Ihe Mathematics EducationVol. 'VI , No. l , Marcb 197:

f r t IMPSAS OF

SECTION B

MAT[, No. 1ANCINNT INDIAN

I'deelakantlra's Rectlttcatlon FormrrlaDJ, R. C. Gupta, Department of Mathcrnatics Birla Institute of Tcchnologlt

P. O. Mesra, Ranchi, Bihar.

( l lccr i led 10 January 1972 )

Neelakantha Somaydji ( f,tora;na dlqqrf; ) was one of the important mathematicians ofrnedieval India. He was born in the year 1443 A. D. and wrote several astronomical worksdrrring his l i fe of about one hundrecl ) ears. His Tantrasangraha ( a;e fq-q ), an erudite treatiseon mathemetical astronortlr rnves r.orlplssfl in A. D. 1500. In another of his works, calledGolas-ara ( q'legq ) Neelakantha gives a rule for computing the length of a small circular arc( or the angle su'lrtended by it at the centre of the circle ) when its Indian Sine and VersedSine 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 distinguishit from the latter. )

The third section of Gola.sira gives 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 ofthe square of rhe Sine ( of the arc ) and the square of the Versed Sine ( of the arc ) togetherwith third part ( o[the latter ) ',.

That is,

Arc=@;6inejzFOr

Ra -y ' ( R sin, 1zq(-ap) ( R vers-6 ; ,

which is equivalent to

q =lsin2 a +( 413 ) | i - cos @ )2

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.

( l )

2 ral r^t'BgraAtro! EDuc,ATrox

Later on Neelakantha guoted the above rule at least at two places in his commentaryon thc Arytabhatcqta, tbe famous Indian work of the elder Aryabhata ( born A. D. 476 ). Inthis commentary Neelakantha also gives a proof of the rule [2].

To check the accuracy of the rule, we easily see that.

' f lnz A+(413)( l -ean @)2=5/29(cos 2 A-16cos A)16=O2-68190+.. .

Thur thc right hand ride of (l)

- 6 - d5/1E0, neglecting higher powen.

Hence we see that Neelakanthatg formula will give result correct to .6a and thereforewill be guite accurate for 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.