The Mathematics Educatron

Vol. IX, No. 2, June 1975

SEC'I ' ION B

GLIMPSES OF ANCIENT INDIAN MATHEMATICS NO. 14

Ttre Lffavati rrle for cornputing sldes ofreE:ular polyEonsl

b2 R. G. Gupta, \Llcnber,International Commis.rion on History of Mathematics) Department of

Methematic.e , Birla Institute of 7'eohnolog P.O. Mesra, RANCHI (India)

(Received lB . \Pr i l 1975)

1. Introduction

Coming from the pen of the famous Bhlskarrcarya (efea<fWd), the Lil. ivati

(dtvf adl) is the most popular work of ancient Indian mathematics. The celebrated author

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 Blrtrskara II (son ofMahedvara) to distinguish him from his name sake Bhlskara I who lived in the seventhcentury of our era. The author of L i l ivat i was born in daka 1036 (or A.D. l l l4) andwrote the work abolt the middle of the twelfth century. Written in lucid Sanskrit, i t isdevoted to arithrnetic, geometry, mensuratiort, and some other topic of elementaryrnathematics.

Ever since its composition, the Lilr ivatl has inspired a number of commentaries,translations, arrd editions in various [ndian languages throughout the past 800 years. Itwas rendered into Persian b1' Faiz i (1587 A.D.) under the patronape of k ind Akbar.Among the Engl ish t ranslat ions of the work, the one by H.T. Coiebrooke (London, l8 l7)is wel l -knowrrt . The recent (1975) edi t ion of the work by Dr. K.V. Sarnra is valuable be-cause it includes an important and elaborate sixteenth century South lndian conrmentarl3,

There is a thr i l l ing story4 according to which LILAVATI ( 'beaut i fu l ' ) \ las the nameof Bhdskra's only daughter and that he tit led the work after her name in the hope of consol-ing her for art accident r,vhich prevented her marriage. But whether rhe romantic storyhas any historical basis or not, it is stated to be found narrated even in the Preface toLildvati 's translation by Faizi (sixtecn century)5.

The Rule For Flnding the Sides.

In the present article we shall discuss a rule from the Lil irvati about the numericalcomputation 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,

26

206-2C8 is as fol lowsG :

THE MATI: ITMATICE EDUG ATION

flaaqe+rfiqcqsg;f: faarqrsetmcefq: I

?erficerqersdqq ts(alqFrr€: fiHr( ltRo!ll

erliEtoerqtlsq kldc;?{qrqt: t

S(Iqsqldsq qf,aqrd ue'r6* slQoetl

tq€?qrrrrd dqt erq;t mRnl gwr: r

1tt;ae+agatqi aatat'd UqTTq{ nRoctl

Tridvyarikagni - nabha"{candraih tribi=,tt. lsta - yuedstabl' ih /

Vedagnibau aidca khi( vaiSca khakhabhr. ibhra-rasaih kramd t I 1206 I I

Banesunakhabinaiica dvidvinandesu-s:lgaraih /

KurlmadaJca'vedaiica vt'ttavyasg sanrihate 1 207 l l

Khakhakhibhrlrka sa4rbhakte labhyante kramio bhujih /

Vltt i inatas-tryasra-pt1rrInirlr navdsantam prthak-prthak I 12081 |

This may be translated thus :

'I l tult iply the diameter of the (given) circle, in order, by (the coefficients) 103923,

84853, 70534,60C00,52055, 45922, and 41031. On div id ing (each of the products just

obtained) by 120000, there aro obtained the sides respect iv ly of thr : (celrr i latcral) t r iangle to

the (regular), nonagon (inscribed in the cilcle separately.'

That is, t ire side of the inscribed regular polygon of n sicles is given by

r" : (D/120000). &" ( l )

where the seven coefficients kn, forn equal to 3 upto 7, are separately given in the above

verbal rule. It is clear from (t) that when D is taken cqual to 120000, v;e shall have sn

equal to &o itself. Thus it may be said that Bh'iskara's ccefficients represent the sides of

regular polygons inscribed in a circle of radius 60u00.

The Li lavat, was equal ly popular in the late Aryabhata School . But the or ig inal

taxt seems to be changed at scveral places apparently to improve rrpon it. It is therefrrre

no surprise that same of the above coefficients have different values in the taxt c f the rule

as published alon.g with the Kriylkramatcari (|fr,+t*'e+'i l) commentary (sixteenth) centrrry

belonging to the School?.

We present the two sets of cofficients in the form of a table rvhich also contain the

corresponding modcrn or actual values for the sake of cornparasion.

