangamagramainKerala,SouthIndiaandincludedamongitsmembers:Parameshvara,NeelakantaSomayaji,Jyeshtadeva,AchyutaPisharati,MelpathurNarayanaBhattathiriandAchyutaPanikkar.Itflourishedbetweenthe14thand16thcenturiesandtheoriginaldiscoveriesoftheschoolseemstohaveendedwithNarayanaBhattathiri(15591632).Inattemptingtosolveastronomicalproblems,theKeralaschoolastronomersindependentlycreatedanumberofimportantmathematicsconcepts.Themostimportantresults,seriesexpansionfortrigonometricfunctions,weregiveninSanskritverseinabookbyNeelakantacalledTantrasangrahaandacommentaryonthisworkcalledTantrasangrahavakhyaofunknownauthorship.Thetheoremswerestatedwithoutproof,butproofsfortheseriesforsine,cosine,andinversetangentwereprovidedacenturylaterintheworkYuktibh(c.1500c.1610),writteninMalayalam,byJyesthadeva,andalsoinacommentaryonTantrasangraha.[73]

TheirdiscoveryofthesethreeimportantseriesexpansionsofcalculusseveralcenturiesbeforecalculuswasdevelopedinEuropebyIsaacNewtonandGottfriedLeibnizwasanachievement.However,theKeralaSchooldidnotinventcalculus,[74]because,whiletheywereabletodevelopTaylorseriesexpansionsfortheimportanttrigonometricfunctions,differentiation,termbytermintegration,convergencetests,iterativemethodsforsolutionsofnonlinearequations,andthetheorythattheareaunderacurveisitsintegral,theydevelopedneitheratheoryofdifferentiationorintegration,northefundamentaltheoremofcalculus.[75]TheresultsobtainedbytheKeralaschoolinclude:

The(infinite)geometricseries: [76]Thisformulawasalreadyknown,for

example,intheworkofthe10thcenturyArabmathematicianAlhazen(theLatinisedformofthenameIbnAlHaytham(9651039)).[77]

Asemirigorousproof(see"induction"remarkbelow)oftheresult: forlargen.Thisresultwasalso

knowntoAlhazen.[73]

Intuitiveuseofmathematicalinduction,however,theinductivehypothesiswasnotformulatedoremployedinproofs.[73]

Applicationsofideasfrom(whatwastobecome)differentialandintegralcalculustoobtain(TaylorMaclaurin)infiniteseriesfor

, ,and [74]TheTantrasangrahavakhyagivestheseriesinverse,whichwhentranslatedtomathematicalnotation,can

bewrittenas:[73]

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where,forr=1,theseriesreducestothestandardpowerseriesforthesetrigonometricfunctions,forexample:

and

Useofrectification(computationoflength)ofthearcofacircletogiveaproofoftheseresults.(ThelatermethodofLeibniz,using

quadrature(i.e.computationofareaunderthearcofthecircle,wasnotused.)[73]

Useofseriesexpansionof toobtainaninfiniteseriesexpression(laterknownasGregoryseries)for :[73]

Arationalapproximationoferrorforthefinitesumoftheirseriesofinterest.Forexample,theerror, ,(fornodd,andi=1,2,3)fortheseries:

Manipulationoferrortermtoderiveafasterconvergingseriesfor :[73]

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Usingtheimprovedseriestoderivearationalexpression,[73]104348/33215forcorrectuptoninedecimalplaces,i.e.3.141592653.

Useofanintuitivenotionoflimittocomputetheseresults.[73]

Asemirigorous(seeremarkonlimitsabove)methodofdifferentiationofsometrigonometricfunctions.[75]However,theydidnotformulatethenotionofafunction,orhaveknowledgeoftheexponentialorlogarithmicfunctions.

