the time according to the Sūrya Siddhānta from A.D. 1101 to 1900. Table XVII. below gives the time-differences for every year from A.D. 300 to 1900, in minutes[1] as well as (for convenience in luni-solar calculations) in the value of a, b, c. The signs + and − as given in Table XVII. are to be used when converting Ārya Siddhānta results to Sūrya Siddhānta results, and when the reverse process is required the signs are to be reversed. This enables anyone to ascertain by either authority the moment of true Mēsha saṁkrānti for any year of the 1600 years of the Tables.
47. Tables I. and XVII., however, give no information as to the exact moment of Mēsha saṁkrānti in years earlier than A.D. 300. For the use of those who may desire to know for such years Note. at what moment this took place according to calculations made with the length of the year as fixed by the Ārya Siddhānta[2]—at what moment, that is, the Mēsha saṁkrānti would have been determined to have occurred if that Siddhānta, or other authority having the same year-length and starting calculations from the same base, had been used—the following note is inserted. The Julian year is 365 d. 6 h. long; the solar year of the Arya Siddhānta is 365 d. 6 h. 12 m. 30 s. long; so that the Ārya Siddhānta year is 12 m. 30 s. longer than the Julian. This excess amounts to exactly five days in 576 years, and at this rate the Ārya Siddhānta Mēsha saṁkrānti is continually advancing on the Julian year. Conversely, that point is, in time, precisely five days earlier in every year removed 576 years backwards. Thus in Table I. of the Indian Calendar it will be observed that in A.D. 600 Mēsha saṁkrānti occurred on March 19th at 5 h. 30 m. Laṅkā time. In A.D. (600 + 576) 1176 it occurred on March 24th at 5 h. 30 m. Laṅkā time, and in A.D. 1752 on March 29th at the same hour. Therefore in A.D. 24 (600 − 576) it occurred on March 14th at 5 h. 30 m. Laṅkā time.
48. The times given in cols. 15, 17 of Table I. of the Indian Calendar are fixed on this principle. The annual increment being 12 m. 30 s., we have tabulated the difference in alternate years as 12 m. and 13 m.; and this same annual increment being 3114 palas, we have tabulated the difference as 31 palas for three years and 32 palas in the fourth year. (See also "Hint" No. 20, below).
49. Thus to find the moment of Mēsha saṁkrānti for any year not given in the Tables, it is only necessary to find what that moment was in a year removed 576 years from it, whether earlier or later, or by any multiple of 576; and while retaining the hours and minutes, to alter the day backwards or forwards by five days for every such period.
50. It must be borne in mind, however, when working for Christian era years B.C., that, since year there was no A.D. 0–1,[3] the interval for such years is to be increased from 576 to 577. For 120 B.C., for instance, we work from the time of Mēsha saṁkrānti as fixed in (577 − 120) A.D. 457. For years of the Kaliyuga era the interval is always 576.
51. As to the week-days; since the same day of the month in consecutive Julian years is always one week-day later in each common year and two week-days later in each leap-year, we have, in an interval of 576 years, an advance of (576 + 144 =) 720 days. To this has to be added the five extra days resulting from the difference between the Julian and Ārya Siddhānta year-lengths in each cycle of 576 years. Total, 725 days, which, divided by 7, leaves remainder 4. So that we subtract four days from, or add them to, the week-day given for the Mēsha saṁkrānti of the base year, for every interval of 576 years removed from it backwards or forwards as the case may be. See, for instance, the examples given above (§§ 47, 50). The week-day corresponding to the Mēsha saṁkrānti day of A.D. 600 being 0, Saturday, the week-day of the Mēsha saṁkrānti day of A.D. 24 was (7 − 4 =) 3, Tuesday; and the similar day in A.D. 1176 was (0 + 4) 4, Wednesday. So the week-day corresponding to the Mēsha saṁkrānti day of A.D. 457 being 2, Monday, the week-day of the Mēsha saṁkrānti day of 120 B.C. was (9 − 4 =) 5, Thursday.
