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  1. 1. The Mathematics Educatron SECIION BVol. I X , N o . 2 , J u n e 1975 GL IM P S E S OF A N C IE N T INDI A N M A T HE M A T I CS .1 4 NO Ttre Lffavati rrle for cornputing sldes of reE:ular polyEonsl b2 R. G. Gupta, Llcnber,International Commis.rion History of Mathematics) on Department of , Birla Instituteof 7eohnologP.O. Mesra, RANCHI (India) Methematic.e ( Re ce ive d lB .P ri l 1975) 1. Introduction Coming from the pen of the famous Bhlskarrcarya (efea<fWd), the Lil.ivati (d tvf adl) is t he m o s t p o p u l a r w o rk o f a n c i e n t Indi an mathemati cs. The cel ebratedauthor belonging to the twelfth century A D., was a great Indian astronomer and mathenratician who wrote several other works alsor: He is now usually designated as BlrtrskaraII (son of Mahedvara) to distinguish him from his name sake Bhlskara I who lived in the seventh ce n t ur y of our er a . T h e a u th o r o f L i l i v a ti w as born i n daka 1036 (or A .D . l l l 4) and wrote the work abolt the middle of the twelfth century. Written in lucid Sanskrit, it is devoted to arithrnetic, geometry, mensuratiort, and some other topic of elementary rn a t hem at ic s . Ever since its composition, the Lilrivatl has inspired a number of commentaries, translations, arrd editions in various [ndian languages throughout the past 800 years. It wa s r ender ed int o Pe rs i a n b 1 F a i z i (1 5 8 7 A .D .) under the patronape of ki nd A kbar. Amo ng t he E nglis h tra n s l a ti o n so f th e w o rk , th e one by H .T. C oi ebrooke (London, l 8l 7) i s well- k nowr r t . T h e re c e n t (1 9 7 5 ) e d i ti o n o f the w ork by D r. K .V . S arnra i s val uabl e be- cause it includes an important and elaborate sixteenth century South lndian conrmentarl3, T her e is a t h ri l l i n g s to ry 4 a c c o rd i n g to whi ch LILA V A TI ( beauti ful ) l as the name of Bhdskras only daughter and that he titled the work after her name in the hope of consol- ing her for art accident r,vhichprevented her marriage. But whether rhe romantic story has any historical basis or not, it is stated to be found narrated even in the Preface to Lildvatis translation by Faizi (sixtecn century)5. The Rule For Flnding the Sides. In the present article we shall discussa rule from the Lilirvati about the numerical computation of the sides of regtrlar polygons (upto nine-sided) inscribed in any circle ol diameter D. The original Sanskrit text as commonly found in the Lil:ivati ksetravyavah?ira,
  2. 2. 26 THE M ATI:ITM ATI C E ED U G AT IO N206-2C8 as followsG is : flaaqe+rfiqcqsg;f:faarqrsetmcefq: I ?erficerqersdqq ts(alqFrr€: fiHr( ltRo!ll erliEtoerqtlsqkldc;?{qrqt: t S(Iqsqldsq qf,aqrduer6* slQoetl tq€?qrrrrddqt erq;t mRnl gwr: r 1tt;ae+agatqi aatatd UqTTq{nRoctl Tridvyarikagni - nabha"{candraih tribi=,tt.lsta yuedstablih / - Vedagnibauaidca khi( vaiSca khakhabhr. ibhra-rasaih kramdt I 1206 II Banesunakhabinaiica dvidvinandesu-s:lgaraih/ KurlmadaJcavedaiica vtttavyasgsanrihate 1 207 l l Khakhakhibhrlrka sa4rbhakte labhyante kramio bhujih / Vlttiinatas-tryasra-pt1rrInirlr navdsantam prthak-prthak I 12081 |This may be translated thus : Iltultiply the diameter of the (given) circle, in order, by (the coefficients) 103923,84853, 70534,6 0 C 0 0 ,5 2 0 5 5 , 5 9 2 2 ,a n d 4 1031. On di vi di ng (each of the products j ust 4o bt ained) by 1 2 0 0 0 0 ,th e re a ro o b ta i n e d th e si desrespecti vl yof