TheworksoftheKeralaschoolwerefirstwrittenupfortheWesternworldbyEnglishmanC.M.Whishin1835.AccordingtoWhish,theKeralamathematicianshad"laidthefoundationforacompletesystemoffluxions"andtheseworksabounded"withfluxionalformsandseriestobefoundinnoworkofforeigncountries."[78]

However,Whish'sresultswerealmostcompletelyneglected,untiloveracenturylater,whenthediscoveriesoftheKeralaschoolwereinvestigatedagainbyC.Rajagopalandhisassociates.TheirworkincludescommentariesontheproofsofthearctanseriesinYuktibhgivenintwopapers,[79][80]acommentaryontheYuktibh'sproofofthesineandcosineseries[81]andtwopapersthatprovidetheSanskritversesoftheTantrasangrahavakhyafortheseriesforarctan,sin,andcosine(withEnglishtranslationandcommentary).[82][83]

TheKeralamathematiciansincludedNarayanaPandit(c.13401400),whocomposedtwoworks,anarithmeticaltreatise,GanitaKaumudi,andanalgebraictreatise,BijganitaVatamsa.NarayanaisalsothoughttobetheauthorofanelaboratecommentaryofBhaskaraII'sLilavati,titledKarmapradipika(orKarmaPaddhati).MadhavaofSangamagrama(c.13401425)wasthefounderoftheKeralaSchool.AlthoughitispossiblethathewroteKaranaPaddhatiaworkwrittensometimebetween1375and1475,allwereallyknowofhisworkcomesfromworksoflaterscholars.

Parameshvara(c.13701460)wrotecommentariesontheworksofBhaskaraI,AryabhataandBhaskaraII.HisLilavatiBhasya,acommentaryonBhaskaraII'sLilavati,containsoneofhisimportantdiscoveries:aversionofthemeanvaluetheorem.NilakanthaSomayaji(14441544)composedtheTantraSamgraha(which'spawned'alateranonymouscommentaryTantrasangrahavyakhyaandafurthercommentarybythenameYuktidipaika,writtenin1501).HeelaboratedandextendedthecontributionsofMadhava.

Citrabhanu(c.1530)wasa16thcenturymathematicianfromKeralawhogaveintegersolutionsto21typesofsystemsoftwosimultaneousalgebraicequationsintwounknowns.Thesetypesareallthepossiblepairsofequationsofthefollowingsevenforms:

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Foreachcase,Citrabhanugaveanexplanationandjustificationofhisruleaswellasanexample.Someofhisexplanationsarealgebraic,whileothersaregeometric.Jyesthadeva(c.15001575)wasanothermemberoftheKeralaSchool.HiskeyworkwastheYuktibh(writteninMalayalam,aregionallanguageofKerala).JyesthadevapresentedproofsofmostmathematicaltheoremsandinfiniteseriesearlierdiscoveredbyMadhavaandotherKeralaSchoolmathematicians.

ChargesofEurocentrism

IthasbeensuggestedthatIndiancontributionstomathematicshavenotbeengivendueacknowledgementinmodernhistoryandthatmanydiscoveriesandinventionsbyIndianmathematiciansarepresentlyculturallyattributedtotheirWesterncounterparts,asaresultofEurocentrism.AccordingtoG.G.Joseph'stakeon"Ethnomathematics":

[Theirwork]takesonboardsomeoftheobjectionsraisedabouttheclassicalEurocentrictrajectory.Theawareness[ofIndianandArabicmathematics]isalltoolikelytobetemperedwithdismissiverejectionsoftheirimportancecomparedtoGreekmathematics.ThecontributionsfromothercivilisationsmostnotablyChinaandIndia,areperceivedeitherasborrowersfromGreeksourcesorhavingmadeonlyminorcontributionstomainstreammathematicaldevelopment.Anopennesstomorerecentresearchfindings,especiallyinthecaseofIndianandChinesemathematics,issadlymissing"[84]

Thehistorianofmathematics,FlorianCajori,suggestedthatheandothers"suspectthatDiophantusgothisfirstglimpseofalgebraicknowledgefromIndia."[85]However,healsowrotethat"itiscertainthatportionsofHindumathematicsareofGreekorigin".[86]

Morerecently,asdiscussedintheabovesection,theinfiniteseriesofcalculusfortrigonometricfunctions(rediscoveredbyGregory,Taylor,andMaclaurininthelate17thcentury)weredescribed(withproofs)inIndia,bymathematiciansoftheKeralaschool,remarkablysometwocenturiesearlier.SomescholarshaverecentlysuggestedthatknowledgeoftheseresultsmighthavebeentransmittedtoEuropethroughthetraderoutefromKeralabytradersandJesuitmissionaries.[87]KeralawasincontinuouscontactwithChinaandArabia,and,fromaround1500,withEurope.Theexistenceofcommunicationroutesandasuitablechronologycertainlymakesuchatransmissionapossibility.However,thereisnodirectevidencebywayofrelevantmanuscriptsthatsuchatransmissionactuallytookplace.[87]AccordingtoDavidBressoud,"thereisnoevidencethattheIndianworkofserieswasknownbeyondIndia,orevenoutsideofKerala,untilthenineteenthcentury."[74][88]

