Square Root

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SquareRootsintheSulbasutraDedicatedtoSriChandrasekharendraSarasvati

whodiedinhishundredthyearwhilethispaperwasbeingwritten.

by

DavidW.Henderson1

DepartmentofMathematics,CornellUniversity

Ithaca,NY,148537901,USA(email:dwh2@cornell.edu)

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.