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Square Root
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http://www.math.cornell.edu/~dwh/papers/sulba/sulba.html 1/9
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.