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Square Roots in the Sulbasutra Dedicated to Sri Chandrasekharendra Sarasvati who died in his hundredth year while this paper was being written. by David W. Henderson 1 Department of Mathematics, Cornell University Ithaca, NY, 148537901, USA (email: [email protected]) In this paper I will present a method for finding the numerical value of square roots that was inspired by the Sulbasutra which are Sanskrit texts written by the Vedic Hindu scholars before 600 B.C.. This method works for many numbers and will produce values to any desired degree of accuracy and is more efficient (in the sense of requiring less calculations for the same accuracy) than the divideandaverage method commonly taught today. Several Sanskrit texts collectively called the Sulbasutra were written by the Vedic Hindus starting before 600 B.C. and are thought 2 to be compilations of oral wisdom which may go back to 2000 B.C. These texts have prescriptions for building fire altars, or Agni. However, contained in the Sulbasutra are sections which constitute a geometry textbook detailing the geometry necessary for designing and constructing the altars. As far as I have been able to determine these are the oldest geometry (or even mathematics) textbooks in existence. It is apparently the oldest applied geometry text. It was known in the Sulbasutra (for example, Sutra 52 of Baudhayana's Sulbasutram) that the diagonal of a square is the side of another square with two times the area of the first square as we can see in Figure 1. Thus, if we consider the side of the original square to be one unit, then the diagonal is the side (or root) of a square of area two, or simply the square root of 2, that is . The Sanskrit word for this length is dvikarani or, literally, "that which produces 2". The Sulbasutra 3 contain the following prescription for finding the length of the diagonal of a square: Increase the length [of the side] by its third and this third by its own fourth less the thirty fourth part of that fourth. The increased length is a small amount in excess (savi´e¸a) 4 . Thus the above passage from the Sulbasutram gives the approximation:

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    SquareRootsintheSulbasutraDedicatedtoSriChandrasekharendraSarasvati

    whodiedinhishundredthyearwhilethispaperwasbeingwritten.

    by

    DavidW.Henderson1

    DepartmentofMathematics,CornellUniversity

    Ithaca,NY,148537901,USA(email:[email protected])

    InthispaperIwillpresentamethodforfindingthenumericalvalueofsquarerootsthatwasinspiredbytheSulbasutrawhichareSanskrittextswrittenbytheVedicHinduscholarsbefore600B.C..Thismethodworksformanynumbersandwillproducevaluestoanydesireddegreeofaccuracyandismoreefficient(inthesenseofrequiringlesscalculationsforthesameaccuracy)thanthedivideandaveragemethodcommonlytaughttoday.

    SeveralSanskrittextscollectivelycalledtheSulbasutrawerewrittenbytheVedicHindusstartingbefore600B.C.andarethought2tobecompilationsoforalwisdomwhichmaygobackto2000B.C.Thesetextshaveprescriptionsforbuildingfirealtars,orAgni.However,containedintheSulbasutraaresectionswhichconstituteageometrytextbookdetailingthegeometrynecessaryfordesigningandconstructingthealtars.AsfarasIhavebeenabletodeterminethesearetheoldestgeometry(orevenmathematics)textbooksinexistence.Itisapparentlytheoldestappliedgeometrytext.

    ItwasknownintheSulbasutra(forexample,Sutra52ofBaudhayana'sSulbasutram)thatthediagonalofasquareisthesideofanothersquarewithtwotimestheareaofthefirstsquareaswecanseeinFigure1.

    Thus,ifweconsiderthesideoftheoriginalsquaretobeoneunit,thenthediagonalistheside(orroot)ofasquareofareatwo,orsimplythesquarerootof2,thatis .TheSanskritwordforthislengthisdvikaranior,literally,"thatwhichproduces2".

    TheSulbasutra3containthefollowingprescriptionforfindingthelengthofthediagonalofasquare:

    Increasethelength[oftheside]byitsthirdandthisthirdbyitsownfourthlessthethirtyfourthpartofthatfourth.Theincreasedlengthisasmallamountinexcess(saviea)4.

    ThustheabovepassagefromtheSulbasutramgivestheapproximation:

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    .

    IuseinsteadofindicatingthattheVedicHinduswereawarethatthelengththeyprescribedisalittletoolong(saviea).Infactmycalculatorgives:

    andtheSulbasutram'svalueexpressedindecimalsis

    SothequestionariseshowdidtheVedicHindusobtainsuchanaccuratenumericalvalue?Unfortunately,thereisnothingthatsurviveswhichrecordshowtheyarrivedatthissaviea.

