the Mathematics EdcationVol. VJI, No l , March 1973

SECTION B

OTIMPSES Otr ANCIENT INDIAN MAT[, NO.5Aryatrhata I's Value ()t n

by R.C, Gupta. Dept. oJ Mathcmatict, Birla Institute oJ Tcchnologlt P. O. Mesra, Ranchi.( Receivecl 25 Janurry 1973 )

l . Introducr ion

The Sanskrit rvork:irtfqe.i4 Aryabhatil 'a (:AB) u'as u-ritten by the well--knorvn Ind-ian astronomer and ruathematic ian . i ryabhata I ( born A. D. 476 ) . According to aninterpretation 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 notjustif ied 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 thearit iQfat.Eta ) rvhich is not extant. This astronomical work was based on the midnight systemof day-reckoning in contrast to the AB in rvhich the day rvas reckoned from one sunrise tothe next. A fresh and detailed study concerning this lost rvork and of quotations from it asfound 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-ached stanzas ( Muktakcr ) is a lso at t t ibuted to the author of the ABt.

2. Approximat ion of n as givcn by Aryabhala I

The AB. II, 10 ( p. 25 ) gives the follorving rule

s$ftTsi {rilqegui dlqFEw?il qqqtqT1q Isrgda{rrssfirrrcqrvq} gqcflsrr(: il lo ll

Caturadhikarir (atami;tagu.larir dvd;as!istathi sahasri4dm rAyutadvaya-vi;kambhasy-isanno vftta parinihall rr l0 rr

'Ifundred plus four rnultiplied by eight and (combined with) sixty-two thousands isthe approximate circurnference of circle of diameter twenty thousand.'That is,

Circunrference, C-(100 +4) x B*6i000 approx, ,whendiameter, D-20000.

So that

11 :ClD-62832/20000 approx:3. t4 l6 t tJ

l8 'rHE MATHEMATTcS EDUcATIoN

The value of n is correct to four decimals and is one of the best approximation for it

used by the ancient peoples any where in the world. What is equally important to note is

that the author states the value to be an approximate one only. This means that he was

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

7f :3+,1, -^l ' l" t+ 16+ l l

This yields the following successive approximations(i) :3 which is the simplest approximation.(ii) =2217 which is called the Archimedean value.(ii i) :3551113 which is called the chinese value or Tsu's number.

(iv) - 392711250 which is simply the reduced form o[ the aB value.

3. Aryabhala'e value as found in other worke.

Lalla ( eight century ) gives a rule according to whichs

C x 62513927-Radius, R.

This implies2tr=39271625

which gives the same value as (1) but in the reduced form :iv) above'

If we take C to be egual to 360x60 parts ( or minutes ), then we have

B-2t600/6.2B32

-3+37'73872 nearly

:3437 44 l9 '4'+66+ eO *tOO

aPProximatelY

By rounding off this value, separately, to the nearest minute or second or third; we

get the norrn or Sinus-Totus (fisl) as found respectively in the AB, the Vate6vara-Siddhinta

( tenth century ), and Govind Svlmin's commentary ( ninth century) on the Mah6''Bhdskar\ta

in connection with the tables of sine6.

A certain astronomer, Puli6a, has also eniployed the same value of 7f as found

ABi .lJtpala ( tenth century ) is also stated to have mentioned the same value in his

mentary on the famous work BShat-Samhita ( q€itiqir )B' Bhdskara rr ( twelfth century

given the same value but in the reduced form (iv)e.

Yallaya (rSth century) in his commentary on the AB has expressed the same value in

thc following chronogram in the Katapayddi systern of Indian numeralsro.

t$afu gr frleflea+ru) arfa il{c I4. Aryabhala's value of ?r Transmitted to the west

Yaqub Ibn Tariq ( Baghdad, eight century ), on the authority of his Indian informant

in thecom-

) has

,t'

R,. C. GUPTA

l256,640,000uni tsandthat i tsdiameter is400,000,000uni tsrr . This impl ies"a value 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 language which, in F. Rosen's

translat ion, is as fo l lorvsr3.

,...,......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 otherrespects also.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 scholars sr,rspect this value to be of Greek origin (for the Greeksalone of a l l people made myriad the rrni t of second order ( Rodet )16.

Fforvever, the choice of a radius of 10000 units may be a matter of convenience guite

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, theAB value, is the following

r -3 * (8/60) + (30i60:):377 l t20 -3.141666.. . . . . . . . (2)

This value is stated to be given by Appllonius (third cenrury B. C.) and by Ptolemy( second century A. D. )to. It is a different in form and magnitude from that found in theAB. By rounding offalso to f<rur decimals we should get, from the above,

Tt:3 'L4I7

Although the AB value is thus not the same as (2), sti l l some scholars insist that thetwo are same.lT

Reference and Notes

l. AB, III (Kalakriya), 10. Se,:Tr1abhaita with the commentary of Parame6vara,edited by H. Kern. Bril l , 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'. GanitaVol. 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.

l9

20 TIrE MATHEMATIcS EDUcATIoN

48Tg4. Other articles on the subject are : Sengupta, P. C. 'Aryabhatta's Lost work', Bullctin

Caleutta Math,Soc.Yol.22 (1930), pp. ll5-120; and Rai, R. N., The Ardharatrika System of

Aryabhata I IndianJ. I j l ist . Science, Vol.6, No.2 (November l97l) , pp:147-152

4, See Shukla, op., cit, pp. 103.104; and T. S. Kuppanna Sastri's edition of the

Mah:a-BhdskariraGovt, Oriental manuscripts Library, Madras, 1957, Introduction, pp. XX-

X xI and p. XLIII.

5. Dvivedi, s. ( eclitor ) : siryadni-urddhiilt (of Lalla ), Graha-Ganita, Iv, 3

Benares, 1886, p. 28.

6. Gupta, R C. : ,,Fractional Parts of Ar1'abha!a's Sines and Certain Rules.........".

IndianJ. Hist . Science, Vol.6, No. I ( l r4ay' l97l ) , PP'51-59'

.; Sachau, E. C. ( translator ) z Alberuni's India. 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 Acadenry, New Delhi, l97t p. l87.

g. Colebrooke, H. T. (ranslator) : Litaaiti. Kitab Mahal, Allahabad, 1967, p. I15.

10. For an exposition of the Katapayidi System see, for example, Datta, B. B. and

Singh, A. N. : History of llindu Mathcmalics. 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{adras Manuscript No. D 13393.

l l . Sachau, E. C. : Op. Cit . , Vol. I , p. 169.

12. Datta, B. B.; "Hindu ( Non-Jaina ) Values of 7T". ./. Asiatic SccieQ of Bengal,Yol. 22 ( 1926 ), p. 27.

13. Quoted by S. N. Sen in Bose, D. M., oP. Cit . , p. lB7.

14. Bond, J. D. : ('The Development of Trigonometric Methods down to the closeof the l5th century". fS/,S, volume 4 (1921-22 )' PP. 313-314.

15. Heath, T. L. : Il istorT of Greck Mathdmatic.c. Oxford, 1965, Vol, I, p.23*.

16. Sengupta, P. C. ( translator ) t Z.r1abheilanl. Dept. of Letters, Calcutta Univ.,Vol XVI (1927), p.17.

17. See Smith, D. E. : History of M.ilhematicr, Dever, New York, 1958, Vol. II, p.308; and Beyer, C. B. : A History of Milhenatics, Wiley, 1968, pp. 158, 187, and233.