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57. All calculations in solar reckoning made by the mean system must have for their starting point the moment of the mean and not of the true Mēsha saṁkrānti. Between these two there is an interval called the śōdhya. (See §§ 16, 39A, &c.) As stated in the Correction III.
Śōdhya-lengths. Indian Calendar (§ 26, pp. 11, 12) we take this interval as a constant. According to the Ārya Siddhānta it consists of 2.1475694̇ d., or 2 d. 8 gh. 51 p. 15 v., or 2 d. 3 h. 32 m. 30 s., and according to the Sūrya Siddhānta of 2.170694̇ d., or 2 d. 10 h. 14 p. 30 v., or 2 d. 4 h. 5 m. 48 s.; the difference between the two amounts to 1 gh. 23 p. 15 v., or 33 m. 18 s., by which amount the latter is longer than the former. In terms of our ${\displaystyle a}$ this difference amounts to 7.83, and therefore we take ${\displaystyle a=8}$ as the working difference between the two Siddhāntas in this respect.

58. In mean calculations on the ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ system we are only concerned with ${\displaystyle a}$, which gives us the mean longitudinal distance of the moon from the sun at any moment. The Ārya Siddhānta śōdhya is, in terms of ${\displaystyle a}$, 727, and the Sūrya Siddhānta śōdhya, in terms of ${\displaystyle a}$, is 734. As before explained every figure of ${\displaystyle a}$ in our method of work by the true system has been reduced by a certain amount to compensate for amounts added to ${\displaystyle b}$ and ${\displaystyle c}$, so that all processes may be additive; and when we have, as in mean calculations, to use ${\displaystyle a}$ independently of ${\displaystyle b}$ and ${\displaystyle c}$, we first find ${\displaystyle a}$ in the usual manner, and then restore it to its proper dimensions by adding the amount by which it was artificially reduced. This amount was for ${\displaystyle b}$ 140.2 and for ${\displaystyle c}$ 60.4, total 200.6. Taken by itself the round figure corresponding to this should be 201, and this is what should properly be used for all direct calculations. But when we are merely converting a result by one of our two Siddhāntas into a result by the other, and are doing so by using the figure of ${\displaystyle a}$ for the difference between the two, which figure I have stated exactly as 7.83 and roughly as 8, the regular addition to the calculated ${\displaystyle a}$ should not be 201, but 200, since 200.6 + 7.83 = 208.43, and the decimal is less than half an unit. On the other hand, when using the Siddhānta difference 7.83 as a minus quantity the proper value of the constant will be 201, since 201 − 8 = 193, which is the nearest whole figure to (200.6 − 7.83) 192.77. For ordinary calculation however, the value of the śōdhya constant may be taken as 200, which is a convenient figure; but for closer work it will be well to use the decimals in the calculation, and then take the nearest whole figure.

59. The time and longitude of the other mean saṁkrāntis is found from the time of mean Mēsha saṁkrānti by adding to the latter the time-difference, or longitudinal (${\displaystyle a}$) difference, given as the collective duration against each saṁkrānti in col. 5 of Table XIX. (A or B, whichever Siddhānta is being utilised),^[1] and if these particulars are required for two consecutive saṁkrāntis, as in calculations for mean intercalated months, then by adding to the ${\displaystyle a}$ of the first the ${\displaystyle a}$ of the interval to the second, as given in col. 6 of those Tables.

60. When the moment of a mean saṁkrānti has been found by the Sūrya Siddhānta it is converted according to the same moment to the Ārya Siddhānta by (1) applying Correction I. (Table XVII., with sign reversed), and Correction III. (in time − 33 m. 18 s., or, in ${\displaystyle a}$, − 8); for the reverse process we apply Correction I. (Table XVII. as it stands) and Correction III. (+ 33 m. 18 s. or, in ${\displaystyle a}$, + 8). (See Examples 17, 36–40.)

61. All this is very simple in practice, as the following example will show: The year A.D. 1096–7 was close to the time when the Sūrya in many parts of India supplanted the Ārya Siddhānta, and we want to know the times of mean Mēsha and mean Vṛishabha saṁkrāntis, as well as the mean moon's distance from the sun, or the value of ${\displaystyle a}$, at each—and these by both the Siddhāntas.

1. ↑ The only differences between these two are in connection with the Kumbha and Mīna samkrāntis. have given the exact figures in Tables XIX.A, XIX.B, cols. 3, 4, as they may be found useful for other purposes, but in practice vipalas and seconds below the half may be ignored.