    Therehavebeenseveralspeculations5astohowthisvaluewasobtained,butnooneasfarasIcandeterminehasnoticedthatthereisastepbystepmethod(basedongeometrictechniquesintheSulbasutram)thatwillnotonlyobtaintheapproximation:

    ,

    butcanalsobecontinuedindefinitelytoobtainasaccurateanapproximationasonewishes.

    Thismethodwillinonemorestepobtain:

    ,

    wheretheonlynumericalcomputationneededis1154=2[(34)(17)1]and,moreover,themethodshowsthatthesquareofthisapproximationislessthan2byexactly

    .

    Theinterestedreadercancheckthatthisapproximationisaccuratetoelevendecimalplaces.

    TheobjectoftheremainderofthispaperisadiscussionofthismethodandrelatedtopicsfromtheSulbasutram.

    BricksandUnitsofLength.

    IntheSulbasutramtheagniaredescribedasbeingconstructedofbricksofvarioussizes.Mentionedoftenaresquarebricksofside1pradesa(spanofahand,about9inches)onaside.Eachpradesawasequalto12angula(fingerwidth,about3/4inch)andoneangulawasequalto34sesameseedslaidtogetherwiththeirbroadestfacestouching6.Thusthediagonalofapradesabrickhadlength:

    1pradesa+4angula+1angula1sesamethickness.

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    Idonotbelieveitispurelybychancethattheseunitscomeoutthisnicely.Noticethatthislengthistoolargebyroughlyonethousandthofthethicknessofasesameseed.Presumablytherewasnoneedformoreaccuracyinthebuildingofaltars!

    DissectingRectanglesandA2+B2=C2

    NoneofthesurvivingSulbasutratellhowtheyfoundthesaviea.However,inBaudhayana'sSulbasutramthedescriptionofthesavieaisthecontentofSutras6162andinSutra52hegivestheconstructionsdepictedinFigure1.MoreoverinSutra54hegivesamethodforconstructinggeometricallythesquarewhichhasthesameareaasanygivenrectangle.IfNisanynumberthenarectangleofsidesNand1hasthesameareaasasquarewithsideequaltothesquarerootofN.ThusSutra54giveaconstructionofthesquarerootofNasalength.Soletusseeifthishintsatamethodforfindingnumericalapproximationsofsquareroots.ThefirststepofBaudhayana'sgeometricprocessis:

    Ifyouwishtoturnarectangleintoasquare,taketheshortersideoftherectangleforthesideofasquare,dividetheremainderintotwopartsand,inverting,jointhosetwopartstotwosidesofthesquare.

    SeetheFigure2.Thisprocesschangestherectangleintoafigurewiththesameareawhichisalargesquarewithasmallsquarecutoutofitscorner.

    Figure2

    InSutra51Baudhayanahadpreviouslyshownhowtoconstructasquarewhichhasthesameareaasthedifferenceoftwosquares.Inaddition,Sutra50describeshowtoconstructasquarewhichisequaltothesumoftwosquares.Sutras50,51and52arerelateddirectlytoSutra48whichstates:

    Thediagonalofarectangleproducesbyitselfboththeareaswhichthetwosidesoftherectangleproduceseparately.

    ThisSutra48isaclearstatementofwhatwaslatertobecalledthe"PythagoreanTheorem"(Pythagoraslivedabout500BC).Inaddition,Baudhayanaliststhefollowingexamplesofintegralsidesanddiagonalforrectangles(whatwenowcall"PythagoreanTriples"):

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    (3,4,5),(5,12,13),(7,24,25),(8,15,17),(9,12,15),(12,35,37),(15,36,39)

    whichtheSulbasutramusedinitsvariousmethodsforconstructingrightangles.

    ConstructionoftheSavieafortheSquareRootofTwo

    IfweapplySutra54totheunionoftwosquareseachwithsidesof1pradesawegetasquarewithside1pradesafromwhichasquareofsidepradesahadbeenremoved.SeetheFigure2.

    Nowwecanattempttotakeastripfromtheleftandbottomofthelargesquarethestripsaretobejustthinenoughthattheywillfillinthelittleremovedsquare.Thepiecesfillinginthelittlesquarewillhavelength1/2andsixoftheselengthswillfitalongthebottomandleftofthelargesquare.Thereadercanthenseethatstripsofthickness(1/6)(1/2)pradesa(=1angula)will(almost)work:

    Figure3

    Thereisstillalittlesquareleftoutoftheupperrightcornerbecausethethinstripsoverlappedinthelowerleftcorner.Noticethat

    .

