1. Madhava of Sangamagrama formulated a rule equivalent to the Gregory series for calculating the inverse tangent function. 2. The rule expresses the inverse tangent as a sum of terms involving sines and cosines, equivalent to the Gregory series. 3. Ancient Indian mathematicians provided proofs of the rule, equivalent to modern proofs through term-by-term integration, showing they understood the concept of infinite series well before Western mathematicians formulated them explicitly.

1. The Mathematics Education SECTION B
Vol. 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 Gregory's 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 famous
Madhava of Saigamagrdma ( circa 1340-1425)s who is also called as Golvid ( .Master of
Spherics') by later astronomers.
Madhava's rules is found the Keralite commentary Krir,trkramakari
quoted in
(fn+rnt+tt ) 1:fffl 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 is
a work of Ndr-;'aLra Vdriar ( circa.l500-6-0) as conje-
( circa 1500-75)a andinot 6f Sar.r[a1a
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 identified to be one called stsatsJyesthadeva (circa 1500-1601)?.
Another Sanskrit stanza which contains the velbal enunciation of an equivalent of
th 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 utumana
Somayaji ( about 1660-1740)e.
Ilel<;rvwe give the Sanskrit text of the M:ldhava's rule, its transliteration, a transla-
tion, its explar.ration modern form, arrd indicate an ancient Indian proof of it.
in
2. Enunciation of the Series :
The Sanskrit text of the rule attributed to M.idlsva i51o
qsesqrfssqqleiarq diaqrcacqq ssq I
gsrifi' E'ffi siflas{ s AIIF{ ll
crrTEri

2. 6B TE! TIII| TTITtCE !I'I'C I| IIO | I
rscTf(so*-qlsq tqr r.'Fdfilt6: I
gs'sqrqtq ierrflqiis'Ats{3l rnE II
qlqrqi dgt<a*<r grcdq qgfts t
<):rlaqheqttq +erdtlfrq €get I
oadtilqqfli (qr;il;qqrFqgg: 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 sequenceof 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) repeatedProcesswill not
tend 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 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

3. N. O. GUPTI 69
x 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)
in
which rvas known to the Indians of the period as is clear from the proof given by them (see
Section 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.r'ill be noted that the r.r'ordingof this stanza is similar to that of the first four
lines of the Sanskrit passageu'hich 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 basically
equivalent to what is implied in the modern formula dT:d ( tan 0 )/( lf tanz 0 ). The
elaborate proof then consistsof stepswhich amount to what, in modern analysis,is called
expansionar,d tcrm-by-term integration. However, it must be remembered that the proof
belongs lo the pre-calculusperiod 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 fervjustificatory remarks", and othersrs. The
itrtelestedreader may refer to their publicatiorrsfor details,
References and Notec
l. C. B. Bo1-er: A lTislor2 of Mathematicr.Wiley, New York, 196'8,pp. 421'22.
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 : "f'he Development of Mathematical Seriesin India after Bhaskara
II". Bull. A'ational 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,r'athi, 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, "Trigonometrical
Seriesin 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 analysed
the views found in the introduction of Nayar's 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 Gregory's Series", ScriptaMathematica, 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 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.