The Mathematics Education

Vol. VI I . No. 3, Sept. 1973

GLIMPSES OF ANCIENT INDIAN MATH. NO. 7

The nnadfrava-Gre€ory SeriesD7 Radha Charan Gupta, Dept. of Matltematics Birla Institute of Technolog7 P,O. Mcsra,

RANCHI ( India )( Received l0July 1973 )

l . Introduction:

fn current mathematical literature the seriesarc tan x:x-xsla- |x, lu- . . . ( l )

is called the Gregory's series for 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.

Madhava's rules is found quoted in the Keralite commentary Krir,trkramakari(fn+rnt+tt ) 1:fff l on Bldskara JI's Lilavati (dtoref,t), the most poptrlar work of

ancient Indian mathematicr. The authorship of KKK has been a matter of con.jecture.

lfowever, according to K. V. Sarma, 'there arerclear evidences ..to show that the KKK isa work of Ndr-; 'aLra ( circa 1500-75 )a andinot 6f Sar.r[a1a Vdriar ( circa.l500-6-0 ) as conje-

ctured by some scholarss. wron S-, @ t {-..- f+it ,,8 u-vL

The Sanskrit verses ( comprising the rule ) which are attributed to Midhava in the

KKK are also found mentioned in the YuktiBh:isa (:YB)0, a popular Malal'alam work

whose authorhas been identif ied to be one called stsatsJyesthadeva (circa 1500-1601)?.

Another Sanskrit stanza which contains the velbal enunciation of an equivalent of

the ser ies ( l ) is found in the Karar la-Paddhat i ( :KP, Chap. VI, Verse lB)8 of Putumana

Somayaji ( about 1660-1740)e.

Ilel<;rv we give the Sanskrit text of the M:ldhava's rule, its transliteration, a transla-

tion, its explar.ration in modern form, arrd indicate an ancient Indian proof of it.

2. Enunciation of the Series :

The Sanskrit text of the rule attributed to M.idlsva i51o

qsesqrfssqqleiarq diaqrca cqq ssq I

crrTEri gsrifi' E'ffi siflas{ s AIIF{ ll

SECTION B

6B TE! TI I I | TTITtCE ! I ' I 'CI | I IO|I

rscTf(so*-qlsq tqr r.'Fdfilt6: Igs'sqrqtq ierrflqiis'Ats{3l rnE I I

qlqrqi dgt<a*<r grcdq qgfts t

<):rlaqheqttq +erdtlfrq €get Ioadtilqqfli (qr;il;qqrFq gg: gt tt

Istajl'd-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 jasalikhl'lbhir 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 first

result. 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 of

the Cosine as the divisor. Divide (the above results) in srder by the odd numbers one' three'

etc. (to get the full sequence of terms). From the sum of the odd terms' subtract the sum of

the 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 the'given Sine'); otherwise, the terms, obtained by the (alrove) repeated Process wil l not

tend to the vanishing-magnitudo.That is, we are asked to form the sequence

(Rlr). (s/c), (n/3). (s/c). (,s/c)" (R/5). (s/c). (sic),. (sic)"..:T\ Tz, Is, ' . .sa] ,

whereR is the radius (norm or sinus totus) of the circle reference and

S:R sin 0

C-R cos 0.

Then, according to the rulevys: (Tt{Tr*. . . ) - ( I r* 7r* . . . )

- -Tr-Tz*Ts-Tr*. . .

That is,

R 0:3 I I , t " 9 j : - x /R sin 0)r R' (R sin 0)r=lllnioJO)' - ; (1( .* 0)' r s. (R .oioF- "'Or

0:tan 0-( tana 0)/3f ( tan5 0)/5-. . .

which is equivalent to (l).

It may be noted that the condition given tolr 'ards the end of the rule amounts to

saying that we should have R sin 0 to be less than R cos 0, 0 being accute. That is, tan 0 or

(2)

N. O. GUPTI

x should be less than trnity which is the condition for the absolute convergence of the series(l ). Incidently this justif ies the re-arrangement of the terms of the sequence in the form (2)which 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 series as found in the IfP, VI, l8 (p. l9)is contained in the verse

aqrwfq {en*rrrgq6: r}aqrcaqtv suE4rqiiur fqflerqrfqqq.d aftc,o set( |

5(ql$'ifegor*l dTg q*6+tftqotl|EftT-

ridoslqgiiee+iq eqfi dtargflcrsat tl tc rl

It r.r'ill be noted that the r.r'ording of this stanza is similar to that of the first fourlines of the Sanskrit passage u'hich we have translated above. The meaning is almost thesame and need not to be repeated.