BothArabandIndianscholarsmadediscoveriesbeforethe17thcenturythatarenowconsideredapartofcalculus.[75]However,theywerenotable,asNewtonandLeibnizwere,to"combinemanydifferingideasunderthetwounifyingthemesofthederivativeandtheintegral,showtheconnectionbetweenthetwo,andturncalculusintothegreatproblemsolvingtoolwehavetoday."[75]TheintellectualcareersofbothNewtonandLeibnizarewelldocumentedandthereisnoindicationoftheirworknotbeingtheirown[75]however,itisnotknownwithcertaintywhethertheimmediatepredecessorsofNewtonandLeibniz,"including,inparticular,FermatandRoberval,learnedofsomeoftheideasof

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theIslamicandIndianmathematiciansthroughsourceswearenotnowaware."[75]Thisisanactiveareaofcurrentresearch,especiallyinthemanuscriptscollectionsofSpainandMaghreb,researchthatisnowbeingpursued,amongotherplaces,attheCentreNationaldeRechercheScientifiqueinParis.[75]

Seealso

ShulbaSutrasKeralaschoolofastronomyandmathematicsSuryaSiddhantaBrahmaguptaBakhshalimanuscriptListofIndianmathematiciansIndianscienceandtechnologyIndianlogicIndianastronomyHistoryofmathematicsListofnumbersinHinduscriptures

Notes

1. ^abEncyclopaediaBritannica(KimPlofker)2007,p.1

2. ^abcd(Hayashi2005,pp.360361)3. ^Ifrah2000,p.346:"ThemeasureofthegeniusofIndiancivilisation,towhichweoweourmodern(number)system,isallthegreaterinthatitwasthe

onlyoneinallhistorytohaveachievedthistriumph.Someculturessucceeded,earlierthantheSouthAsiancultures,indiscoveringoneoratbesttwoofthecharacteristicsofthisintellectualfeat.Butnoneofthemmanagedtobringtogetherintoacompleteandcoherentsystemthenecessaryandsufficientconditionsforanumbersystemwiththesamepotentialasourown."

4. ^Plofker2009,pp.4447

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5. ^Bourbaki1998,p.46:"...ourdecimalsystem,which(bytheagencyoftheArabs)isderivedfromHindumathematics,whereitsuseisattestedalreadyfromthefirstcenturiesofourera.Itmustbenotedmoreoverthattheconceptionofzeroasanumberandnotasasimplesymbolofseparation)anditsintroductionintocalculations,alsocountamongsttheoriginalcontributionoftheHindus."

6. ^Bourbaki1998,p.49:Modernarithmeticwasknownduringmedievaltimesas"ModusIndorum"ormethodoftheIndians.LeonardoofPisawrotethatcomparedtomethodoftheIndiansallothermethodsisamistake.ThismethodoftheIndiansisnoneotherthanourverysimplearithmeticofaddition,subtraction,multiplicationanddivision.RulesforthesefoursimpleprocedureswasfirstwrittendownbyBrahmaguptaduring7thcenturyCE."Onthispoint,theHindusarealreadyconsciousoftheinterpretationthatnegativenumbersmusthaveincertaincases(adebtinacommercialproblem,forinstance).Inthefollowingcenturies,asthereisadiffusionintotheWest(byintermediaryoftheArabs)ofthemethodsandresultsofGreekandHindumathematics,onebecomesmoreusedtothehandlingofthesenumbers,andonebeginstohaveother"representation"forthemwhicharegeometricordynamic."

7. ^ab"algebra"2007.BritannicaConciseEncyclopedia(http://www.britannica.com/ebc/article231064).EncyclopdiaBritannicaOnline.16May2007.Quote:"Afullfledgeddecimal,positionalsystemcertainlyexistedinIndiabythe9thcentury(CE),yetmanyofitscentralideashadbeentransmittedwellbeforethattimetoChinaandtheIslamicworld.Indianarithmetic,moreover,developedconsistentandcorrectrulesforoperatingwithpositiveandnegativenumbersandfortreatingzerolikeanyothernumber,eveninproblematiccontextssuchasdivision.SeveralhundredyearspassedbeforeEuropeanmathematiciansfullyintegratedsuchideasintothedevelopingdisciplineofalgebra."

