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                ...
78                                              The Mathe matics Education H er e ( 2c ll ) is th e c l i ffe re n c eb e ...
R. C. Gupta                                                   79Slvrng                                                    ...
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Gupta1972f

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Gupta1972f

  1. 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 1t - ? 7 f i t IIflP S ilSOT ANCIEFiTI} i NI Ai. { } I A1iEI lATI CS NO . 3 ; BaucnEralraxi€as 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.rell 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 Baudblyanas treatise glves tbe following ru l e l . -A cqtri 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 IV 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 girenby 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 laxf )
  2. 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 entGreekso. 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 befound by thc processo[ successive cor rection as already explained by Gurjars. For inetancclfle 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. Itmay be pointed out that lhe processgiven by Neugebauerofor arriving at Euccestiveterms ismat 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 aayapproxinration 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)t12and tliis explains as to why the approxirnation (6) is the average of the given approximation aand ("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 ave 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-+ ^606026ot 19.
  3. 3. R. C. Gupta 79Slvrng 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 iaccurate 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.yanas 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

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