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the Mathematics Edcation SECTIONB Vol. VJI, No l, March 1973 O T I M P SES ANCIENT Otr INDIANMAT[, NO.5 Aryatrhata Is 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.i4Aryabhatila (:AB) uas 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 aninterpretation of the staternentl made in the AB it-self, the nork 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 notjustified 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 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-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 thousandsisthe 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
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l8 rHE MATHEMATTcSEDUcATIoN it The value of n is correct to four decimals and is one of the best approximation forused by the ancient peoples any where in the world. What is equally important to note is wasthat the author states the value to be an approximate one only. This means that heaware of the fact that the value is not exact, although it is close (d.sanna) to the truevalue. 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 Tsus number. (iv) 392711250 which is simply the reducedform o[ the aB value. - 3. Aryabhalaevalue 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+3773872nearly 44 l94 :3437+66+ aPProximatelY eO*tOO By rounding off this value, separately,to the nearest minute or second or third; weget the norrn or Sinus-Totus(fisl) as found respectivelyin the AB, the Vate6vara-Siddhinta( tenth century ), and Govind Svlmins commentary ( ninth century) on the Mah6Bhdskarta 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. Aryabhalas value of ?r Transmitted to the west Yaqub Ibn Tariq ( Baghdad,eight century ), on the authority of his Indian informant
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,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. Rosens tra n sl a t ion, is as f ollo rv s r3 . ,...,......Multiply the diameter by sixty-two thousand eight hundred and thirty-twoand 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 theeleventh century Spain in the r,vork of Az-Zarqali who followed the Indians in many otherrespectsalso.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 Greeksal 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 convenienceguitesuitable to the Indian coinputational methods in decimal scale, and for attaining the desiredaccuracy 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 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 theAB. 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 thetwo 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 Is 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.
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20 TIrE MATHEMATIcS EDUcATIoN48Tg4. Other articles on the subjectare : Sengupta,P. C. Aryabhattas Lost work, Bullctin Math,Soc.Yol.22 (1930), ll5-120; and Rai, R. N., The Ardharatrika SystemofCaleutta 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 Sastris edition of theMah: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 Ar1abha!as Sinesand Certain Rules.........".IndianJ. Hist. Science,Vol.6, No. I ( lr4ayl97l ), PP51-59 .; Sachau, E. C. ( translator ) z AlberunisIndia. S. Chand and Co., New Delhi,,l96t; Vol. I, p. 168. g. Bose,D. N4.and others( editors): AConcise Hiilor7 of Scicnccin Ind,ia. IndianNational 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 ofSingh, A. N. : History llindu Mathcmalics. of Asia Publishing Ffouse,Bombay, 1962, VolumeI, pp. 69-72. For the Chronogram, see Yallayas 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, ofYol. 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 ofof 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.
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