8. ^(Pingree2003,p.45)Quote:"Geometry,anditsbranchtrigonometry,wasthemathematicsIndianastronomersusedmostfrequently.Greekmathematiciansusedthefullchordandneverimaginedthehalfchordthatweusetoday.HalfchordwasfirstusedbyAryabhatawhichmadetrigonometrymuchmoresimple.Infact,theIndianastronomersinthethirdorfourthcentury,usingaprePtolemaicGreektableofchords,producedtablesofsinesandversines,fromwhichitwastrivialtoderivecosines.Thisnewsystemoftrigonometry,producedinIndia,wastransmittedtotheArabsinthelateeighthcenturyandbythem,inanexpandedform,totheLatinWestandtheByzantineEastinthetwelfthcentury."

9. ^(Bourbaki1998,p.126):"Asfortrigonometry,itisdisdainedbygeometersandabandonedtosurveyorsandastronomersitistheselatter(Aristarchus,Hipparchus,Ptolemy)whoestablishthefundamentalrelationsbetweenthesidesandanglesofarightangledtriangle(planeorspherical)anddrawupthefirsttables(theyconsistoftablesgivingthechordofthearccutoutbyanangle onacircleofradiusr,inotherwordsthenumber theintroductionofthesine,moreeasilyhandled,isduetoHindumathematiciansoftheMiddleAges)."

10. ^Filliozat2004,pp.14014311. ^Hayashi1995

12. ^abEncyclopaediaBritannica(KimPlofker)2007,p.613. ^Stillwell2004,p.173

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14. ^Bressoud2002,p.12Quote:"ThereisnoevidencethattheIndianworkonserieswasknownbeyondIndia,orevenoutsideKerala,untilthenineteenthcentury.GoldandPingreeassert[4]thatbythetimetheseserieswererediscoveredinEurope,theyhad,forallpracticalpurposes,beenlosttoIndia.Theexpansionsofthesine,cosine,andarctangenthadbeenpasseddownthroughseveralgenerationsofdisciples,buttheyremainedsterileobservationsforwhichnoonecouldfindmuchuse."

15. ^Plofker2001,p.293Quote:"ItisnotunusualtoencounterindiscussionsofIndianmathematicssuchassertionsasthattheconceptofdifferentiationwasunderstood[inIndia]fromthetimeofManjula(...inthe10thcentury)[Joseph1991,300],orthat"wemayconsiderMadhavatohavebeenthefounderofmathematicalanalysis"(Joseph1991,293),orthatBhaskaraIImayclaimtobe"theprecursorofNewtonandLeibnizinthediscoveryoftheprincipleofthedifferentialcalculus"(Bag1979,294)....Thepointsofresemblance,particularlybetweenearlyEuropeancalculusandtheKeraleseworkonpowerseries,haveeveninspiredsuggestionsofapossibletransmissionofmathematicalideasfromtheMalabarcoastinorafterthe15thcenturytotheLatinscholarlyworld(e.g.,in(Bag1979,285))....Itshouldbeborneinmind,however,thatsuchanemphasisonthesimilarityofSanskrit(orMalayalam)andLatinmathematicsrisksdiminishingourabilityfullytoseeandcomprehendtheformer.TospeakoftheIndian"discoveryoftheprincipleofthedifferentialcalculus"somewhatobscuresthefactthatIndiantechniquesforexpressingchangesintheSinebymeansoftheCosineorviceversa,asintheexampleswehaveseen,remainedwithinthatspecifictrigonometriccontext.Thedifferential"principle"wasnotgeneralisedtoarbitraryfunctionsinfact,theexplicitnotionofanarbitraryfunction,nottomentionthatofitsderivativeoranalgorithmfortakingthederivative,isirrelevanthere"

