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16

INDIAN CHRONOGRAPHY.

Year.	Brāhma Siddhānta.		Siddhānta Śirōmaṇi.
Year.	By a series of approximations.	By direct calculation.	By a series of approximations.	By direct calculation.
K.Y. 50000	2.171973	2.171971	2.171973	2.171971
K.Y. 1000	2.172227	2.172226	2.173821	2.173819
K.Y. 2000	2.172482	2.172481	2.175669	2.175668
K.Y. 3000	2.172736	2.172736	2.177517	2.177516
K.Y. 4000	2.172991	2.172991	2.179365	2.179364
K.Y. 5000	2.173245	2.173246	2.181213	2.181213

Year.	Second Ārya Siddhānta.		Original Sūrya Siddhānta.
Year.	By a series of approximations.	By direct calculation.	No calculation attempted owing to insufficient data.
K.Y. 50000	2.171973	2.171972	Sh. B. Dikshit's valuation for K.Y. 0, viz., 2.23972, stands. (Indian Antiquary, Vol. XIX., for 1890, p. 49).
K.Y. 1000	2.172218	2.172217
K.Y. 2000	2.172462	2.172462
K.Y. 3000	2.172707	2.172707
K.Y. 4000	2.172951	2.172952
K.Y. 5000	2.173196	2.173197

The difference in 5000 years in the case of the First Ārya Siddhānta (the Āryabhaṭīya) amounts to 1 m. 26 s., in that of the Sūrya Siddhānta to 1 m. 27 s., in that of the Brāhma Siddhānta to 1 m. 51 s., and in that of the Siddhānta Śirōmaṇi to the much larger amount of 13 m. 19 s. For purposes of my Tables for Jupiter, however, this difference is immaterial since they are based on the values at K.Y. 0 alone.

39E. The following shows Dr. Schram's method of direct calculation. Let $x$ be the sun's equation of the centre at the instant of true Mēsha saṁkrānti. As the true longitude of the sun at this instant is 0° or 360° its mean longitude will be $360^{\circ }-x$ . At the instant of mean Mēsha saṁkrānti the sun's mean longitude is 0° or 360°, and the difference between the mean longitude of the sun at the instants of true and mean Mēsha saṁkrānti is equal to $x$ . Now, if the mean daily motion in longitude of the sun is called $m$ , the sun takes $\textstyle {\frac {x}{m}}$ days to travel over this difference $x$ , and $\textstyle {\frac {x}{m}}=$ the śōdhya . Hence—

(i.) The sun's equation of the centre at the instant of true Mēsha saṁkrānti $=x$

The sun's daily motion in mean longitude $m$

Śōdhya in days $\textstyle {\frac {x}{m}}$

The equation of the centre has to be separately calculated for each Siddhānta, and we require for each the sine of the sun's anomaly, or the angle $\alpha$ , completing the anomaly to 360°. The anomaly of the sun is the difference between the sun's mean longitude and the mean longitude of the sun's apsis.