thr: (cel rri l atcral )tri angl e tothe (regular), nonagon (inscribed in the cilcle separately. That is, tire side of the inscribed regular polygon of n sicles given by is r" :(D /1 2 0 0 0 0 ). & " (l )where the seven coefficientskn, forn equal to 3 upto 7, are separately given in the aboveverbal rule. It is clear from (t) that when D is taken cqual to 120000, v;e shall have snequal to &o itself. Thus it may be said that Bhiskarasccefficients represent the sides ofregular polygons inscribed in a circle of radius 60u00. T he Lila v a t, w a s e q u a l l y p o p u l a r i n the l ate A ryabhata S chool . B ut the ori gi naltaxt seemsto be changed at scveral places apparently to improve rrpon it. It is therefrrreno surprise that same of the above coefficientshave different values in the taxt c f the ruleas published alon.gwith the Kriylkramatcari (|fr,+t*e+il) commentary (sixteenth) centrrrybelonging to the School?. We present the two sets of cofficients in the form of a table rvhich also contain thecorresponding modcrn or actual values for the sake of cornparasion.
  3. 3. R. C. GUPT A 27 TABI,E (Sides polygorrs of inscribed a circleof redius in 60(,00)No . o f O r iginz ri Kri y z k ra makari Modern valuesi de s Lr llv at i re a d i n g ( t o r r , a r e s t i n t t g e r ) v alue.l 103923 I 0 3 92 2, k r 03923+ 8+853 s am e same5 70i3+ s am e sameo 6000rJ s am e same 5205s 52C67 s2066a 45922 s ant e same9 41031 a. lt L2 41042 J us t af t er s t a ti n g th e a b o v e ru l e , th e a u th or has gi ven and w orked out the fol i ow i ngexample : I n a c ir c le of d i a me te r 2 0 0 0 , te l l me s e p aratel ythe si des of the (i nscri bed) equi l a-teral triangle and etc . 3. Retionales of the Rule T he r net hod o f d e ri v i n g th e s ec o e ffi c i e n t s not gi ven i n the Li l i vati . The comi nel -ta to r Ganc s a ( 1545 ) me n ti o n s tw o m e th o d s o f obtai ni ng them (pp. 207-208). The fi rst i sb a se dor r r r s inga t ab l e o f Si n e s to o b ta i n k" - 120000sin (190ln) (2) F or th i s p u rp ()s e n e s a u s e d Gava l u e s f i om t he t abl e o f S i n e s (fo r rl te ra d i u s 343t1) w hi ch i s founds i n B h:tskar:rII sastrrnomical work called Siddhzint;r iiromani ifvarr;afwrlq|qr). But the cooefficientso b ta i ned ir r t his lv ay (u s i n g l i n e a r i n te l p o l a ti o n w here neceesary) are so rought that mastof them do not agree lvith tlrr;segiven in the original text. Horvever, a secondorcier inter-polation does help in this resf,ect (seebelow). Ofcoures, more :rccrlrateSine tables can be rrsedto derive the values of the coffici-ents to the supposedor irnlied degree olaccuracy. But it is doubtul whether Bhiskara IIhad anv such table ready at hi-sdisposal although he knewe a method of constructing a tableof 90 Sines (that is, with a tabular interval of one degree) which could serve the prrrpose.Moreover, two of the coefficientsare far from being accrrrateto the same dcgree as others.This indicates the possibility of sonre different method. The second method given by the commer.tator GaneSa consistsof finding the sidesT hc las t d i g i t r c a c l i n g d vi ( two .; is sta tcd to b e co lr cctcd to tri (three) i n one of the manuscri pts :