16. ^Pingree1992,p.562Quote:"OneexampleIcangiveyourelatestotheIndianMdhava'sdemonstration,inabout1400A.D.,oftheinfinitepowerseriesoftrigonometricalfunctionsusinggeometricalandalgebraicarguments.WhenthiswasfirstdescribedinEnglishbyCharlesMatthewWhish,inthe1830s,itwasheraldedastheIndians'discoveryofthecalculus.ThisclaimandMdhava'sachievementswereignoredbyWesternhistorians,presumablyatfirstbecausetheycouldnotadmitthatanIndiandiscoveredthecalculus,butlaterbecausenoonereadanymoretheTransactionsoftheRoyalAsiaticSociety,inwhichWhish'sarticlewaspublished.Thematterresurfacedinthe1950s,andnowwehavetheSanskrittextsproperlyedited,andweunderstandthecleverwaythatMdhavaderivedtheserieswithoutthecalculusbutmanyhistoriansstillfinditimpossibletoconceiveoftheproblemanditssolutionintermsofanythingotherthanthecalculusandproclaimthatthecalculusiswhatMdhavafound.InthiscasetheeleganceandbrillianceofMdhava'smathematicsarebeingdistortedastheyareburiedunderthecurrentmathematicalsolutiontoaproblemtowhichhediscoveredanalternateandpowerfulsolution."

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17. ^Katz1995,pp.173174Quote:"HowclosedidIslamicandIndianscholarscometoinventingthecalculus?IslamicscholarsnearlydevelopedageneralformulaforfindingintegralsofpolynomialsbyA.D.1000andevidentlycouldfindsuchaformulaforanypolynomialinwhichtheywereinterested.But,itappears,theywerenotinterestedinanypolynomialofdegreehigherthanfour,atleastinanyofthematerialthathascomedowntous.Indianscholars,ontheotherhand,wereby1600abletouseibnalHaytham'ssumformulaforarbitraryintegralpowersincalculatingpowerseriesforthefunctionsinwhichtheywereinterested.Bythesametime,theyalsoknewhowtocalculatethedifferentialsofthesefunctions.SosomeofthebasicideasofcalculuswereknowninEgyptandIndiamanycenturiesbeforeNewton.Itdoesnotappear,however,thateitherIslamicorIndianmathematicianssawthenecessityofconnectingsomeofthedisparateideasthatweincludeunderthenamecalculus.Theywereapparentlyonlyinterestedinspecificcasesinwhichtheseideaswereneeded.Thereisnodanger,therefore,thatwewillhavetorewritethehistorytextstoremovethestatementthatNewtonandLeibnizinventedthecalculus.Theywerecertainlytheoneswhowereabletocombinemanydifferingideasunderthetwounifyingthemesofthederivativeandtheintegral,showtheconnectionbetweenthem,andturnthecalculusintothegreatproblemsolvingtoolwehavetoday."

18. ^Sergent,Bernard(1997),Gensedel'Inde(inFrench),Paris:Payot,p.113,ISBN222889116919. ^Coppa,A.etal.(6April2006),"EarlyNeolithictraditionofdentistry:Flinttipsweresurprisinglyeffectivefordrillingtoothenamelinaprehistoric

population"(http://www.nature.com/nature/journal/v440/n7085/pdf/440755a.pdf),Nature440(7085):7556,doi:10.1038/440755a(https://dx.doi.org/10.1038%2F440755a),PMID16598247(https://www.ncbi.nlm.nih.gov/pubmed/16598247).

20. ^Bisht,R.S.(1982),"ExcavationsatBanawali:197477",inPossehl,GregoryL.(ed.),HarappanCivilisation:AContemporaryPerspective,NewDelhi:OxfordandIBHPublishingCo.,pp.113124

21. ^A.Seidenberg,1978.Theoriginofmathematics.ArchiveforthehistoryofExactSciences,vol18.22. ^(Staal1999)

23. ^ab(Hayashi2003,p.118)

24. ^ab(Hayashi2005,p.363)25. ^Pythagoreantriplesaretriplesofintegers(a,b,c)withtheproperty:a2+b2=c2.Thus,32+42=52,82+152=172,122+352=372,etc.26. ^(Cooke2005,p.198):"ThearithmeticcontentoftheulvaStrasconsistsofrulesforfindingPythagoreantriplessuchas(3,4,5),(5,12,13),

(8,15,17),and(12,35,37).Itisnotcertainwhatpracticalusethesearithmeticruleshad.Thebestconjectureisthattheywerepartofreligiousritual.AHinduhomewasrequiredtohavethreefiresburningatthreedifferentaltars.Thethreealtarsweretobeofdifferentshapes,butallthreeweretohavethesamearea.Theseconditionsledtocertain"Diophantine"problems,aparticularcaseofwhichisthegenerationofPythagoreantriples,soastomakeonesquareintegerequaltothesumoftwoothers."

