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  • 1. The Mathematics Education SECTION BVol. V I I . N o . 3 , S e p t . 1 97 3 G L I M PS E S OF A N C IE N T INDIAN M ATH. NO. 7 The nnadfrava-Gre€ory Series D7Radha Charan Gupta, Dept. of MatltematicsBirla Institute of Technolog7 P,O. Mcsra, RANCHI ( India ) ( Re ce ive d l0 Ju ly 1973 )l. Introduction: fn current mathematical literature the series a rc ta n x :x -x s l a -| x ,l u -... (l )is called the Gregorys seriesfor the inverse tangent function after the Scottish mathemat-cianJames Gregory ( 1638-75)1 who knew it about the year 1670. In India, an equiva:lent of the series(l) is found enunciated in a rule which is attributed to the famousMadhava of Saigamagrdma ( circa 1340-1425)s who is also called as Golvid ( .Master ofSpherics) by later astronomers. Madhavas rules is found the Keralite commentary Krir,trkramakari quoted in (fn+rnt+tt ) 1:fffl on Bldskara JIs Lilavati (dtoref,t), the most poptrlar work ofancient Indian mathematicr. The authorship of KKK has been a matter of con.jecture.lfowever, according to K. V. Sarma, there arerclear show that the KKK isa work of Ndr-;aLra Vdriar ( circa.l500-6-0) as conje- ( circa 1500-75)a andinot 6f Sar.r[a1actured by some scholarss. wron S-, @ t {-..- f+it ,,8 u-vL The Sanskrit verses( comprising the rule ) which are attributed to Midhava in theKKK are also found mentioned in the YuktiBh:isa (:YB)0, a popular Malalalam workwhose authorhas been identified to be one called stsatsJyesthadeva (circa 1500-1601)?. Another Sanskrit stanza which contains the velbal enunciation of an equivalent ofth e se r ies( l) is f ound i n th e Ka ra rl a -Pa d d h a ti(:K P, C hap. V I, V erse l B )8 of P utumanaSomayaji ( about 1660-1740)e. Ilel<;rvwe give the Sanskrit text of the M:ldhavas rule, its transliteration, a transla-tion, its explar.ration modern form, arrd indicate an ancient Indian proof of it. in2. Enunciation of the Series : The Sanskrit text of the rule attributed to M.idlsva i51o qsesqrfssqqleiarq diaqrcacqq ssq I gsrifi Effi siflas{ s AIIF{ ll crrTEri
  • 2. 6B TE! TIII| TTITtCE !IIC I| IIO | I rscTf(so*-qlsq tqr r.Fdfilt6: I gssqrqtq ierrflqiisAts{3l rnE II qlqrqi dgt<a*<r grcdq qgfts t <):rlaqheqttq +erdtlfrq €get I oadtilqqfli (qr;il;qqrFqgg: gt tt Istajld-trijyayor gh tet kotyaptarp prathamaqr phalarp I Jyavargaqr gurlakarl kltvf kotivargap ca harakam ll Pratham:1di-phalebhyo tha ney:t phalatatatir muhuh I Eka-tryldyo jasalikhllbhir bhaktesv etesv-anukramf,t ll Ojana p sapyutes-tyaktvf, yugma-yogarp dhanur-bhabet I Dohkotyor-alpam-eveha kalpaniyam-iha smrtamll Labdhinam-avasdnar.n sy-nninyathSpi muhuh krite I We may translate the above as follows:lr The product of the given Sine and the radius divided by the Cosine is the firstresult. From the first, (and then, second, third,) etc., results obtain (successively)a seque-nce of results by taking repeatedlythe square of the Sine as the multiplier and the square ofthe Cosine as the divisor. Divide (the above results) in srder by the odd numbers one threeetc. (to get the full sequenceof terms). From the sum of the odd terms subtract the sum ofthe even terms. (The results) becomes the arc. In this connection, it is laid down that the(Sine of the) arc or ( that of ) its complement, which ever is smaller, should be taken here(as thegiven Sine); otherwise, the terms, obtained by the (alrove) repeatedProcesswill nottend to the vanishing-magnitudo. That is, we are asked to form the sequence(Rlr). (s/c), (,s/c)" (s/c). (n/3). (R/5). (s/c). (sic)".. (sic),.:T T z , I s , . . s a ],whereR is the radius (norm or sinus totus) of the circle reference and S:R sin 0 C -R c o s 0 .Then, according to the rule v y s : ( T t { Tr* ...) - (Ir* 7 r* ...) - - T r - T z* T s -T r* ... (2)That is, /R sin0) r R ( Rsin r 0) =lllnioJO): - x R 0 :3 II ,t" 9 j ; (1( 0) r s. (R .oioF- " .*Or 0 :ta n 0 -(ta n a 0 )/3 f (ta n 5 0)/5-... which is equivalent to (l). It may be noted that the condition given tolrards the end of the rule amounts tosaying that we should have R sin 0 to be less than R cos 0, 0 being accute. That is, tan 0 or
