Tire i l lathematics Eclucat ion f I ; ' -CIION B
Vol. VI , No 3, SePt. 1972 ,7f 1 ' t - ? 7
f i t I I f lPSi lS OT ANCIEFiT I} i NIAi . { } IA1' ; iEI lATICS NO.3BaucnEralraxi€a's VaExre ()f tz*
ayR.C.Gupta, AssistantProfessor,Bir l t lnr t i tuteof Technol lgJP.O. l - Iesro, Ranchi lndia.
( l lccc:r 'c<i 2G Ju' ,c 1972 )
There is a smal l c lass of Sanskr i t l i terature cal leci Sulbasrr t ra ig"<Ta). These Sulba.
st t rer , or s imply Sulbas, are rnanuals for the ccnstr t tct ion of vedic al tars anr] rnay be tLken
to be the oldest geometr ical t reat ises of Indi : r . In them tre get gl i ' r rp les of rn, : rerr t Indian
geometry and a few otber subjects of maihematical interest'
At present many Sulha manuals are extani . The,\past;rmba ( i , ; IT(- ;4 ) , Baudl idyana
(dJqrqa ), Katyeyana ( +rcetel ), and l\4inava Srrlbi,sutras are r.r 'ell known" But exact d.rtes of
their composi t ion are not ktrown. That of Blrrdhyiyena is regarded to be the cr ldest o l t i rem
and may bc placed betrveea 800 B. C. to 4OO n. C.
The 6lst aphorism in the first chapter of Baudblyana's treatise glves tbe following
rulel . -A
cqt'ri EdlA-d c'.ltn-{ agtfaun<ftaa)ia r
Pramir.rarh tgtiyena vardhayet-tat-c;r catur[hena-irtrrra-catustridr(onena. 'Increase the
measure ( tbat is, the given side of a square ) by its t l i ird part and again b-y th-r fourth part-*, . ,* - I
V ;,|;;sfiijJi.l,
is, of -the fourth part ). ( We get tfe approxiruate valu: cf the diagonal ,'t J
( Baucih. I ,6t ) .
Taking uni ty to be the side of thr sguare, the above r i r le i .npl ies
{z - l + l+ ^t . - - l= . ( l )3 ' 3.4 3A. i+
Same rule is founcl in the Sulba manuals of Apastau:ba and l{ i r ty i lana2. The approximation
( l) givert / 2 -5771408= l . t l42l , 5€86
the actual value being gi'ren by
t / z =1.41421, 356
G. Tbibaut and B. B. Dattas havc given rather compl icated
shal l give a simple explrnat ioo herea.
The l inear ioterpolat ion method or the Rule of
ancient India, yields the two term approximation
(a2+x)t 12=clx l r2c* l )
aCc( li, /a/h,nr^,1(4 1/-',, /At ;dl la'xf )
(2)
(3)
derivat ions of ( l ) . We
' lhree, rvhictr ivas very popular in
(4)
78 The Mathe matics Education
Here ( 2cl l ) is thecl i f ference betwecn the scluares of c an, l tbe next posi t ive integer(c* l ) .I f r is 0 u 'e ge t thc exact square root c and when x is (2cl l ) wc again get the exacr squareroot (6+l) . I lence,for anyotber intermecl iaryvalueof r n 'ctrkexpartsof the f ract ionl l (2c+ I )and add i t to c to get (4). I l lur t rat ing th is argument numerical ly, we have
( i ) / I ={ l?+o)t i r=l+o/(2r- . lz) .=1.
( i i ) rz f : (12+ l ) t l r= I + i /3 as in ( l ) .
( i i i ) . /3 -(1r42;r l r==l- f213 as found in t i re valne of ru 5 etatcd by Dattas.Last ly ( iv) 1/ ty -=(t?+ r) t /n=l +3/3-2.
Simi lar ser ies of values can be given betrvcen any t ,ao succest ive square numbers.
Thus v;e shal l ha, ;e
{7 -( .22 +3)r / ' -2+31 .3 ' - 22)=13, ' . - r .
I t may bc pointed out that the approximat\on (1) is not found among the ancient
Greekso. By above argument v.'e shali also have, similarly,
(a3*x t ' l t = a + x l (3az* 3a * l )
which rras given by S. Stcvin (about 1590 A' D')?
Once we get the two term approximat ion, the four term approximat ion ( l ) may be
found by thc process o[ successive cor rection as already explained by Gurjars. For inetancc
lfle aSsurle
\ / - , I + ( l /3){e (5)
Squaring both s ides and neglect ing e2 we easi ly get r to be equal to l /12 which, wh en,
put in (5), g ives the thi rd term of ( l ) '
If we now apply the process once more v;e shall get the required approximation. It
may be pointed out that lhe process given by Neugebauero for arriving at Euccestive terms is
mathematical ly ec1' . r ivaleut to the above process of repeated correct ions. For, let a be aay
approxinration to the squarc root of "lf, t lren the next approxirnation by tbe abcve process,
af ter assrrrning
\ 'Jr \ f =a+
"wil l be
t/ {:o * (ff- oz)12u,
which can be wri t ten as
{N:{o*(Nla)t12and tl i is explains as to why the approxirnation (6) is the average of the given approximation aand ("M/a).
Before c losing this art ic le, i t may be pointed out that the Babylonians also gave a
very good value for y ' [ which may be wri t ten asro
\ /2-t +T +!I-+ 19.' ^ '60'602'6ot
(6)
R. C. Gupta
t / z ' :gos+U 2i6oo:1.41421, 29tThe Indian value, in addi t ion to being expressed in a qui le di f ferent menner, is less
accurate than the Babylonian value. Evcn their f irst fractional terms do noi agree. More-over, there is no negat ive term in the Babylonian value. Also the Indian value is in ex, :ess,and the Babyltnian value in defect, of the aciual value.
R.eferences
l. Bauclhd.yana's Sulbashtram ed. by S. Prakash and R. S. Sbarma, I ' {ew Delhi , 1968, p.61.
2. See Apastamba Sulbasttra ed. by D. Srinivasachar andS. Narasimhach:r, N{ysore, 1931,p.26 and Kir tyayana Sulbasutram ed. by Vidyadhar Sharma, I{ashi, 1928, p. I7.
3. Datta, B. B. : The Science of the Sulba. Calcul ta, 1932, pp. 189-194.
4. Gupta, R. C, : "Some Importaut Indian Mathematical Methods as Conccivcd in Sanskri tLanguage." An invi ted paper presentcd at the fnternat ional Saaskri t Conference, NewDe lhi, Irdarch 1972, pp, 7-8,
5. Datta, B. 8, , op. c i t , , p. I95.
6. Smith, D. E. : History of l r {athematics. New York, lg58, Vol. I I , p. 254.
7. Smith, D. E., op. c i t . , p. 255.
B. Gurjar, L. V. : Ancient Indian i \ {arhematics and Vedha. Poona, lgi t7, p. 39,9. Neugebauer,0. : The Exact Sciences in Ant iqui ty. I . {erv York, 1962, p.50.
10. Neugebauer, 0. : op. c i t . , p. 35.
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