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27. ^(Cooke2005,pp.199200):"Therequirementofthreealtarsofequalareasbutdifferentshapeswouldexplaintheinterestintransformationofareas.AmongothertransformationofareaproblemstheHindusconsideredinparticulartheproblemofsquaringthecircle.TheBodhayanaSutrastatestheconverseproblemofconstructingacircleequaltoagivensquare.Thefollowingapproximateconstructionisgivenasthesolution....thisresultisonlyapproximate.Theauthors,however,madenodistinctionbetweenthetworesults.Intermsthatwecanappreciate,thisconstructiongivesavalueforof18(322),whichisabout3.088."

28. ^abc(Joseph2000,p.229)

29. ^ab(Cooke2005,p.200)30. ^Thevalueofthisapproximation,577/408,istheseventhinasequenceofincreasinglyaccurateapproximations3/2,7/5,17/12,...to2,the

numeratorsanddenominatorsofwhichwereknownas"sideanddiameternumbers"totheancientGreeks,andinmodernmathematicsarecalledthePellnumbers.Ifx/yisoneterminthissequenceofapproximations,thenextis(x+2y)/(x+y).Theseapproximationsmayalsobederivedbytruncatingthecontinuedfractionrepresentationof2.

31. ^Neugebauer,O.andA.Sachs.1945.MathematicalCuneiformTexts,NewHaven,CT,YaleUniversityPress.p.45.32. ^MathematicsDepartment,UniversityofBritishColumbia,TheBabyloniantabledPlimpton322(http://www.math.ubc.ca/~cass/courses/m446

03/pl322/pl322.html).

33. ^Threepositiveintegers formaprimitivePythagoreantripleifc2=a2+b2andifthehighestcommonfactorofa,b,cis1.IntheparticularPlimpton322example,thismeansthat135002+127092=185412andthatthethreenumbersdonothaveanycommonfactors.HoweversomescholarshavedisputedthePythagoreaninterpretationofthistabletseePlimpton322fordetails.

34. ^ab(Dani2003)

35. ^ab(Fowler1996,p.11)

36. ^ab(Singh1936,pp.623624)

37. ^ab(Filliozat2004,p.137)38. ^(Pingree1988,p.637)39. ^(Staal1986)

40. ^abc(Filliozat2004,p.139)

41. ^abcde(Filliozat2004,pp.140141)42. ^(Yano2006,p.146)

43. ^abc(Filliozat2004,pp.143144)

44. ^ab(Pingree1988,p.638)

45. ^abc(Hayashi2003,pp.122123)

46. ^abc(Hayashi2003,p.119)

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47. ^abPlofker2007,p.39548. ^Plofker2007,p.395,Plofker2009,pp.474849. ^(Hayashi2005,p.366)

50. ^abcPlofker2009,p.45

51. ^abcdPlofker2009,p.46

52. ^abcdePlofker2009,p.4753. ^(Pingree1978,p.494)

54. ^abPlofker2009,p.48

55. ^abc(Hayashi2005,p.371)56. ^(Datta1931,p.566)57. ^(Ifrah2000,p.464)Quote:"TogivethesecondorfourthcenturyCEasthedateofthisdocumentwouldbeanevidentcontradictionitwouldmean

thatanorthernderivativeofGuptawritinghadbeendevelopedtwoorthreecenturiesbeforetheGuptawritingitselfappeared.GuptaonlybegantoevolveintoShradstylearoundtheninthcenturyCE.Inotherwords,theBak(h)shalimanuscriptcannothavebeenwrittenearlierthantheninthcenturyCE.However,inthelightofcertaincharacteristicindications,itcouldnothavebeenwrittenanylaterthanthetwelfthcenturyCE."

