1. Aryabhata, an Indian astronomer and mathematician from the 5th century AD, approximated pi (π) to 3.1416 in his famous work the Aryabhatiya. This approximation correct to four decimal places was one of the most accurate approximations of pi used anywhere in the ancient world. 2. Aryabhata expressed pi as a fraction 62832/20000, which can be expressed in continued fractions as 3 + (4/16). This value of pi was later used and referenced by many Indian and foreign mathematicians and astronomers over subsequent centuries. 3. Scholars debate whether Aryabhata's value of pi was influenced by the Greeks

1. the Mathematics Edcation SECTIONB
Vol. VJI, No l, March 1973
O T I M P SES ANCIENT
Otr INDIANMAT[, NO.5
Aryatrhata I's Value ()t n
by R.C, Gupta. Dept.oJ Mathcmatict,Birla InstituteoJ Tcchnologlt O. Mesra, Ranchi.
P.
( Receivecl Janurry 1973)
25
l . In tro d u c r i on
The Sanskrit rvork:irtfqe.i4Aryabhatil'a (:AB) u'as u-ritten by the well--knorvn Ind-
i a n a str onom er and ru a th e m a ti c i a n .i ry a b h a ta I ( born A . D . 476 ). A ccordi ng to an
interpretation of the staternentl made in the AB it-self, the n'ork lvas composed by the aut-
hor at a very young age of 23 years. However, Sengupta?,who agreed with the above inter-
pretation in the beginning, gave another interpretation later on and said that ((we are not
justified in concluding that the AB was composed rvhen,ryabhata was only 23 years old"'
Aryabhata. is knorvn to be the author of another rvork ( which may be called the
aritiQfat.Eta) rvhich is not extant. This astronomical work was based on the midnight system
of day-reckoning in contrast to the AB in rvhich the day rvas reckoned from one sunrise to
the next. A fresh and detailed study concerning this lost rvork and of quotations from it as
found in some of the later works has been recently carrie,J out by Dr. K. S. Shukla3.
In addition to the tlvo rvorks mentioned above, the comosition of some free or det-
ach e d stanz as( M uk t a k c r ) i s a l s o a ttti b u te d to th e author of the A B t.
2 . Ap p ro x i ma ti o n o f n a s gi vcn by A ryabhal a I
The AB. II, 10 ( p. 25 ) gives the follorving rule
s$ftTsi {rilqegui dlqFEw?il qqqtqT1q I
gqcflsrr(: il lo ll
srgda{rrssfirrrcqrvq}
Caturadhikarir (atami;tagu.larir dvd;as!istathi sahasri4dm r
Ayutadvaya-vi;kambhasy-isanno vftta parinihall rr l0 rr
'Ifundred plus four rnultiplied by eight and (combined with) sixty-two thousandsis
the approximate circurnference of circle of diameter twenty thousand.'
Th a t i s,
Cir c unr f er ence , C -(1 0 0 + 4 ) x B* 6 i 0 0 0 approx,,
whendiameter, D-20000.
So that
11 :ClD-62832/20000 approx
:3. t4l6 ttJ

2. l8 'rHE MATHEMATTcSEDUcATIoN
it
The value of n is correct to four decimals and is one of the best approximation for
used by the ancient peoples any where in the world. What is equally important to note is
was
that the author states the value to be an approximate one only. This means that he
aware of the fact that the value is not exact, although it is close (d.sanna) to the true
value.
Using the theory of continued fractions the value (l) can be expressed as
" :3+,1,
7f t+ ll
-^l + 'l
16
This yields the following successiveapproximations
(i) :3 which is the simplest
approximation.
(ii) =2217 which is called the Archimedeanvalue.
(iii) :3551113which is calledthe chinesevalue or Tsu's number.
(iv) 392711250 which is simply the reducedform o[ the aB value.
-
3. Aryabhala'evalue as found in other worke.
Lalla ( eight century ) givesa rule accordingto whichs
C x 62513927-Radius,R.
This implies
2tr=39271625
which gives the samevalue as (1) but in the reducedform :iv) above'
If we take C to be egual to 360x60 parts ( or minutes), then we have
B-2t600/6.2B32
-3+37'73872nearly
44 l9'4
:3437'+66+ aPProximatelY
eO*tOO
By rounding off this value, separately,to the nearest minute or second or third; we
get the norrn or Sinus-Totus(fisl) as found respectivelyin the AB, the Vate6vara-Siddhinta
( tenth century ), and Govind Svlmin's commentary ( ninth century) on the Mah6''Bhdskarta
in connection with the tables of sine6.
A certain astronomer, Puli6a, has also eniployed the same value of 7f as found in the
ABi .lJtpala ( tenth century ) is also stated to have mentioned the same value in his com-
mentary on the famous work BShat-Samhita ( q€itiqir )B' Bhdskara rr ( twelfth century ) has
given the same value but in the reduced form (iv)e.
Yallaya (rSth century) in his commentary on the AB has expressedthe same value in
thc following chronogram in the Katapayddi systern of Indian numeralsro.
t$afu grfrleflea+ru) il{c I
arfa
4. Aryabhala's value of ?r Transmitted to the west
Yaqub Ibn Tariq ( Baghdad,eight century ), on the authority of his Indian informant

