1) The ancient Indian text Baudhayana's Sulba Sutra from around 800-400 BC contains a rule for approximating the square root of two. The rule increases the side of a square by one-third and one-fourth parts to get an approximation of 1.41421. 2) This approximation method of linear interpolation was popular in ancient India. The rule can be explained through a two-term interpolation formula. 3) Repeated use of this interpolation process yields the four-term approximation found in the Sulba Sutra, equivalent to methods used by later mathematicians like Neugebauer. While an improvement, the Indian approximation was still less accurate than the sexagesimal

1. fI;'-C IION B
Tire i l l a t h e m a t i c s E c l u ca tio n
Vo l . V I , No 3, S eP t.1 9 7 2,7 f 1't - ?
7
f i t IIflP S ilSOT ANCIEFiTI} i NI Ai. { } I A1'iEI lATI CS NO . 3
;
BaucnEralraxi€a's VaExre ()f tz*
l-Iesro, Ranchilndia.
TechnollgJP.O.
ayR.C.Gupta, AssistantProfessor,Birl tlnrtituteof
( llccc:r ' c< i 2 G Ju',c 1972 )
T her e is a s ma l l c l a s s o f S a n s k ri t l i te rature cal l eciS ul basrrtra i g" < Ta). These S ul ba.
sttrer , or s im ply Su l b a s , a re rn a n u a l sfo r th e ccnstrttcti onof vedi c al tars anr] rnay be tLken
to b e t he oldes t ge o m e tri c a l tre a ti s e so f In d i :r . In them tre get gl i ' rrpl esof rn,:rerrt Indi an
geometry and a few otber subjects of maihematical interest'
A t pr es ent m a n y Su l h a ma n u a l s a re e xtani . The,past;rmba ( i ,;IT(-;4 B audl i dyana
),
(dJqrqa), Katyeyana ( +rcetel ), and l4inava Srrlbi,sutrasare r.r'ell known" But exact d.rtesof
th e i r c om pos it ion a re n o t k tro w n . T h a t o f Bl r rdhyi yena i s regardedto be the crl destol ti rem
a n d may bc plac ed b e trv e e a8 0 0 B. C . to 4 OO . C . n
The 6lst aphorism in the first chapter of Baudblyana's treatise glves tbe following
ru l e l .
-A
cqt'ri EdlA-dc'.ltn-{ agtfaun<ftaa)ia r
Pramir.rarhtgtiyena vardhayet-tat-c;r catur[hena-irtrrra-catustridr(onena. 'Increase the
measure ( tbat is, the given side of a square ) by its tliird part and again b-y th-r fourth part-*
-the ,. ,* -
valu: cf the diagonal ,'t I
V is, of fourthpart ). ( We gettfe approxiruate
J
;,|;;sfiijJi.l,
( B auci h. I,6t ).
T ak ing unity to b e th e s i d e o f th r s g u a re, the above ri rl e i .npl i es
{ z -l + l 3 ' 3.4.- 3A.i+
+ ^t -l= . ( l)
Samerule is founclin the Sulbamanuals Apastau:ba l{irtyilana2. The approximation
of and
(l) giver
t/ 2 -5771408= l.tl42l, 5€86 (2)
the actual value being gi'renby
356 (3)
t/ z =1.41421,
derivations (l).
G. Tbibaut and B. B. Dattashavc givenrathercomplicated of We
shall give a simpleexplrnatiooherea.
T he linear i o te rp o l a ti o n m e th o d o r th e R ul e of 'lhree, rvhictr ivas very popular in
ancient India, yields the two term approximation
( a2+ x ) t 12= c l x l r2 c * l (4)
)
aCc( li, /a/h,nr^,1
(4 1/-',, ;dl
/At la'xf )

