AYANA-VALANA 171 proportion: When the Rversed-sine of the hour angle amounts to R, the Rsine of the aksa-valana equals the Rsine of the latitude, what then would be the value of the Rsine of the akṣa-valana corresponding to the Rversed- sine of the desired hour angle? Hence the above formula. The above formula can be easily seen to be incorrect. It was first modified by Brahmagupta, who replaced Rversin H by Rsin H.¹ Better and accurate formulae were given by Bhaskara II.³ The rules for the direction of the akṣa-valana can be seen to be true by means of a diagram. A rule for the determination of the magnitude and direction of the ayana-valana: 45. The (Sun's) declination determined from the Rversed- sine of the longitude of the Sun or Moon as increased by three signs (treated as the Rsine of the bhuja) (is the ayana-valana). Its direction in the eastern half (of the disc of the eclipsed body) is the same as that of the ayana (of the Sun or Moon); ³ in the other half, it is contrary to that.* That is, Rsin (ayana-valana) Rsin Ex Rversin (λ+90°) R where e denotes the obliquity of the ecliptic and the sayana longitude of the eclipsed body (the Sun or Moon). This formula also is based on inference. The proportion used is the following: "When Rversin (+90°) is equal to R, the Rsine of the ayana-válana is equal to the Rsine of the obliquity of the ecliptic, what then would be the Rsine of the ayana-valana corresponding to the desired value of the Rversed-sine ?" 1. See Br Sp.Si, iv. 16. 2 See SiSi, I, v. 20-21(i); II, viii. 68; and II, viii. 66(ii)-67. 3 The ayana of a planet is north or south according as it is in the half-orbit beginning with the (sāyana) sign Capricorn or in that beginning with the (sayana) sign Cancer. ..
- This rule occurs also in ŚiDVṛ, I, iv. 25 and Sise, v. 20.
