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(B) The shorter rule. The value of the śōdhya according to the Ārya Siddhānta being a constant, viz. 2 d. 3 h. 32 m., and the śōdhya according to the Brāhma Siddhānta being taken by us as a constant, viz. 2 d. 4 h. 9 m., we observe that the difference between these two values amounts in every year with which we have any concern, to 37 m., by which period a saṁkrānti calculated by the Brahma was always earlier than the same as calculated by the Ārya Siddhānta. We therefore take the time of true Mēsha saṁkrānti of the given year according to the Ārya Siddhānta from Table I., deduct the time-difference given in the Table in § 167A for the number of the expired year of the Kaliyuga, and deduct also, for the śōdhya-difference, 37 m. The result is the time of true Mēsha saṁkrānti according to the Brāhma Siddhānta. The time of mean Mēsha saṁkrānti is obtained by adding to this the amount of the śōdhya by that authority, as stated, viz., 2 d. 4 h. 9 m.

Working by the same year, K.Y. 4101 expired, the process is as follows:—

This result is the same as that obtained by the longer method.

62F. The Siddhānta Śirōmaņi, written by Bhāskarāchārya in A.D. 1150, K.Y. 4251 expired, gives the length of the solar year as identical with that of the Brāhma Siddhānta, with which also it agrees as to the moment of the epoch of the Kaliyuga era; so that the Table given below in § 167A applies equally to both these authorities in their respective relations to the Ārya Siddhānta. It also agrees with the Brāhma Siddhānta as to the value of the śōdhya in K.Y. 0.; but reference to Dr. Schram's Table shows that in later centuries there was a change, and that the two authorities differ. Consequently, while these two authorities are entirely in agreement as to the position of Jupiter at any time (§ 163), in calculating which we have to consider the value of the śōdhya only in K.Y. 0., they are not always in agreement as to the moment of occurrence of Mēsha saṁkrānti, because their śōdhya-values in later years are different. That is to say, we can calculate the time of occurrence of Mēsha saṁkrānti in any year by the Siddhānta Śirōmaņi partly with the aid of the Table in § 167A, but in completing that calculation we must use a śōdhya-value different from that of the Brāhma Siddhānta.

62G. Since the Siddhānta Śirōmaņi came into existence in K.Y. 4251 expired, we need consider the length of the śōdhya interval according to it only for the period K.Y. 4000 to 5000. Dr. Schram has settled for us its value at either end of that millennium. In K.Y. 4000 it was 2.179364 days, or 2 d. 4 h. 18 m. 17.05 s., and in K.Y. 5000 it was 2.181213 days, or 2 d. 4 h. 20 m. 56.8 s. The total change during the thousand years in question was 2 m. 39.75 s. If we divide this thousand years into centuries, for convenience of calculation, we find that the centennial change amounts to + 15.975 s. I give below, therefore, a Table showing in col. 3 the exact value of the Siddhānta Śirōmaņi śōdhya in the years noted in cols. 1 and 2; and I have added in col. 4 the figures to be used when working by the longer, and in col. 5 the figures to be used when working by the shorter, rule.