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  1. 1. fhe Mathematics Educatrou SECTION BV ol, IX , N o , l , M a r c h 1 9 7 5 GL IM PSE S OFA N C IE N T IN O I A N M A T HE illA T I CSNO . 3 1 Sorne Ancient Values of, Pi and Tlrelr uae In Indla P.O. Mesra, RANCHI (Bihar) Birla Institttteof Technologyb2 R.G. Gupta* Dept. of Mathematics, ( Re ce ive d 2 2 January 1975) The study and investigation of the relation between the circunrlerence C and the diameter D of a circle has always been a part of arrcient and med jtval mathematical a ctiv it y in all parts o f th e rv o rl d . T h e p l o b l e m of sqrral i ng the ci rcl e, that i s, fi ndi ng of a square whose area is the same as that of a giverr circle, t1,as one of the most famous geome- trical problems of r,ntiquitv. The corrstarrtratio CID is n(,w universally denoted by the Greek lerrer rs (Pi) bur the first writer to use the symbol definitely in this sensewas William .fo nes( 1706) r . Various ar.cient and medieval Indian works contain rules which are taken to imply certain values of n . Bclow we present some of those ancient approximations which were used outside Irrdia also. (1) The Sinplest Approximation, [I : 3. T his is an Ol d Ba b y l o n i a n (c . 1 7 0 0 B .C .) val uez. Iti sfound i n theB i bl e and the Talmuds. The same is used in ancient Chinese works such as tlte Chiu ChangSuanSiu (compl- eted in the first century A;D. ) and the ChouPei-in which the following is statedr At the winter solsticethe suns orbit has a diameter 47600 (Chinese) miles, the circumferenceof the orbit being 142800miles. Somewhat similar statements are found in Indian Epic and Puranic literature. !or XII, 44 statess Bh,smaparua, irrstance,a published version of the Maha-Bhdrata lrIQIHTI<T), qd<aaal q€rtlfqt i stt4 gt?rffI I fsctgtqr d(i {rqq q{sii f{€dr qrr[ nyyrl Sitrvastvastau sahasripi dve clnye Kurunarrrdana/ Viskambhet"tt 1a1sldjan mallJalali trirlisati samam ll44ll 0 Kurunandala ! The diameter ol the sun is eight-plus-two thousand (yojanas) whence (its) circular periphery,0 king, is equal to thirty (thousandTojanas). : Th is im plies n : 3 0 0 0 0 /1 0 0 0 0 3 *Mcmber, International Commissiort on l{istory ol Mathematics
  2. 2. IEE Itr IIIEEM IT IOB E D U C ITION As another example, we quote the following stanza from the Eltg11q Vilu purdnas nsqlqrsrq(i FsTilrrt qR*((qg I qf"qrQlsq F<rar<thgttsitrcrl c(eqeT || The diameter of the sun is nine thousandltojanas . Three times the diameter is the cilcumference of its peripheral circle,. T he B audht )a n a{ u l b a s i tra ,r,l l 2 -Il 3 [ c.500 B .c. ?] al so i mpl y the same approxi m-ationT. Even some of the later writers who knew better values, mentioneclthe above appro-ximation for crude and quick calculatiorrs. For example, Brahmagupta (628 A.D.) in hisqlg€Safgar;d Brifuna-sphuta-siddhtnta (: BSS), XII,40 says8 6qr€a4rrltd5ft qflrfqqil E{Trstif(+ hgti r Vyr{sa-vyrsirrdha-kr.ti parid},i-phale vyivaharike trigu ne / The diameter and the square of the semi-diarneter (separately)multiplied by threegivc the practical (or gross) circumference and area (of the circle) (respectively).,lh:it is, C:3D (r) Area : 3(Dl2)t (2) (n) TheJaina Value, n - i; I n Chinat his v a l u e w a s g i v e n b y C h a rr g H eng (c.130A .D .) ancl by C hhi n C hi u-S hao(c.1 2 50) 0.I t has be e n c a l l e d th e J a i n a V a l u e because t i s fi equentl y usedi nJai l a vr.o1ks. iThere are plenty of references but the controversies regarding clateand authorship of someof these works makes it almost impossibleto decide with certainty the earlicst r,se..,tthevalue in India. The canonical work {Rqcq(fff, S|.ria-ltanr.