पृष्ठम्:महाभास्करीयम्.djvu/१९४

FINDING THE RSINE OF AN ARC 109 The method for calculating the Rsine or Rversed-sine of an arc, as stated in the text, may be explained by means of an example as follows: Example. Calculate Rsin 32º and Rversin 32º. Reducing 32 degrees to minutes, we get 1920'. Dividing this by 225, we get 8 as the quotient and 120 as the remainder. (1) The sum of the first 8 Rsine-differences is 1719'. Multiplying the remainder 120 by the 9th Rsine-difference (viz. 191') and dividing the product by 225, we get 101' 52". Adding this to the previous sum, we get 1820' 52" or 1821' approx. This is the value of Rsin 32º. (2) The sum of the first 8 Rversed-sine-differences is 460'. Multi- plying the remainder 120 by the 9th Rversed-sine-difference (viz. 119') and dividing that product by 225, we get 63' 28". Adding this to the previous sum, we get 523' 28" or 523' approx. This is the value of Rversin 32°.¹ The above method for finding the Rsine of a given arc is evidently based on the simplest law of interpolation, viz. that of proportion. In later works, we come across more elegant methods of interpolation. We state here two of them. 1. Brahmagupta's formula. If <225' and t be an integer, then Rsin (225't+0)=sum of t Rsine-differences + 1 0' 225 { tth Rsine-diff. +(t+1)th Rsine-diff, 2 tth Rsine-diff. - (t+1)th Rsine-diff. 2 0' 225 = sum of t Rsine-differences 0' + •{(t+1)th Rsine-difference} 225 -}. 1 +7.225 (225-1) ((t+1)th Rsine-difference -tth Rsine-difference}. (2) Using modern four-figure tables and assuming that one radian= 206265", we get Rsine 32° 30° 21' 43" approx. and Rversin 32°=8°42′ 22" approx. This shows that the values derived from Aryabhaţa I's table give fairly good approximations to the Rsines and Rversed-sines up to minutes of arc.