Still another Sanskrit stanza which gives the game series is found in the Sadratna-mala of Sankala Varma (A.D. te23;tr .3. Derivation of the Series :

An ancient fndian derivation of the Mldhava-Gregory series is found in the YB (pp.

l l3-16). 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 consists of steps which amount to what, in modern analysis, is called

expansion ar,d tcrm-by-term integration. However, it must be remembered that the proofbelongs lo the pre-calculus period in the modern sense.

The YII derivation has been published in various presentations by scholars such ar

C. T. Rajagopala, according to whom the proof "would even today be regalded as

satisfactory except for the abscnce of a ferv justif icatory remarks", and othersrs. The

itrtelested reader may refer to their publicatiorrs for details,References and Notec

C. B. Bo1-er : A lTislor2 of Mathematicr.Wiley, New York, 196'8, pp. 421'22.

K. V. Sarma ! Historl of Kerala School of Hindu Astronoml,t ( in Perspectittcz

\ : ishveshvaranand fnst . , Hoshiarpur, 1972, p ' 51.

T.A. Sara:watl i : "f 'he Development of Mathematical Series in India after Bhaskara

II". Bull. A'ational Inst. of Sciences of India, No. 2l ( 1963 ),p.337; arrd Sarma, OP. c i t . , p.20.

1. Sarma, OP. c i t . , pp. 57-59.

5. K. Kunjunni Raja : "Astronomy and Mathematics in Kerala (an Account of the Litera-

t t r re)" Adyar Li l - r rary Bul l , No. 27 (1963) ' pp. 154-55; and Sara-

sr,r 'athi, oP. cit., p.320.

6. The Yukti-EI:asa (in Malalalam). Part I, edited with noted by Rama Varma Maru

69

l .

2.

70 TIIE X.[TI IEMII I ICg EDUCITION

Thampuran and A.R. Akhileswar Aiyar, Mangalodayam press,

Trichur, 1948, pp. l l3-14. Also see the Garlita-Yukti-Bhasa edited byT. Chandrasekharan and others, lvf adras Government OrientalI\4anuscripts Selies No. 32, Madras, 1953, pp. 52-53 (the text asedited here is corrupt).Op. cit., pp. 59-60.

Paddhati: edited by K. Sambasiva Sastri, Trivandrum Sanskrit Series No.126, 'Ir ivandrum, 1937, p. 19. Also see the KP along rvith twoMalayalam commentaries edited by S.K. Nayar Government Orie-ntal Manuscripts Library, Madras; 1956, pp. 196-97.Op. cit., pp. 68-69.

7. Sarma,B. The Kararla

9. Sarma,It may be noted here that the arguments given by A. K. Bag, "Trigonometrical

Series in 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 because he has not fully analysedthe views found in the introduction of Nayar's edition cited above and also those summari-zed by Raja, op. cit., etc.lC. See references under 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 see C. T. Rajagopal and T. V. Vedamurthi Aiyar, "Or the HinduProof of Gregory's Series", Scripta Mathematica, Vol. l7 (1951), p. 67; Or Sarma,

" op. cit., pp.20-21 where the translation of Rajagopal and Ai1'ar has bee reproduced.It may alsobe noted here that these two joint authors mention the nrle (quoted in theYB) 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 ( see ref. 6 above ) the work TS is

mentioned within brackets after one more rule given, besides the one which we have

quoted. The Ganita-YB does not montion TS at this place. Moreover the printed

TS (edited by S. K. Pil lai, Trivandrum, l35B), which seems to be complete in itself,

does not contain the l ines. 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, see Sarma, oP. cit., p. 78.

13. Some references are :(i) C. T. Rajagopal : "A Neglected Chapter of Hindu Mathematics". Soipta Math.,

Vol . l5 (1949), pp. 201-209.(i i) Rajagopala andVedanurthi Aiyar, op. cit., pp.65-74.(iii) C. N. Srinivasiengar' : The Historlt of Ancient Indian Mathematics, World Press,

Calcutta, 1967, pp. 146-47.Horvever, the reader should be careful about the dates of the concerned Indian works

as given in the above three refcrences.