-ralr (Saskrit, qdwfia Shryaproj?t6p1;1 igstated to have adoPted the rulelo c:"/Too (3)for getting the circumference of the suns innermost orbit as 315(lB9 l,rjana.rplus little morefro m or bit s m ent io n e d d i a rn e tt r o f 9 9 6 4 0y o j a n as, and for getti ng di rrrerrsi ons the suecessi - olve orbits, but the work is clated variously in the range from abour 5((, B.C. to about500 A.D. As a sample of an origin:rl text for the nrle (3) we quote the following frorn a c()rnm-entary on the (t4lSiftf q q qa T attuart hridhigama-sutr aLr lqorqcrti$gqlqr {d qrqftiq: r Viskambha-klter-da6agunayii rnularlr vrta-pariksepah . T he square-root of ten times tl.resqrrare of the diameter is the circrrmference ofth e c ir c le. T he alaovervork and the commentary are both attributed to gqf(1lfld Umtisvali (fi1s1co :tUr y A D. ) but t: rv i r.rl : rn .rrti r (i :rc l i r di ug date) i s not free from some seri orrs
  3. 3. R, C. GUP|IA 2.controversiesl There are several other direct and indirect reference to (3) or its equivalentrs. Amongthe early non-Jaina rvorks nhich mention this value are the Sitrla-Siddhdnta,I,58 (K.S.shukla,s edition, p.l9; Lucknow, 1957); Panca--siddhantika, lY, l, of Varrhamira (sixthce n ru r y A . n. ) ; B r ah ma g u p ta sB SS , X It,4 0 (s e condhal f) w here i t i s gi ven to be an accurate(s[&srna)value; etc. fhis value appearedrin the eleventh centur)Spain in a work of Az--Zarcpli otArzachel (under Indian influence). It is rurprising to note that even very late mathematicians continued to stick to theva l rre ( pos s iblybec a rrs eo f tra d i ti o n a n d th e e legant form of the val ue) r.vhenmuch more:rcculat{jvalues were ucll-kn()rr.For instance, the Sidd}tintatatlba-uiuefta (f€af;Adtrfle*O),II, l +7 ( B enar esedi ti o n , 1 9 2 4 , p .5 0 ) o f Ka ma l rrkara (1658) gi ves i t. In C hi na i t w as sti l lb e i n g us ed in t he m id d l e o f th e l Bth c e n tu ry rs . (III) The Archirnedean Value, II : 2217. Archimede-s (c.225 B.C.) had showrr that the ratio CID is lessthan 2217 and greaterthen 223171, but after his trme the value 2217became recognizedas a satisfactory approxim-a ti o n l 6 . I t n as giv enrz b ),H e ro n o f A l e x a n d ri a (f i rst century A .D .) and by R abbi N ehemi ah(c. 1 50) . I t was gen e ra l l y u s e d i n C h i n a i n th e fi frh century A .D . but Tsu-C hhung - C hi h(4 3 0 -5 01) c ons ider e d i t i n a c c u ra te a n d g a v e th e famous C hi neseval ue (seebel ow )l 8. A l- B ir I ni ( ele v e n th c e n tu ry ) h a d c re d i t ed B rahrnagupta for know i ng the aboveva l tte r e. Lalla ( eigh th c e n tu rl ,) g i v e s th e d i a n re t er of the earth as 1050uni ts and i ts ci rcum-fercnce to be as 3300 units which imply the :rbove valuer0. fr,ryabhataII (950) has giveni t e xp l ic it ll as an ac c rrri rte a l u e i n th e fo l l o .,v i n grvords zr v aqrgrsstidqrd]ssafq q+q cf(fq: r rg Rr r €d: {qci T he diam et er (w h e n ) n i u l ti p l i e d b y 2 2 and di vi dcd by 7 becomes the accurateci rcttm f er en e , c Subseqtrently, the :rbr.,ve vaiue is found mentioned by several lndian arrd foreign wri-te rs i n clr r ding B his k a ra II (1 1 5 0 ) w h o c o n s i d e re di t as grosszz. (IV) The Chinese Value, tI : 355/113. As alle;idy tnentioned, this was given by the Chinese Isu in the 1fifth century. In theDhat,ala(tfaor) commentary by elrlc Virasena (c.800) is found a Sanskrit stanza whichgives the rule23 c:3D + (l6D + t 6 )1 1 r3 (4 )And this would have yielded the Chinesevalue of r if the illogical absolute constant numberl 6 we re not t her e in i t. The value is stated to occur in the Tantra-samuccala2{ Nlrziyarta (c. 1450). It is used ofin the Tantra-saitgraha(1500) of Nilakarltha Somayrji (f,tq6€ eltatfe)tu. It is also found in