R. C. GUPTA

TABI,E(Sides of polygorrs inscr ibed in a circ le of redius 60(,00)

27

No. ofs id es

Originzr i

Lr l lvat ivalue

Kriyzkramakar ireading

Modern value

(to rr , 'arest int t 'ger)

.l

+

5

o

a

9

103923

8+853

70i3+

6000rJ

5205s

45922

41031

I0 3 92 2,k

same

same

same

52C67

sante

a. l t \L2

r 03923same

same

same

s2066

same

41042

Just af ter stat ing the above rule, the author has given and worked out the fo l iowing

example :

' In a c i rc le of d iameter 2000, te l l me separately the s ides of the ( inscr ibed) equi la-teral triangle and etc .

3. Retionales of the Rule

The rnethod of der iv ing these coeff ic ients not given in the Li l ivat i . The cominel-tator Gancsa (1545) ment ions two methods of obtaining them (pp. 207-208). The f i rst isbased orr r rs ing a table of Sines to obtain

k" - 120000 sin (190ln) (2)For th is purp()se Ganesa used

values f i 'om the table of Sines ( for r l te radius 343t1) which is founds in Bh:tskar:r I I 'sastrr'nomical work called Siddhzint;r i iromani ifvarr;afwrlq|qr). But the cooefficients

obtained i r r th is lvay (using l inear intelpolat ion where neceesary) are so rought that mastof them do not agree lvith tlrr;se given in the original text. Horvever, a second orcier inter-polation does help in this resf,ect (see below).

Ofcoures, more :rccrlrate Sine tables can be rrsed to derive the values of the coffici-ents to the supposed or irnlied degree ol'accuracy. But it is doubtul whether Bhiskara IIhad anv such table ready at hi-s disposal 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 coefficients are far from being accrrrate to the same dcgree as others.This indicates the possibil i ty of sonre different method.

The second method given by the commer.tator GaneSa consists of f inding the sides

Thc last d ig i t rcacl ing dvi ( two.; is statcd to be colrcctcd to t r i ( three) in one of the manuscr ipts :

28 I I IE I . f ,TIr I I I ITtOg ED go^|r IOX

of the inscribed triangle, sguare, hexagon, and octagon geometrically by the usual methodofemploying tl,e so-called Pythagoream theorem (see Colebrooke's 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 findin61 of 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 cases points 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 diff icult 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 Brahmagupta's (A.D. 62S)technique of second order interpolation which he knew and which is equivalent to themodern Newton-Stirl ing intcrpolation formula upto the second orderrr. By tbis methodthe result for z equal to 9 (nonagon) tall ies almost fully, but in the only remaing case ofheptagon (n equal to 7), the most tedious onc where even the argumental angle is notexpressible in whole degrees or minutes, a small difference is f,rund.

t .

Refcrenceg and Notcs

For a brief description of his works, see R.C. Gupta,"Bhiskara II 's 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.

C,rlebrooke's English translation, with nots by H.C. Banerji, has been recently reprinted

by M/s Kitab Mahal, Allahabad, 1967.

K.V. Sarma leditor): Lll;ruatl with Krilt-,kramakari of sa,rkara and Nitilar.ta, Vishveshara-

nand Vedic Research Institute, Hoshiarpur, 1975.

Edna E. Krarner ; The Main Strearn of Malhematics. Oxford univ. Press, N.Y., 1951,

Pp. 3-5.

Also see the present author's note on LILAVATI published in Tfu Hindustan Timet,

New Delhi , Vol .5 l , No. l2 l , p. 5 (deted the l9th May 1974).

5. R.E. Moritz : On Mathematics, p, 164. Dover, New York, 1958.

6. See the Lil ivatr with the commentaries of Ga4eSa and Mahidhara edite<i by D.V. Apte,

2.

3.

4.

R. G. GUPTT

Part II, pp. 207-208, Poona, 1937 (Anandasram Sanskrit Series No. 107). In Cole-

brooke's t ranslat iorr (p. 120), these stanzas are 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-siddanta is slightly different.

9. 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. l l),Thc

Malhemalics Edacatinn, Vol. VIII, No.3 (September 1974), Sec. B, pp. +3-46.

10. See his Jlotpatl i, verses 7-B in Bapudeva Sastri (editor), op. cit., p. 28l.

It. Sid. i i ir. Graha Ganita, II, l6 (Bapudeva's edition cited above, p.42). For details see

R. C. Gupta, ttSeconder Order Interpolation in Indian Mathematics etc.", Indian J.Htst. Sciencc, Vol. 4 (1969), pp. 86-89.

29

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