  4. 4. 28 IIIE I.f,T Ir IIIIT tOg E D go^|rIOXof the inscribed triangle, sguare, hexagon, and octagon geometrically by the usual methodofemploying tl,e so-called Pythagoream theorem (see Colebrookes translation, pp.120-12l). However, he remarks that the proof of the sides of the regular pentagon, hep-tagon, and nonagon cannot be given in a similar (simple and elementarr) manner. This method is cssentially equivalent to findin61of Sines of the type (2) geometricallyfor n equal to3r 4,6, and B in which cases the exact values can be easily obtained byemploying elementary mathematical operations upto the extraction of square roots. Theaccuracy of the text values in these casespoints out that it was possibly this very methodwhich was followed by Bhakara II. He also knew the exact value of the Sine of 36 degreeswhich explains the accuracy of his cofficient for n equal to 5 (pentagon)r0. The ramaining cases(septagon and nonagon) are difficult and the lack of knowledgeof the exact solutions is reflected in the much less accurate text-values in these two cases.But how did Bh{skara got even these approximate values ? One possibility is that he usedhis tabular Sines (as indicated by Ganeda) but employed Brahmaguptas (A.D. 62S)technique of second order interpolation which he knew and which is equivalent to themodern Newton-Stirling intcrpolation formula upto the second orderrr. By tbis methodthe result for z equal to 9 (nonagon) tallies almost fully, but in the only remaing case ofheptagon (n equal to 7), the most tedious onc where even the argumental angle is notexpressiblein whole degreesor minutes, a small difference is f,rund. Refcrenceg and Notcst. For a brief description of his works, see R.C. Gupta,"Bhiskara IIs Derivation for the Surface of a Sphere" (Glimpses of Ancient Indian Mathcmatics No.6), The Malhema- tics Education, Vol. VIII, No. 2 (June, 1973), sec. B, pp. 49-52.2. C,rlebrookesEnglish translation, with nots by H.C. Banerji, has been recently reprinted by M/s Kitab Mahal, Allahabad, 1967.3. K.V. Sarma leditor): Lll;ruatl with Krilt-,kramakariof sa,rkaraand Nitilar.ta, Vishveshara- nand Vedic Research Institute, Hoshiarpur, 1975.4. Edna E. Krarner ; The Main Strearn of Malhematics. Oxford univ. Press, N.Y., 1951, P p. 3- 5. Also see the present authors note on LILAVATI published in Tfu Hindustan Timet, New Delh i , V o l .5 l , N o . l 2 l , p . 5 (d e ted the l 9th May 1974).5. R.E. Moritz : On Mathematics, 164. Dover, New York, 1958. p,6. See the Lilivatr with the commentaries of Ga4eSa and Mahidhara edite<iby D.V. Apte,
  5. 5. R. G. GUPTT 29 Part II, pp. 207-208,Poona, 1937 (Anandasram Sanskrit Series No. 107). In Cole- b ro ok e st r ans lat io rr(p . 1 2 0 ), th e s es ta n z a sa re numbered as 2(9-211.7. Sarma (editor), op. cit., pp. 204-206.B, Bapudeva Sastri (editor) t Siddhinta Siromani, Graha Ganita, II, 3-6, pp. 39-40 (Benares,1929). This table appeared earlier h tbe Mah,isidhanta Aryabhata II (960 A. D). fhe Sine table of Aryabhata I (born 476 A.D) and ,SltrTa-siddantaslightly different. is9. See R.C. Gupta. "Addition and Subtraction Theorems for the Sine aud their use ir.r computing Tabular Sines" (Glimpses of Ancient Indian Mathematics No. ll),Thc Malhemalics Edacatinn, Vol. VIII, No.3 (September 1974), Sec. B, pp. +3-46.10. See his Jlotpatli, verses 7-B in Bapudeva Sastri (editor), op. cit., p. 28l.It. Sid. iiir. Graha Ganita, II, l6 (Bapudevas edition cited above, p.42). For details see .tl R. C. Gupta, ttSeconder Order Interpolation in Indian Mathematics etc.", Indian J. or Htst. Sciencc,Vol. 4 (1969), pp. 86-89. 1n of th nd is rry C S, It ar. t7) be- ;3. me ;ol- ory to i cal r ol i ra,