58. ^(Hayashi2005,p.371)Quote:"ThedatessofarproposedfortheBakhshaliworkvaryfromthethirdtothetwelfthcenturiesCE,butarecentlymadecomparativestudyhasshownmanysimilarities,particularlyinthestyleofexpositionandterminology,betweenBakhshalworkandBhskaraI'scommentaryontheryabhatya.Thisseemstoindicatethatbothworksbelongtonearlythesameperiod,althoughthisdoesnotdenythepossibilitythatsomeoftherulesandexamplesintheBakhshlworkdatefromanteriorperiods."

59. ^(Ifrah2000,p.464)

60. ^abAnton,HowardandChrisRorres.2005.ElementaryLinearAlgebrawithApplications.9thedition.NewYork:JohnWileyandSons.864pages.ISBN0471669598.

61. ^(Neugebauer&Pingree(eds.)1970)62. ^Cooke,Roger(1997),"TheMathematicsoftheHindus",TheHistoryofMathematics:ABriefCourse,WileyInterscience,p.197,ISBN0471

180823,"ThewordSiddhantameansthatwhichisprovedorestablished.TheSulvaSutrasareofHinduorigin,buttheSiddhantascontainsomanywordsofforeignoriginthattheyundoubtedlyhaverootsinMesopotamiaandGreece."

63. ^Katz,VictorJ.(1995),"IdeasofCalculusinIslamandIndia",MathematicsMagazine68(3):163174,doi:10.2307/2691411(https://dx.doi.org/10.2307%2F2691411).

64. ^(Hayashi2005,p.369)

65. ^abcd(Hayashi2003,pp.121122)66. ^(Stillwell2004,p.77)

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67. ^(Stillwell2004,p.87)

68. ^abcdef(Stillwell2004,pp.7273)69. ^(Stillwell2004,pp.7476)70. ^Gupta,R.C.(2000),"HistoryofMathematicsinIndia"(http://books.google.co.uk/books?id=

xzljvnQ1vAC&pg=PA329&lpg=PA329&dq=Virasena+logarithm#v=onepage&q=Virasena%20logarithm&f=false),inHoiberg,DaleRamchandani,Indu,Students'BritannicaIndia:Selectessays,PopularPrakashan,p.329

71. ^abSingh,A.N.,MathematicsofDhavala(http://www.jainworld.com/JWHindi/Books/shatkhandagama4/02.htm),LucknowUniversity

72. ^abJoseph(2000),p.298300.

73. ^abcdefghi(Roy1990)

74. ^abc(Bressoud2002)

75. ^abcdefg(Katz1995)76. ^Singh,A.N.Singh(1936),"OntheUseofSeriesinHinduMathematics",Osiris1:606628,doi:10.1086/368443

(https://dx.doi.org/10.1086%2F368443).77. ^Edwards,C.H.,Jr.1979.TheHistoricalDevelopmentoftheCalculus.NewYork:SpringerVerlag.78. ^(Whish1835)79. ^Rajagopal,C.Rangachari,M.S.(1949),"ANeglectedChapterofHinduMathematics",ScriptaMathematica15:201209.80. ^Rajagopal,C.Rangachari,M.S.(1951),"OntheHinduproofofGregory'sseries",Ibid.17:6574.81. ^Rajagopal,C.Venkataraman,A.(1949),"ThesineandcosinepowerseriesinHindumathematics",JournaloftheRoyalAsiaticSocietyofBengal

(Science)15:113.82. ^Rajagopal,C.Rangachari,M.S.(1977),"OnanuntappedsourceofmedievalKeralesemathematics",ArchivefortheHistoryofExactSciences18:

89102.83. ^Rajagopal,C.Rangachari,M.S.(1986),"OnMedievalKeralaMathematics",ArchivefortheHistoryofExactSciences35(2):9199,

doi:10.1007/BF00357622(https://dx.doi.org/10.1007%2FBF00357622).84. ^Joseph,G.G.1997."FoundationsofEurocentrisminMathematics."InEthnomathematics:ChallengingEurocentrisminMathematicsEducation

(Eds.Powell,A.B.etal.).SUNYPress.ISBN0791433528.p.6768.85. ^Cajori,Florian(1893),"TheHindoos",AHistoryofMathematicsP86,Macmillan&Co.,"Inalgebra,therewasprobablyamutualgivingand

receiving[betweenGreeceandIndia].WesuspectthatDiophantusgothisfirstglimpseofalgebraicknowledgefromIndia"86. ^FlorianCajori(2010)."AHistoryofElementaryMathematicsWithHintsonMethodsofTeaching(http://books.google.com/books?

id=gZ2Us3F7dSwC&pg=PA94&dq&hl=en#v=onepage&q=&f=false)".p.94.ISBN1446022218

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86.^Bourbaki,Nicolas(1998).ElementsoftheHistoryofMathematics.Berlin,Heidelberg,andNewYork:SpringerVerlag.46.ISBN3540647678.

