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