This document summarizes a rule from the ancient Indian mathematics text Lilavati for computing the sides of regular polygons inscribed in a circle. The rule provides coefficients that when multiplied by the circle's diameter and divided by 12,000 yield the side lengths of triangles, squares, pentagons, etc. up to nonagons. While the rationales for some coefficients are clear, others are less accurate, possibly due to limitations in available sine tables or knowledge of exact solutions. The document discusses methods that may have been used to derive the coefficients and compares the original values to more accurate modern calculations.

1. The Mathematics Educatron SEC'I'ION B
Vol. 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 7'eohnologP.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 Bhdskra's 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
Lildvati's 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. 26 THE M ATI:ITM ATI C E ED U G AT IO N
206-2C8 as followsG
is :
flaaqe+rfiqcqsg;f:faarqrsetmcefq:
I
?erficerqersdqq
ts(alqFrr€: fiHr( ltRo!ll
erliEtoerqtlsqkldc;?{qrqt: t
S(Iqsqldsq qf,aqrdue'r6* slQoetl
tq€?qrrrrddqt erq;t mRnl gwr: r
1tt;ae+agatqi aatat'd UqTTq{nRoctl
Tridvyarikagni - nabha"{candraih tribi=,tt.lsta yuedstabl'ih /
-
Vedagnibauaidca khi( vaiSca khakhabhr. ibhra-rasaih kramdt I 1206
II
Banesunakhabinaiica dvidvinandesu-s:lgaraih/
KurlmadaJca'vedaiica vt'ttavyasgsanrihate 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
4
o 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 to
the (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 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'sccefficients represent the sides of
regular 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 nal
taxt seemsto be changed at scveral places apparently to improve rrpon it. It is therefrrre
no surprise that same of the above coefficientshave different values in the taxt c f the rule
as published alon.gwith the Kriylkramatcari (|fr,+t*'e+'il) 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.

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 value
si 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 same
5 70i3+ s am e same
o 6000rJ s am e same
5205s 52C67 s2066
a 45922 s ant e same
9 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 ng
example :
' 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 s
b 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
Ga
va 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' s
astrr'nomical work called Siddhzint;r iiromani ifvarr;afwrlq|qr). But the cooefficients
o 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 mast
of 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 ol'accuracy. But it is doubtul whether Bhiskara II
had anv such table ready at hi-sdisposal although he knewe a method of constructing a table
of 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 sides
T 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. 28 IIIE I.f,T Ir IIIIT tOg E D go^|rIOX
of the inscribed triangle, sguare, hexagon, and octagon geometrically by the usual method
ofemploying 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 findin61of Sines of the type (2) geometrically
for n equal to3r 4,6, and B in which cases the exact values can be easily obtained by
employing elementary mathematical operations upto the extraction of square roots. The
accuracy of the text values in these casespoints out that it was possibly this very method
which was followed by Bhakara II. He also knew the exact value of the Sine of 36 degrees
which 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 knowledge
of 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 used
his 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 the
modern Newton-Stirling intcrpolation formula upto the second orderrr. By tbis method
the result for z equal to 9 (nonagon) tallies almost fully, but in the only remaing case of
heptagon (n equal to 7), the most tedious onc where even the argumental angle is not
expressiblein whole degreesor minutes, a small difference is f,rund.
Refcrenceg and Notcs
t. 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.
2. C,rlebrooke'sEnglish 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 author's 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. 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.
is
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. 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 (Bapudeva's 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,