3. ,t'
R,. C. GUPTA l9
l 2 5 6 ,6 40, 000unit s an d th a ti ts d i a me te ri s 4 0 0 ,0 0 0 ,000uni tsrr. Thi s i mpl i es" a val ue of ?r
which same as that found in the AB,
Another Arab author, Al-khwarizmi ( ninth century ), recorded the original form of
Aryabhata, value 62832/20000 and remarked it as being due to the Indian astronomerstt.
He reproduced the AB value in his r{lgaDrc almost in the same languagewhich, in F. Rosen's
tra n sl a t ion, is as f ollo rv s r3 .
,...,......Multiply the diameter by sixty-two thousand eight hundred and thirty-two
and then divide the product by twenty thousandl the quotient is the periphery."
Exactly the sarne form of the Indian value, 6283212000; of ?r appears in the
eleventh century Spain in the r,vork of Az-Zarqali who followed the Indians in many other
respectsalso.l a
5. The so.called Greek inf luence on Aryrbha.ta value of lr
Since, in the statement of the rule giving tf, AB takes a radius equal to one aluta
( myriad in Greek ), some scholarssr,rspectthis value to be of Greek origin (for the Greeks
al o n e o f all people m a d e my ri a d th e rrn i t o f s e c o ndorder ( R odet )16.
Fforvever,the choice of a radius of 10000units may be a matter of convenienceguite
suitable to the Indian coinputational methods in decimal scale, and for attaining the desired
accuracy to four decimals but at the same time avoiding the use of fractional parts.
The only Greek value of r which comes very near to, but is not exactly equal to, the
AB value, is the following
r -3 * (8/60) (30i60:)
+
:377 lt20 (2 )
-3.141666..... . . . .
This value is stated to be given by Appllonius (third cenrury B. C.) and by Ptolemy
( second century A. D. )to. Itis a different in form and magnitude from that found in the
AB. By rounding offalso to f<rur decimals we should get, from the above,
Tt:3 ' L 4 I7
Although the AB value is thus not the same as (2), still some scholarsinsist that the
two are same.lT
Reference and Notes
l. AB, III (Kalakriya), 10. Se,:Tr1abhaita with the commentary of Parame6vara,
edited by H. Kern. Brill, Leiden, 1874, p. 58.
2. Sengupta, P. C. : The Khanda Kht.d1aka of Brahmgupta, translated into English,
University of Calcutta, Calcutta, 1934. introduction p. XIX.
3. Shukla; K. S. : 'Aryabhata I's Astronomy rvith midnight day reckoning'. Ganita
Vol. lB, No. I (June 1967 ), pp. 83-106. Hindi verson of this article appeared in Sri C.8.
Gupto Abhinandon Gronlha ( edited by D. D. Gupta ). S" Chand and Co., New Delhi, 1966, pp.

4. 20 TIrE MATHEMATIcS EDUcATIoN
48Tg4. Other articles on the subjectare : Sengupta,P. C. 'Aryabhatta's Lost work', Bullctin
Math,Soc.Yol.22 (1930), ll5-120; and Rai, R. N., The Ardharatrika Systemof
Caleutta pp.
AryabhataI IndianJ. Ijlist.Science, Vol.6, No.2 (November l97l), pp:147-152
4, SeeShukla, op., cit, pp. 103.104;and T. S. Kuppanna Sastri's edition of the
Mah:a-BhdskariraGovt, Oriental manuscriptsLibrary, Madras, 1957, Introduction, pp. XX-
X xI and p. XLIII.
5. Dvivedi, s. (of Lalla ), Graha-Ganita, Iv, 3
( eclitor) : siryadni-urddhiilt
p.
1886, 28.
Benares,
6. Gupta, R C. : ,,FractionalPartsof Ar1'abha!a's Sinesand Certain Rules.........".
IndianJ. Hist. Science,Vol.6, No. I ( lr4ay'l97l ), PP'51-59'
.; Sachau, E. C. ( translator ) z Alberuni'sIndia. S. Chand and Co., New Delhi,,
l96t; Vol. I, p. 168.
g. Bose,D. N4.and others( editors): AConcise Hiilor7 of Scicnccin
Ind,ia. Indian
National Science New Delhi, l97t p. l87.
Acadenry,
g. Colebrooke, T. (ranslator) : Litaaiti. Kitab Mahal, Allahabad, 1967, I15.
H. p.
10. For an exposition the Katapayidi Systemsee,for example,Datta, B. B. and
of
Singh, A. N. : History llindu Mathcmalics.
of Asia Publishing Ffouse,Bombay, 1962,
Volume
I, pp. 69-72. For the Chronogram, see Yallaya's comntentary available in a transcript
(p. l9 ), at the Lucknow University, of lv{adrasManuscriptNo. D 13393.
ll. Sachau, C. : Op. Cit., Vol. I, p. 169.
E.
12. Datta, B. B.; "Hindu ( Non-Jaina) Valuesof 7T". ./. Asiatic
SccieQ Bengal,
of
Yol. 22 ( 1926 p. 27.
),
13. Quotedby S. N. Senin Bose, M., oP. Cit., p. lB7.
D.
14. Bond,J. D. : ('The Development Trigonometric Methodsdown to the close
of
of the l5th century". fS/,S,volume 4 (1921-22)' PP. 313-314.
15. Heath, T. L. : IlistorT of Greck Vol, I, p.23*.
Oxford, 1965,
Mathdmatic.c.
16. Sengupta, C. ( translator) t Z.r1abheilanl. Dept. of Letters,CalcuttaUniv.,
P.
Vol XVI (1927), p.17.
17. SeeSmith, D. E. : Historyof M.ilhematicr, Dever, New York, 1958,Vol. II, p.
308;and Beyer, C. B. : A History of Milhenatics, Wiley, 1968,pp. 158,187, and233.