  4. 4. !HE Itr IIIIIEM A!IOB ED gOAI!IOIIhis Golasdra, [II, l2 as a close approximation in the following wordsto les,ioqclaqrs: cftT: cqls$fqKrgqqFr: r Vidvaika-samo-vydsah paridheh prdyortha-bha-gunabhegah I (When) the diameter equals ll3, the circumference has 355 parts nearly. Some European writers of the sixteenth century also gave the valuez7. (V) The Egyptian Value II : (16/.19)r This value is implied in a rule fourrd in the Ahmes Papyrus (c. 1600 B.C.;za. In India, it is found in the Md.naaa-Sulba-sitrar2s in the Triloka sara of Nemican- andd r a ( lO t h c ent u ry )3 0 . We have already published a separatearticle on Aryabhata Is (born 476 A.D.) Valueo f P i ( 3. 1416)in th e p re s e n tSE R IE SS r. References and Notesl. D.E. Smith, Historlt of MathematicsVolumeII, p, 312 ( Dover, New York, l95B).2. B.L. Van der Waerden, Science Awakening, pp. 32-312 and 75 (Wily Science Ed., N. Y . 1963 ).3. Smith, op,cit.,p. 302 where the following passageis c1ur.,tt,d nr tht Bible (l Kings fir vii, 23) : "And he made a molten sea, ten cubits fi<.mone brim to the; it was round all about . . . an d a l i n e o f th i rty c u b i ts d i d compassi t round about" .4. Y. Mikami, The Deuelopment Mathemalics ChinaandJapan, p. B (Chelsea,N. Y., 196l). of in J. Needham, Science andCiailization in China VolumeIII t Mathematicsand etc., p.99 (Cambridge lJniversty Press, 1959, places two works in the Han Period (202 B.C. to220 A.D.) and gives other references.5. The Mahdbhdratawith Hindi translation, Part III, p.2572 (Geeta Press,Gorakhpur).6. S r i Ram Sh a rma s e d i ti o n u i th H i n d i transl ati on, P art I, p.357 (B arei l y, 1967). Almost sirnilarly worded stanzais ft,trr,d in the ffltrma-Purirla (sec Sharmas edition, P ar t [ , p. 4 5 6 ; Ba re i l y 1 9 7 0 ).7. B.B. Datta, The Science thc Sulba, 149 (Calcutta, 1932). of p.8. T he B S S ed i te d b y R S Sh a rma a n d o thcrs, V ol . III p. 857 (N ew D el hi , 1966).9. M ik m i, p. 7 0 ; Smi th p . 3 0 9 ;N e e d a m, p . 100.lC. H.R. Kapadia (editor), Ga ita-tilaka (Baroda, 1937), Introduction, p. XLV; and B. Datta, "IheJaina School of Mathenralics", Bullelin Calcutta Math. Soc., Voh.m, XXI ( 1929) pp. I l 5 -l a 5
  5. 5. R. C. GUPTA edited by Khubachandraji, p.170 (Bombay, 1932).ll. The Sabhasla-Tattuirlhi.dhigama-sitra12. For a few controversies, see R.C.Gupta, "Circumference of the Jambudvipa in Jaina Cosmography". Paper presented at Seminar on Lord Mahavira and His Heri- ta g e , N ew Delhi, 197 3 . However, D. Pingree, Censusof the Exact Scicnccin Sanskrit,series A, Volume I, p.5B (Philadelphia, 1970) agrees with the authorship and date as given by us in the present paper.13. Few more references are given by us in the paper just cited. We also propose to p u b l i s h a s epar at epa p e r o n th e J a i n a V a l u e o f Pi .14. J. D. Bond, "The Development of Trigonometric lethods etc.", .I,S/,S Volume.l (1921- -22), p. 314.15. Ne e d h am , op. c it . s e es e ri a l N o .4 a b o v e ), p . 1 0 2 (16 . Smi th, op. c it . , p. 30717 " Smi th, p. 3O 7;and W a e rd e n , p . 3 3 .i8. Ne e d h am , p. l0l India,Yol. I, p. 168 (Two parts in one, Delhi, 1964).19. E.C. Sachau (translator), Alberunis20. Tlre Jisladhlutdhida(fucttflgtZ<), Grahaganita I,56 (S. Dvivedis edition, Benares,1886, p .l 0 ).21. Th e tr t ah. is iddhant ra tfv a r;d ), XV ,9 2 (a (S . D v i vedi s edi ti on, B enares,1910,p. 172).22. Smi th , pp. 307- 310a n d B o n d , p . 3 1 4 .23. HiralalJains edition of the ,q1ftft44Jz1ganra (Uqra{et{S) with Dhauali commentary, Vo l u m e I V , pp. 42 a n d 2 2 1 (A rn a ra v a ti , 1 9 4 2 ).24. D.M. Bose and others (editors), AConciseHistor2 of Science India, p. 147 (New Delhi in l e Tl ).25. The Tantra-samgraha,Il,fTrivandrum, 1958, p. lB).26. The Golas-trar l9 (K.V. Sarmas edition), Hoshiarpur, 1970). p.27. Smi th, p. 310 and N e e d l ra m, p . I0 l .28. Smi th, p. 30229. Datta. Scicnce Sulba,p. 149. of30. g a th i l8 ( M anohar L a l s e d i ti o n , Bo mb a y , l 9 l B , p. l 0).31. Glimpses of Ancient Indian l[athematics No. 5, The Mathematics Education Volume VII, No . I ( M ar c h l973 ),S e c ti o n B, p p . l 7 -2 0 .