Page:Sewell Indian chronography.pdf/30

This gives us for the former (Ex. 9, Ārya Siddhānta) 9.354525, which, when taken as days (Table XXXVI.) = 9 d. 8 h. 30 m. 31 s.; and for the latter (Ex. 10, Sūrya Siddhānta) 8.772306, which, when taken as days = 8 d. 18 h. 32 m. 7 s. Both these results are almost similar to those given by the more complicated process in the two examples quoted. If fully worked the results exactly agree.

39A. In very accurate calculations for tropical saṁkrāntis the annual change in the position of the line of apsides of the earth's orbit ought strictly to be taken into account, and the consequent slight change in the śōdhya interval, or the time difference between the moments of true and mean Mēsha saṁkrānti; though even in the 5000 years embraced by our Tables the change in the śōdhya has been exceedingly small.

39B. Since, however, the amount of this difference as it existed in K.Y. 1 current or the epoch of the Kaliyuga is of very great importance later on when we come to deal with the saṁvatsaras of Jupiter, it will be well that I should here, once for all, consider the subject in detail. The word "śōdhya" means in mathematics something to be subtracted or cleared off with the view of rectifying a stated quantity; but, following the Indian Calendar, I apply the name to the time-difference between the mean and true sun (or the sun's place as measured by his mean and his true longitude) at the first point of Mēsha, or longitude 0°, the point from which all celestial longitudes are measured in Hindū astronomy. It represents the time-equivalent of the equation of the centre, or the difference between the time at which the sun, moving with his true velocity in the ecliptic, reaches the first point of Mēsha and the time at which he would reach it if his velocity were constant. Both true and mean suns are together on the perigee and apogee points of the orbit, but by the time the true sun, having left the perigee point, reaches longitude 0° or 360° the mean sun is behind it. The difference between them is the śōdhya. Exactly 180° from the perigee point is the apogee point, and a line joining them is the "line of apsides." Now there is a slight annual shift in this line of apsides, and consequently a slight change in the longitude of the apogee and the perigee. Since the velocity of the true sun, and therefore the time at which it reaches longitude 0°, depends on the position of the line of apsides, it follows that, in proportion to the shift of that line or the change in the longitude of the apogee point, there must be a change in the śōdhya interval. But this change is exceedingly slow. According to the Sūrya Siddhānta the annual change in the position of the sun's apogee is only 0.1161″,^[1] while the Ārya Siddhānta does not teach any change at all; and since by the Sūrya Siddhānta the difference of anomaly (Ind. Cal., p. 5, note 4) in 5000 years only amounts to 9′ 13″, it is evident that the difference in the śōdhya in years divided even by so long a period would amount to very little. I have therefore retained for the initial point of Kaliyuga reckoning (otherwise called the "epoch of the Kaliyuga," or mean Mēsha saṁkrānti of K.Y. 0 expired or 1 current) the śōdhya value as calculated for K.Y. 4238, or A.D. 1137 (see Ind. Cal., §§ 24, 26, p. 11), i.e., its value for longitude of apogee 77° 16′. It amounts by the Sūrya Siddhānta to 2.170694̇ days, and by the First Ārya Siddhānta to 2.1475694̇ days. I admit that the time results of my Tables for Jupiter by the Sūrya Siddhānta may be wrong by a minute or two in consequence of my making no change in the śōdhya value for these Siddhantas; and if the beginning of a saṁvatsara should be found to occur within a minute or two of Mēsha saṁkrānti the case may be specially examined. The value for the sun's anomaly for different centuries is given in Jacobi's (Epig. Ind., Vol. I.) Table XIII., and the corresponding equation of the centre in his Table XXIV. For the Original Sūrya Siddhānta I have taken the śōdhya value as declared by Sh. B. Dikshit in Ind. Ant. 1890, p. 49, viz. 2.23972̇ d., or 2 d. 5 h. 45 m. 12 s. For the rest I have accepted Dr. Schram's valuation.

1. ↑ The authors of E. Burgess's Sūrya Siddhānta stated the longitude of the apogee in K.Y. 1 current as 77° 7′ 48″, and over 4200 years later it has been calculated by the same authority as 77° 16′. Prof. Jacobi states the annual change by the Second Ārya Siddhānta as 0.1383″, and by the Brāhma Siddhānta as 0.144″. (Epig. Ind., I., 441.)