1. Ihe MathematicsEducation SECTION B
V ol. ' V I, No . l, Marcb 19 7:
f rt I M P S A S OF ANCINNT I NDI AN 1
MAT[, No.
I'deelakantlra's Rectlttcatlon Formrrla
DJ,R. C. Gupta, Department Mathcrnatics
of Birla Institute Tcchnologlt
of
P. O. Mesra,Ranchi,Bihar.
( llccr ile d 1 0 Ja n u a ry 1972 )
Neelakantha Somaydji ( f,tora;na dlqqrf; ) was one of the important mathematicians of
rnedieval India. He was born in the year 1443A. D. and wrote several astronomical works
drrring his life of about one hundrecl ears. His Tantrasangraha a;e fq-q an erudite treatise
(
) ),
on mathemetical astronortlr rnves r.orlplssfl in A. D. 1500. In another of his works, called
Golas-ara q'legq Neelakantha gives a rule for computing the length of a small circular arc
( )
( or the angle su'lrtended it at the centre of the
by circle ) when its Indian Sine and Versed
Sine are kntrwn ( an Inclian trigonometric function is equal to the radius times the correspon-
ding modern trigenometric funcrion and is usually written with a capital letter to distinguish
it from the latter. )
The third section of Gola.siragives the rule as follows []
s'-'lati..gcqiq q14rria3t( q{ qg: ciq: I
"The leogth of an arc ( of a circle ) is approximately the square-root of the sum of
the square of rhe Sine ( of the arc ) and the square of the Versed Sine ( of the arc ) together
with third part ( o[the latter )',.
That is,
Arc=@;6inejzF
Or
R a - y'( R sin, 1zq(-ap)( R vers -6; ,
which is equivalent to
q =lsin2 a +( 413 ) | i - cos @ )2 (l)
Formula r I ) which is the modern form of Neelakantha's rule will enable us to compute
approximately the angle when its sine i or cosine ) is given. Ia fact Neelakantha's pupil
Shankar has explained this very use in his oommentary on his guru's Tantrasangraha.

2. 2 ral r^t'BgraAtro! EDuc,ATrox
Later on Neelakanthaguoted the above rule at least at two placesin his commentary
on thc Arytabhatcqta, famous Indian work of the elder Aryabhata ( born A. D. 476 ). In
tbe
this commentary Neelakantha also gives a proof of the rule [2].
To check the accuracy of the rule, we easily seethat.
A+(413)( l-ean @)2=5/29(cos 2 A-16cos A)16=O2-68190+...
'flnz
Thur thc right hand ride of (l)
- 6 - d5/1E0,
neglectinghigher powen.
Hence we seethat Neelakanthatgformula will give result correct to .6aand therefore
will be guite accuratefor practical purpose.
Referencer
[] Golasira ed. by K. V. Sarma, Horhiarpur, 1970: P. 17.
t2l Sce Trivandrum ed ( 1930) of Neelakantha'r Commentary on the AryabhateeYar
PP.
63- I 10.