1. The Mathematics Education SECTION DV ol. V I I . No.4, Dec. 1973 G L I M P S E S OF A N C IT N T INDIAN M ATH. NO.8Nhr-ayarlas Method for Evaluattng Gluadratic Surds D7R. C. Gupta, Department Mathematics, Birla Instituteof Teclmologl,P, O, Mesra, of IIAJVCHI, India. (Received 16 October 1973) 1. Introduction We frequently come across the name Ndr;1anu (rr<fqq) among Hindu rvriters on se-cular sciences(including Astrology, Astronomy and l{athematics). The one r,rith whom rveare concerned here is called i{a-rdyartaPaldita. He was the son NarasiEha (or Nrsirpha)and flourished about the middle of l4th Century of our era. 6r:iyana Pan{ita is an important autlror of ancient Indian Mathematics. Two worksfrom lris pen are now well known. Ire composed the qfq6 slg{t Ganita Kaumudl (:GK) inthe vear 1356 according to the colophonic verse of the work itself. The Gfi, rvhich is devotedto elementary mathematics (arithmetic, geometry, series,magic squares, and etc.), has beeneditedl. lr{IrZya as other work is called S}g.ffqfafqtq Bija-Ganitavatagsa (:BG) and is devo-ted to algebra. The BG is taken to be written eariier than the G,tf sincethe authors R.ja-Ganita(Algebra) is mentioned in the commentarv part of the GK (part I, p. lg). Dr. K. s. Suk-las edition of a part of the BG, which is based on an incomplete manuscript, has been recen-tly published2, According to Dr Shukla (,8G, Intlodnction, p. iv), Narayana introduced new slbje-cts of treatment and made his own contributions to the existing ones. He is the first mathe-matician to have dealt with the subjectof N4agicSquares(Bhadra-garlita). Sorneof his copt-ributions are indeed rer:rarkableand deservespecial credit.........the so-called Fermat,sfacto-rizatiorr method was given by N:i1ziy3na in his GK (Part II, pp. 245-47) altout three cen-turies before it struck the mind of the French mathematicians. There are quite a few prob-lems which were proposed and solved for the first time by Ndrdyana. Ihe GK is indeed a very significant rvork on ancient and medieval Indian mathema-tics. M. M. Sudhakara Divevedi during his last days is stated to have said that if he werealive for few more years he would have enthroned G.6 in place of the L lzivati, the mostpopular work of ancient Indian Mathematicsa.
2. THE M AITHEM ATICS ED U C AT ION In this article we shall descril:eNxrlyanas method of finding the approximate valueof a quadratic surd by using thc theory and solutions of indeterminate equation of the secondo rd e r . A s per S hu k l a th e m e th o d w a s g i v e n b y N i rdyaua for the fi rst ti me (B G, Intr., p.i v).What is meant lty srrch a statement is tlrat an earlier referenceto the method is not knounto us at the plesent mornent. 2. Narryauas Method for Approximating yN-. The indeterminate equation Jtfyzl6---yz (l)is called Varga-praklti (Square-Nature) in Hindu mathematics". .f is called the prakrtior gunaka (N{ultiplier), c the ksepa (ka) orInterpcilator, and x andy, the Lesser Rootand the GreaterRoot respectivelr. For finding an approximate valtre of the square-rootof the non-squ:rre nttrnber"Ai theBG,I, 86 (p. aa) as well as the GK, X, l7 (Part ll, p.244; girresthe follorving rr.rle. {(i TTF q€q q d(cdc} ca dr I slea F?€qc4T q € qE qt;{oqrq?4q tl pade tatra I M[larp srrhyat1ryas)aca tadritpaksepaje,ob,ai,he "fr:-ffi il:H::,*:H,::T, re. is, roo,s,be determined "/::r;l-]il ^ roo,o (as the Multiplier) and trnity as the Interpolatot, and then divide the Greater (Root) by the Smaller Root. (The resrrlt isl an approximation to the sqr-tare-root" That is, if a, b are a pair of roots of the equation N)cz+l:12 (2) so that N42+ l -_ b 2 (3) then { N:olo approximately (4) One of the cxamples given both jn the .BG and the G.ft is that of finding the sqlrare- roote of 10. The three pairs of solution menlioned by the attthor (6, l9), (228,721) and (8658, 27379). So that e have the approximations r t o: t g /e :3 .1 6 6 6 7 n e al l y +rto:lztpze :3.1622807 nearly { t o: z z z l g i B6 5 8 :3 .I G2 2 7 7 GG2 arty neThe last value being correct to eight decimal places. The above example is important because yiD-was one of the approxirnations of rc wid-ely used in the ancient world6. The other example given in the BG as rvell as in the G.K is that of finding the square ro ot of l/ 5.
