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52. To show that the above processes yield accurate results and are not mere rough-and-ready approximations, I may state that at my request Mr. Sh. B. Dikshit specially worked out the times of the Mēsha saṁkrāntis in some years by the elements of the Ārya Siddhānta itself, in order that we might have a quite independent check on the figures which we obtained by the method just explained. One of these years was A.D. 499–500 (Śaka 422 current, K.Y. 3601 current). He found that by this independent calculation the moment of true Mēsha saṁkrānti of that year by the Ā.S. was Friday, March 19th, A.D. 499, at 6 gh. 11 p. Table I. has 6 gh. 9 p. on the same day.^[1] The difference between the two amounts to 48 seconds, of which half at least, if not the whole, is attributable to fractions of vipalas.

53. Similar calculation direct by the Ā.S. for the Mēsha saṁkrānti in A.D. 78, Śaka 1 current, yielded Sunday, March 15th, at 11 gh. 55 p. Calculation by the process I have explained gives 11 gh. 52 p. 30 vip. on the same day, showing a difference of 1 m. The slight difference is not material, and on consulting Mr. Dikshit he advised a systematic adherence to our own system, in preference to direct calculation from the Siddhānta, as more likely to yield correct results.

The rule, then, for finding the true Mēsha saṁkrānti time by the Ārya Siddhānta of any year not already given in col. 1 of the Indian Calendar is as follows: Look in Table I. and take the day, week-day, hour and minute of the similar moment (cols. 13 to 17) in a Rule for finding Mēsha
saṁkrānti for any year
not given in Table I.,
by the Ārya Siddhānta. year removed 576 years or a multiple of 576 years from the given year. For the day of the month deduct five days for every period of 576 years in the case of years earlier, or add the same for a similar period in the case of years later, than the year taken from Table I. For the week-day deduct or add four days similarly. Retain the hours and minutes given in Table I. for the year taken as base. The result is the moment of "true" Mēsha saṁkrānti for the given year. For mean Mēsha saṁkrānti add the śōdhya interval (Corr. III. below') by the Siddhānta concerned.^[2]

55. The Table on p. 10 of the Indian Calendar shows the lengths of true solar months according to both the Siddhāntas concerned, and these differ from one another. In consequence, even if there were no difference in the astronomical beginning of the solar year, the Correction II.
Month-lengths. astronomical beginnings of subsequent solar months, i.e., the moments of the saṁkrāntis, would not be the same by the two authorities. Table III. (p. cvii.) of the Ind. Cal. gives the collective duration of months from the Mēsha to each subsequent saṁkrānti by the two Siddhāntas; and have repeated this in my new Tables, XVIII.A and XVIII.B, adding the ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ corresponding to that collective duration, and giving besides the difference in the lengths of each solar month between the saṁkrāntis, with the corresponding ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ for each. This helps to simplify calculations of all kinds; for when we have found the ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ for Mēsha saṁkrānti of any year we simply add to them the ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ given in the new Table for the interval to the required saṁkrānti, obtaining the desired result in one process instead of having to make two separate calculations. The ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ for any subsequent interval of days is obtained from Table IV. of the Indian Calendar, and of hours and minutes from Table V.

56. For the purpose of converting a "true" measurement result arrived at by the Sūrya Siddhānta into a similar result by the Ārya Siddhānta, and vice versa, we apply, for true Mēsha saṁkrānti, the correction (I.) given in Table XVII.; and for all subsequent saṁkrāntis we add the correction (II.) given in Table XVIII.C, which gives the time and ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ difference for each saṁkrānti between the two authorities. The first process alone must be used for the Mēsha saṁkrānti; both processes are required for all subsequent saṁkrāntis. (See Examples 13, 14, below.)

1. ↑ (See § 162 below, para. 2, and Note.) It being a fact that in A.D. 499 mean Mēsha saṁkrānti by the Ārya Siddhānta took place precisely at noon on March 21st, it is also a fact that the true Mēsha saṁkrānti of that year occurred precisely at 2 h. 27 m. 30 s., or in Hindū reckoning at 6 gh. 8 p. 45 vip. Lankā time, on March 19th. Consequently, our tabulation in Table I. of 6 gh. 9 p. is correct.
2. ↑ In Table XXXVIII.C. I have given the moments of occurrence of mean Mēsha saṁkrānti by the Ārya Siddhānta for period of 576 years, viz., from A.D. 499 to 1075.