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    Gupta1972d Gupta1972d Document Transcript

    • Ihe MathematicsEducation SECTION BV ol. V I, No . l, Marcb 19 7: f rt I M P S A S OF ANCINNT I NDI AN 1 MAT[, No. Ideelakantlras 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 ofrnedieval India. He was born in the year 1443A. D. and wrote several astronomical worksdrrring 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, calledGolas-ara qlegq Neelakantha gives a rule for computing the length of a small circular arc ( )( or the angle sulrtended it at the centre of the by 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.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 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 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 Neelakanthas rule will enable us to computeapproximately the angle when its sine i or cosine ) is given. Ia fact Neelakanthas pupilShankar has explained this very use in his oommentary on his gurus Tantrasangraha.
    • 2 ral r^tBgraAtro! EDuc,ATrox Later on Neelakanthaguoted the above rule at least at two placesin his commentaryon thc Arytabhatcqta, famous Indian work of the elder Aryabhata ( born A. D. 476 ). In tbethis 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+... flnzThur thc right hand ride of (l) - 6 - d5/1E0, neglectinghigher powen. Hence we seethat Neelakanthatgformula will give result correct to .6aand thereforewill be guite accuratefor practical purpose. Referencer[] Golasira ed. by K. V. Sarma, Horhiarpur, 1970: P. 17.t2l Sce Trivandrum ed ( 1930) of Neelakanthar Commentary on the AryabhateeYar PP. 63- I 10.