3. R. C. GU P TII 95 The rationale of the method is simple and follows from the equation (2) which gives {fi:{@_ll1* :1f x nearly,rvh e n T ( s o als o x ) is l a rg e , 3. Sorne Rernarks orr the rnethod It can be easily seenthat if (a, b) and {.a, b) arc any two solutions of the equation(2 ), then x : ab, *ba , ):N a - a l b b also form a solution of (2). This additive solution (sam=isa-bh.iv2nd)may be called Brahm-aguptas Lemma (A. D. 628)?. Thtrs trsing the solutions (6, l9) and (228, 721) we get r:6 x 7 2 1 + 1 9x 2 2 8 :8 6 5 8 t : lox 6 x 2 2 8 + 1 9x 7 2 1:2 7 3 7 5gir.ing the thircl solution (8658, 27379)noted above. Taking two indenlical solutions, that is 4 :4 b: brc get the tulya-bhlvanE solution, namely x : 2 ab ):N az tr.bzrvlriclr is formed fr.om the single solution (a, b). Tl.us from the approximate solution (4) weget the better appro-ximation .Jlr:1"u I bz)l2ab :("lr+ u2)12a. (5)rvhere a stands for the first approximatton bfa. T hes olut ion (5 )te l l s u s o f a w o n d e rfu l e q ui val enceofthe above method w i th someother ancient methods. For example, if a better approximation is assumed to be8. ctrt:d] € (6)then the small correction e can be found by squaring both sides of (6) and neglecting et. Thevalue of c thud found, when put in (6), leads to the same solution (5). Of course the same result is got (as should be expected) by applying the so.calledNewton-Raphson method by taking ( x ) : N- x " fthe better root being given by u-f (e)l f (e.)
4. 90 TIII M .tr IIHEMT T IC S ED IC II!IO N Lastly, as alreadv shollrr b1 the plesent ,1ritere, same result is also obtained lry. the taking the averageof the approximation a and another approximation.A/a. This last algo- rithm r.ras krrorvtrto the Babvlonians of antiquity and rvas later on rrsed by the Greeks and o t her s inc ludir g In d i a n s l o . Thrrs, thesepr-ocedures all arrlount to the result (5) uhich can be obtained dilectly by u s ing t he s ir npleb i n o mi a l a p p l o x i m a ti o n ! ( ot t *r ) :d i rl 2 u where :42+r References and Notes l. Tlu Ganita A-attnrudi(rvith the authors outn commentary rvhich may be said to form an integral part of the work itself) edited by Padmakara Dvivedi. The princessof Wales Sarasvati Bhavana Texts No. 57, Govt. Sanskrit Liltrary, Benarcs (Varanasi); Part I, 1936 and Part II, 1942. The rnorkis also called Ganitapati Kaumudi and is written in the st1leof otirel ancient pati-ganita norks of India. 2. N,irayana p2ntlitas Bija Gaqitavatarpsa,Part I, edited by K. S. Shukla and published by the Akhila Bharatiya Sanskrit Parished in their journal called $taqr, Lucknow, 1970. of The Catalogue Publicalions the Varanaseya Sanskrit University dated l968 mentions of an earlier edition, by Chandrabhanu Pandeya, of the BG published from (Varanasi.) In our article the page referencesare according to Shuklas edition. 3. For details seeK. S. Shukla, Hindu Methods for Finding Factors or I)ivisors of a Num- ber " . G anit a , V o l . 1 7 , N o . 2 (D e c . 1 9 6 6),pp. 109-117. 4. R. C.Jha : Ilidoaduiluti(in Hindi), Chorvkhamba Sanskrit Series Office, Varanasi, 1959,p. 68. 5. For details see B. Datta and A. N Singh : History of Hindu lvlathcntatics, Single Voltrme E dit ion, A s ia P rrb l i s h i n gH o u s e , B o m b a y , P art II, pp. l 4l -181. 6. The author of the present article proposesto publish a separatepaper on this (Jaina) /alue of Pi. 7. Datta and Singh, Op, cit., part II, pp. 146-47. 8. R. C. Gupta, "Barrdhdyanas Valtte of {i". The Mathematics Education, Vol. VI, No.3 ( S ept . 1972) ,t). 7 8 . 9. G upt a, ( ) P , c i t., p ,7 8 .10. c. B. Boyer : A rlistory of A,fathematics,John wiley, New york, 1968, pp. 30-31; and B. Datta, NSrZialtasMethod for Finding Approximate Value of a Surdo, Bull. Calcutta M at h. S oc . , Vo l . 2 3 , N o . 4 (1 9 3 t), p p . l 8 7 -194.
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