    Wecangetdirectlyto byconsideringthefollowingdissection:

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    Figure4

    Wenowhavethattwosquarepradesasareequaltoalargesquareminusasmallsquare.Thelargesquarehassideequalto1pradesaplus1/3ofapradesaplus1/4of1/3ofapradesa,or1pradesaand5angulasandthesmallsquarehassideof1angula.Tomakethisintoasinglesquarewemayattempttoremoveathinstripfromtheleftsideandthebottomjustthinenoughthatthestripswillfillinthelittlesquare.Sincethesetwothinstripswillhavelength1pradesaand5angulasor17angulaswemaycuteachinto17rectangularpieceseach1angulalong.Ifthesearestackeduptheywillfillthelittlesquareifthethicknessofthestripsis1/34ofanangula(or pradesa).Withoutamicroscopewewillnowseethetwosquarepradesasasbeingequalinareatothesquarewithside pradesa.Butwithamicroscopeweseethatthestripsoverlapinthelowerleftcornerandthusthatthereisatinysquareofside stillleftout.

    Figure5

    Thus

    isstillalittleinexcess.Wecannowperformthesameprocedureagainbyremovingaveryverythinstripfromtheleftandbottomedgesandthencuttingtheminto

    pradesalengthsinordertofillintheleftoutsquare.Ifwistwicethenumberoflengthsin pradesa,thenthestripsweremovemusthavewidth

    pradesa.Wecancalculateweasilybecausewealreadynotedthattherewere17segmentsoflength andeachofthesesegmentswasdividedinto34piecesandthenoneofthesepieceswasremoved.Thusw=2[34(17)1]=1154and

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    witherrorexpressedby

    .

    Iwrite"21"insteadof"2"toremindusthatforBaudhayana(and,infact,formostmathematiciansupuntilneartheendofthe19thCentury)that denotedtheside(alength)ofasquarewitharea2.

    Ifweagainfollowthesameprocedureofremovingaverythinstripfromtheleftandbottomedgesandcuttingtheminto lengthpieces,thenthereadercancheckthatthenumberofsuchpiecesmustbe

    2[1154(1154/2)1]=(1154)22=1,331,714

    andthusthatthenextapproximation(saviea)is

    .

    Thedifferencebetween21andthesquareofthissavieais

    .

    ThismethodwillworkforanynumberNwhichyoucanfirstexpressastheareaofthedifferenceoftwosquares,N1=A2B2,wherethesideAisanintegralmultipleofthesideB.Forexample,

    .

    IfindthattheeasiestwayformetoseethattheseexpressionsarevalidistorepresentthemgeometricallyinawaythatwouldalsohavebeennaturalforBaudhayana.Toillustrate:

    Figure6

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    Figures3and4giveotherexamples.Thereadershouldtryoutthismethodtoseehoweasyitistofindsavieasforthesquarerootsofothernumbers,forexample,3,11,2.

    FractionsintheSulbasutram

    Youhaveprobablynoticedthatallthefractionsaboveareexpressedasunitfractions,butthisisnotalwaysthecaseintheBaudhayana'sSulbasutram.Forexample,inSutra69hediscusseshowtofindalengthwhichisanapproximationtothediagonalofasquarewhosesideisthe"thirdpartof"8prakramas(whichequals240angulas).Hedescribestheconstruction:

    ...increasethemeasure[the8prakramas]byitsfifth,dividethewholeintofivepartsandmakeamarkattheendoftwoparts.

    InmoremodernnotationifweletDequal8prakramas,thenthisgivestheapproximationofthediagonalofasquarewithside(1/3)Das

    .

    Thisisequivalentto beingapproximatedby1.44.

    Ifyouattempttofindthesavieasforothersquarerootsyouwillfinditconvenienttousenonunitfractions.Forexample,bystartingwiththispicture:

    Figure7

    youcanmakeslightmodificationsintheabovemethodtofind:

    .

    ComparingwiththeDivideandAverage(D&A)Method

    Todaythemostefficientmethodusuallytaughttofindsquarerootsiscalled"divideandaverage".ItisalsosometimescalledNewton'smethod.IfyouwishtofindthesquarerootofNthenyoustartwithaninitialapproximationa0andthentakeasthenextapproximationtheaverageofa0andN/a0.Ingeneral,ifanisthenthapproximationofthesquarerootofN,thenan+1=(an+(N/an)).Theinterestedreadercancheckthatifyoustartwith[1+(1/3)+(1/12)]=[17/12]=1.416666666667asyourfirstapproximationof ,thenthesucceedingapproximationsarenumericallythesameasthosegivenbyBaudhayana'sgeometricmethod.

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    However,Baudhayana'smethodusessignificantlylesscomputations(inaddition,ofcourse,tothedrawingseitheronpaperorinone'smind).Forexample,lookatthefollowingtablewhichcomparesthemethodsforthefirstfourapproximations.ForBaudhayana'smethodatthenthstageletkndenotethenumberofthinpiecesaddedintothemissingsquareandletcndenotethecorrectiontermthatisadded.