87.^BritannicaConciseEncyclopedia(2007),entryalgebra

SourcebooksinSanskrit

Keller,Agathe(2006),ExpoundingtheMathematicalSeed.Vol.1:TheTranslation:ATranslationofBhaskaraIontheMathematicalChapteroftheAryabhatiya,Basel,Boston,andBerlin:BirkhuserVerlag,172pages,ISBN3764372915.Keller,Agathe(2006),ExpoundingtheMathematicalSeed.Vol.2:TheSupplements:ATranslationofBhaskaraIontheMathematicalChapteroftheAryabhatiya,Basel,Boston,andBerlin:BirkhuserVerlag,206pages,ISBN3764372923.Neugebauer,OttoPingree(eds.),David(1970),ThePacasiddhntikofVarhamihira,Neweditionwithtranslationandcommentary,(2Vols.),Copenhagen.Pingree,David(ed)(1978),TheYavanajtakaofSphujidhvaja,edited,translatedandcommentedbyD.Pingree,Cambridge,MA:HarvardOrientalSeries48(2vols.).Sarma,K.V.(ed)(1976),ryabhayaofryabhaawiththecommentaryofSryadevaYajvan,criticallyeditedwithIntroductionandAppendices,NewDelhi:IndianNationalScienceAcademy.Sen,S.N.Bag(eds.),A.K.(1983),TheulbastrasofBaudhyana,pastamba,KtyyanaandMnava,withText,EnglishTranslationandCommentary,NewDelhi:IndianNationalScienceAcademy.Shukla,K.S.(ed)(1976),ryabhayaofryabhaawiththecommentaryofBhskaraIandSomevara,criticallyeditedwithIntroduction,EnglishTranslation,Notes,CommentsandIndexes,NewDelhi:IndianNationalScienceAcademy.Shukla,K.S.(ed)(1988),ryabhayaofryabhaa,criticallyeditedwithIntroduction,EnglishTranslation,Notes,CommentsandIndexes,incollaborationwithK.V.Sarma,NewDelhi:IndianNationalScienceAcademy.

87. ^abAlmeida,D.F.John,J.K.Zadorozhnyy,A.(2001),"KeraleseMathematics:ItsPossibleTransmissiontoEuropeandtheConsequentialEducationalImplications",JournalofNaturalGeometry20:77104.

88. ^Gold,D.Pingree,D.(1991),"AhithertounknownSanskritworkconcerningMadhava'sderivationofthepowerseriesforsineandcosine",HistoriaScientiarum42:4965.

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Externallinks

ScienceandMathematicsinIndia(http://www.indohistory.com/science_and_mathematics.html)AnoverviewofIndianmathematics(http://wwwgap.dcs.stand.ac.uk/~history/HistTopics/Indian_mathematics.html),MacTutorHistoryofMathematicsArchive,StAndrewsUniversity,2000.IndexofAncientIndianmathematics(http://wwwgroups.dcs.stand.ac.uk/~history/Indexes/Indians.html),MacTutorHistoryofMathematicsArchive,StAndrewsUniversity,2004.IndianMathematics:Redressingthebalance(http://wwwhistory.mcs.standrews.ac.uk/history/Projects/Pearce),StudentProjectsintheHistoryofMathematics(http://wwwhistory.mcs.stand.ac.uk/Projects).IanPearce.MacTutorHistoryofMathematicsArchive,StAndrewsUniversity,2002.IndianMathematics(http://www.bbc.co.uk/programmes/p0038xb0)onInOurTimeattheBBC.OnlinecoursematerialforInSIGHT(http://cs.annauniv.edu/insight/insight/maths/history/index.htm),aworkshopontraditionalIndiansciencesforschoolchildrenconductedbytheComputerSciencedepartmentofAnnaUniversity,Chennai,India.

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