2. 78 The Mathe matics Education
H er e ( 2c ll ) is th e c l i ffe re n c eb e tw e c n th e scl uares c an,l tbe next posi ti ve i nteger(c* l
of ).
If r is 0 u' e get t h c e x a c t s q u a re ro o t c a n d w hen x i s (2cl l
) w c agai n get the exacr square
root ( 6+ l) . I l e n c e ,fo r a n y o tb e r i n te rm ecl i aryval ueof r n' ctrkexpartsof the fracti on
l l (2c + I ) and a d d i t to c to g e t (4 ). Il l u rtrati ng thi s argument numeri cal l y, w e have
( i ) / I = { l ? + o )ti r= l + o /(2 r-. l z ).= 1.
( ii ) r z f :(1 2 + l )t l r= I + i /3 a s i n (l ).
( iii) . / 3 -(1 r4 2 ;r l r= = l -f2 1 3 a s fo und i n ti re val ne of ru 5 etatcd by D attas.
L as t ly ( iv ) 1/ t y -= (t? + r)t/n = l + 3 /3 -2 .
S im ilar s e ri e so f v a l u e sc a n b e g i v e n betrvcen any t,ao succesti ve
square numbers.
Th us v ; e s hall ha ,;e
{ 7 -(.2 2 + 3 )r/' -2 + 3 1 .3 ' - 22)= 13,' .-r.
I t m ay bc p o i n te d o u t th a t th e a p proxi maton (1) i s not found among the anci ent
Greekso. By above argument v.'eshali also have, similarly,
( a3*x t ' lt = a + x l (3 a z * 3 a * l )
whic h r r as giv en b y S . S tc v i n (a b o u t 1 5 9 0A ' D ' )?
O nc e we g e t th e tw o te rm a p p ro x i mati on, the four term approxi mati on (l ) may be
found by thc processo[ successive cor rection as already explained by Gurjars. For inetancc
lfle aSsurle
/- , I + (l/3){e (5)
S quar ing b o th s i d e sa n d n e g l e c ti n ge 2 w e easi l y get r to be equal to l /12 w hi ch, w h en,
p u t in ( 5) , giv es th e th i rd te rm o f (l )'
If we now apply the process once more v;e shall get the required approximation. It
may be pointed out that lhe processgiven by Neugebauerofor arriving at Euccestiveterms is
mat hem at ic ally e c 1 ' .ri v a l e u t th e a b o v e p ro cess
to of repeated correcti ons. For, l et a be aay
approxinration to the squarc root of "lf, tlren the next approxirnation by tbe abcve process,
a fter as s r r r ning
' J rf a +
=
"
will be
t/ {:o * (ff- oz)12u, (6)
whichcan be written as
{N:{o*(Nla)t12
and tliis explains as to why the approxirnation (6) is the average of the given approximation a
and ("M/a).
B ef or e c l o s i n g th i s a rti c l e , i t ma y b e poi nted out that the B abyl oni ans al so gave a
ve r y good v alue f o r y ' [ w h i c h m a y b e w ri tten asro
/2-t+T +!I-+
'^'60'602'6ot
19.

3. R. C. Gupta 79
Slvrng
9t
t / z ':g o s+Ui 6 o o :1 .4 1 4 2 1,
2
T he I ndian v a l u e , i n a d d i ti o n to b e i n g expressedn a qui l e di fferent menner, i s l ess
i
accurate than the Babylonian value. Evcn their first fractional terms do noi agree. More-
o ve r, t her e is no neg a ti v e te rm i n th e Ba b y l o n i an val ue. A l so the Indi an val ue i s i n ex,:ess,
and the Babyltnian value in defect, of the aciual value.
R.eferences
l. Bauclhd.yana's
Sulbashtram by S. Prakash
ed. and R. S. Sbarma, p.61.
I'{ewDelhi, 1968,
2. SeeApastamba Sulbasttra by D. Srinivasachar
ed. andS. Narasimhach:r, N{ysore,1931,
p.26 and Kirtyayana
Sulbasutram by VidyadharSharma,
ed. p.
I{ashi, 1928, I7.
3. Datta, B. B. : The Science the Sulba. Calculta,1932,pp. 189-194.
of
4. Gupta,R. C, : "Some ImportautIndian Mathematical
Methods Conccivcd Sanskrit
as in
Language." An invited paperpresentcd the fnternational
at New
SaaskritConference,
Delhi, Irdarch1972,pp, 7-8,
5. Datta, B. 8,, op. cit,, p. I95.
6. Smith,D. E. : History of lr{athematics.New York, lg58,Vol. II, p. 254.
7. Smith,D. E., op. cit., p. 255.
B. Gurjar, L. V. : AncientIndian i{arhematics Vedha.Poona,
and lgit7,p. 39,
9. Neugebauer,0. The ExactSciences Antiquity. I.{erv
: in York, 1962,p.50.
10. Neugebauer, : op. cit.,p. 35.
0.
-r