    D&Acalculator D&AFractions Baudhayana'sMethoda1=1.416666667 17/12 1+(1/3)+(1/4)(1/3)

    a2=(a1+(2/a1))=

    1.414215686

    [(17/12)+2(12/17)]=

    (577/408)

    k2=2[(34)+4+1]=34

    c2=(1/34)(1/4)(1/3)a3=(a2+(2/a2))=

    1.414213562

    [(577/408)+2(408/577)]=

    (665857/470832)

    k3=(34)2=1154

    c3=(1/1154)(1/34)(1/4)(1/3)

    a4=(a3+(2/a3))=

    1.414213562

    [(665857/470832)+2(470832/665857)]

    =(886731088897/627013566048)

    k4=(1154)22=1331714

    c4=(1/1331714)c3

    Noticethatthe(10digit)calculatorreachesitsmaximumaccuracyatthethirdstage.AtthisstagetheBaudhayanamethodobtainedmoreaccuracy(itcanbecheckedthatitisaccurateto12digits)andtheonlycomputationrequiredwas(34)2=1154whichcaneasilybeaccomplishedbyhand.Baudhayana'sapproximationsarenumericallyidenticaltothoseattainedintheD&Amethodusingfractions,butagainwithsignificantlylesscomputations.Ofcourse,Baudhayana'smethodhasthisefficiencyonlyifyoudonotchangeBaudhayana'srepresentationoftheapproximationintodecimalsorintostandardfractions.AtthefourthstagetheBaudhayanamethodisaccuratetolessthan2[(133171422)(1331714)(1154)(34)(4)(3)]1orroughly24digitaccuracywiththeonlycalculationneededbeing(1154)22=1331714.

    NoticethatinBaudhayana'sfourthrepresentationofthesavieaforthesquarerootof2:

    ,

    theunitisfirstdividedinto3partsandtheneachofthesepartsinto4partsandtheneachofthesepartsinto1154partsandeachofthesepartsinto133174parts.NoticethesimilarityofthistostandardUSAlinearmeasurewhereamileisdividedinto8furlongsandafurlonginto220yardsandayardinto3feetandafootinto12inches.Othertraditionalsystemsofunitsworksimilarlyexceptforthemetricsystemswherethedivisionisalwaysby10.Also,somecarpentersIknowwhentheyhaveameasurementof inchesarelikelytoworkwithitas

    ,or2inchesplusahalfinchminusaneighthofthathalfthisisaclearerimage

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    toholdontoandworkwith.FromBaudhayana'sapproximationitiseasiertohaveanimageofthelengthof thanitisfromtheD&A's(886731088897/627013566048).

    Conclusions

    Baudhayana'smethodcannotcomeevenclosetotheD&Amethodintermsofeaseofusewithacomputeranditsapplicabilitytofindingthesquarerootofanynumber.However,theSulbasutracontainsmanypowerfultechniques,which,inspecificsituationshaveapowerandefficiencythatismissinginmoregeneraltechniques.Numericalcomputationswiththedecimalsystemineitherfixedpointorfloatingpointformhasmanywellknownproblems.7Perhapswewillbeabletolearnsomethingfromthe(apparently)firstappliedgeometrytextintheworldanddevisecomputationalproceduresthatcombinegeometryandnumericaltechniques.

    1ThisarticlegrewoutofresearcheswhichwerestartedduringmyJanuary,1990,visittotheSankaracharyaMuttinKonchipuram,Tamilnadu,India,whereIwasgivenaccesstotheMutt'slibrary.IthankSriChandrasekharendraSarasvati,theSankaracharya,andallthepeopleoftheMuttfortheirgeneroushospitality,inspirationandblessings.

    2Seeforexample,A.Seidenberg,TheRitualOriginofGeometry,ArchivefortheHistoryoftheExactSciences,1(1961),pp.488527.

    3BaudhayanaSulbasutram,i.612.ApastambaSulbasutram,i.6.KatyayanaSulbasutram,II.13.

    4Thislastsentenceistranslatedbysomeauthorsas"Theincreasedlengthiscalledsaviea".Ifollowthetranslationof"saviea"givenbyB.Dattaonpp.196202inTheScienceoftheSulba,UniversityofCalcutta,1932seealsoG.Joseph(TheCrestofthePeacock,I.B.Taurus,London,1991)whotranslatesthewordas"aspecialquantityinexcess".

    5SeeDattaOp.cit.foradiscussionofseveralofthese,someofwhicharealsodiscussedinG.Joseph,Op.cit.

    6BaudhayanaSulbasutram,i.37.

    7See,forexample,P.R.Turner's"Willthe'Real'RealArithmeticPleaseStandUp?"inNoticesofAMS,Vol.34,April1991,pp.298304.