172 ECLIPSES This formula also is incorrect. It was modified by Brahmagupta,¹ who replaced Rversin (+90°) in the formula by Rsin (+90°). An accurate expression for the ayana-valana was given by Bhaskara II (1150 A. D.).³ A rule for finding the value of the resultant valana (spasta-valana) for the circle drawn with half the sum of the diameters of the eclipsed and eclipsing bodies as radius: 46-47. When they (i.e., the akşa-valana and the ayana- valana) are of unlike directions, take the difference of their arcs; in the contrary case, take their sum.³ Multiply the Rsine of that (sum or difference) by half the sum of the diameters of the eclipsed and eclipsing bodies and divide (the product) by the radius. Add whatever is thus obtained to the (Moon's true) nati, provided that they are of like directions; in the contrary case, take their difference: the resulting sum or difference is the valana. The sum or difference of the akşa-valana and the ayana-valana accord- ing as they are of like or unlike directions gives the so called spasta-valana, i.e., the amount of deflection of the ecliptic from the prime vertical on the horizon of the eclipsed body. When the Rsine of that is multiplied by half the sum of the diameters of the eclipsed and eclipsing bodies and the product divided by radius, we the corresponding deflection on the circumference of the circle drawn with half the sum of the diameters of eclipsed and eclipsing bodies as radius. The sum or difference of this and the Moon's true nati according as the two are of like or unlike direc- tions gives the distance of the centre of the eclipsing body from the east- west line passing through the centre of the eclipsed body. So has been assumed in the above rule. The direction in the rule for adding or taking the difference of the reduced spasta-valana and the Moon's true nati is wrong. The two quanti- ties should be kept separately and laid off properly one after the other (in the projected figure). ¹ See BrSp.Si, iv. 17. & See Sisi, I, v. 21(ii)-22(i). s This rule occurs also in BrSpSi, iv. 18 (i) and SiDV, I, iv. 26.