  • 3. N. O. GUPTI 69x should be less than trnity which is the condition for the absolute convergence of the series(l ). Incidently this justifies the re-arrangement of the terms of the sequence the form (2) inwhich rvas known to the Indians of the period as is clear from the proof given by them (seeSection 3 belor.r). The statement of the Midhava-Gregory seriesas found in the IfP, VI, l8 (p. l9)is contained in the verse aqrwfq {en*rrrgq6: r}aqrcaqtv su E4rqiiur fqflerqrfqqq.d aftc,o set( | 5(ql$ifegor*l dTg q*6+tftqotl|EftT- ridoslqgiiee+iq eqfi dtargflcrsat tl tc rl It r.rill be noted that the r.rordingof this stanza is similar to that of the first fourlines of the Sanskrit passageuhich we have translated above. The meaning is almost the same and need not to be repeated. Still another Sanskrit stanza which gives the game seriesis found in the Sadratna- ma l a of S ank ala V a rma (A .D . te 2 3 ;tr.3. Derivation of the Series : An ancient fndian derivation of the Mldhava-Gregory seriesis found in the YB (pp. ll3-16). The proof starts rvith a geornetrical derivation of the rule which is basicallyequivalent to what is implied in the modern formula dT:d ( tan 0 )/( lf tanz 0 ). Theelaborate proof then consistsof stepswhich amount to what, in modern analysis,is calledexpansionar,d tcrm-by-term integration. However, it must be remembered that the proofbelongs lo the pre-calculusperiod in the modern sense. The YII derivation has been published in various presentations by scholars such arC. T. Rajagopala, according to whom the proof "would even today be regalded assatisfactory except for the abscnce of a fervjustificatory remarks", and othersrs. Theitrtelestedreader may refer to their publicatiorrsfor details, References and Notecl. C. B. Bo1-er: A lTislor2 of Mathematicr.Wiley, New York, 1968,pp. 42122.2. K. V. Sarma ! Historl of Kerala School of Hindu Astronoml,t ( in Perspectittcz :i s h v e s h v a ra n a n d s t., H oshi arpur, 1972,p 51. fn T.A. Sara:watl i : "fhe Development of Mathematical Seriesin India after Bhaskara II". Bull. Aational Inst. of Sciences India, No. 2l ( 1963 ), of p .3 3 7 ; a rrd S a rm a , OP. ci t., p.20.1. Sar m a, OP. c i t., p p . 5 7 -5 9 .5. K. Kunjunni Raja : "Astronomy and Mathematics in Kerala (an Account of the Litera- ttrre )" Ad y a r L i l -rra ryBul l , N o. 27 (1963) pp. 154-55; and S ara- sr,rathi, cit., p.320. oP.6. The Yukti-EI:asa (in Malalalam). Part I, edited with noted by Rama Varma Maru
  • 4. 70 T IIE X.[T IIEM IIIICg E D U C ITION Thampuran and A.R. Akhileswar Aiyar, Mangalodayam press, Trichur, 1948,pp. ll3-14. Also seethe Garlita-Yukti-Bhasa edited by T. Chandrasekharan and others, lvfadras Government Oriental I4anuscripts Selies No. 32, Madras, 1953,pp. 52-53 (the text as edited here is corrupt).7. Sarma, Op. cit., pp. 59-60.B. The Kararla Paddhati: edited by K. Sambasiva Sastri, Trivandrum Sanskrit SeriesNo. 126, Irivandrum, 1937, p. 19. Also seethe KP along rvith two Malayalam commentariesedited by S.K. Nayar Government Orie- ntal Manuscripts Library, Madras; 1956, pp. 196-97.9. Sarma, Op. cit., pp. 68-69. It may be noted here that the arguments given by A. K. Bag, "TrigonometricalSeriesin KP and the probable date of the text", IndianJ. Hist. Sci., vol.l (1966), pp. 102-105,for a much earlier date of the work cannot be accepted becausehe has not fully analysedthe views found in the introduction of Nayars edition cited above and also those summari-zed by Raja, op. cit., etc.lC. See referencesunder serial nos. 3 and 6 above. We have followed the text as given in the YB in the Malayalam script. The text given by Sarma is slightly different.I l. We have tried to give our own translation which is more or less a literal one. For a different translation seeC. T. Rajagopal and T. V. Vedamurthi Aiyar, "Or the Hindu Proof of Gregorys Series", ScriptaMathematica, Vol. l7 (1951), p. 67; Or Sarma," op. cit., pp.20-21 where the translation of Rajagopal and Ai1ar has bee reproduced. It may alsobe noted here that these two joint authors mention the nrle (quoted in the YB) as a quotation from the Tantra-Samgraha (:TS, 1500 A.D.). So also Sarasr,,,a. thi, op. cit., p. 337. Of course in the printed YB ( seeref. 6 above ) the work TS is mentioned within brackets after one more rule given, besidesthe one which we have quoted. The Ganita-YB does not montion TS at this place. Moreover the printed TS (edited by S. K. Pillai, Trivandrum, l35B), which seemsto be complete in itself, does not contain the lines. According to Sarma, op. cit., p. lB, the information, given to Saraswathi, op. cit., p.32+, foot-note 9, that the printed TS is not complete, is not likely to be correct.12. Govt. Oriental Manuscripts Libray, Madras, lvls. No. R 4448, Ch. III, verse 10. For date, seeSarma, oP.cit., p. 78.13. Some referencesare : (i) C. T. Rajagopal : "A Neglected Chapter of Hindu Mathematics". Soipta Math., V ol. l5 ( 19 4 9 ), p p . 2 0 1 -2 0 9 . (ii) Rajagopala andVedanurthi Aiyar, op. cit., pp.65-74. (iii) C. N. Srinivasiengar : The Historlt of Ancient Indian Mathematics, World Press, Calc ut t a, 19 6 7 ,p p . 1 4 6 -4 7 . Horvever, the reader should be careful about the dates of the concerned Indian works as given in the above three refcrences.