Page:Sewell Indian chronography.pdf/29

many ghaṭikās^[1] before or after the nirayaṇa one as there are years between (for Ārya Siddhānta calculation) K.Y. 3624 current, or (for Sūrya Siddhānta calculation) between K.Y. 3601 current and the year next following or next preceding the given year. This rough calculation involves an error which, beginning at the base-year, increases yearly to as much as one day by about the year A.D. 1900. It amounts to about 1 m. 1.7 s. per annum, or about one hour every ⁠58+1/3⁠ years, and it is partly caused by the fact that the estimate of the sun's motion as 1 ghațikā in 1′ of longitude is only an approximation.

37. For more accurate purposes the following is the process. Having found the number of years intervening between (Ārya) K.Y. 3624 current, or (Sūrya) K.Y. 3601 current, and the given year, multiply that number by, in the case of the Ārya ⁠1/60⁠, or, in the case of the Sūrya, by ⁠3/200⁠. Take the product as the amount of precession in degrees accumulated by the beginning of the given year since, or prior to, the coincidence which has been assumed to have occurred in the base-year. These degrees of precession are called ayanāṁśas. The product gives the longitudinal distance of each equinoctial saṁkrānti point from the corresponding sidereal (nirayaṇa) saṁkrānti point. With this knowledge we proceed to find the difference in time between the sun's arrival at those two points, or the interval between them in days. Those days, in years succeeding the base-year of coincidence, belong sidereally to the month preceding the saṁkrānti with which we are concerned; though in years prior to the base-year, they belong sidereally to the same month. Thus, if we are working for a tropical Kanyā saṁkrānti, the interval of days between tropical Kanyā and sidereal Kanyā belong to sidereal Siṁha in years following the base-year; but in years prior to the base-year they belong to sidereal Kanyā. We know the lengths of those apparent months by Table XVIII.A or XVIII.B, according to the Siddhanta by which we are calculating; and we have only to multiply the time occupied by the sun in travelling 1° during such a month by the amount of the intervening ayanāṁśas to obtain in days the required difference in time between the sun's arrival at the sidereal or tropical saṁkrānti-point concerned. The rule therefore is: Multiply the exact length of that month by the amount of ayanāmśas found, and, dividing by 30, take the product as days; and by so many days, hours and minutes will the sāyana saṁkrānti precede or follow the moment of the nirayaṇa saṁkrānti according as the given year is after or before the base-year.

38. A little further explanation is still necessary. In finding the ayanāmśas for the given year by the Ārya Siddhānta the number of intervening years is divided by ⁠1/60⁠ because the numerator 1 represents 1°, and ⁠1/60⁠° = 1′, which is the amount of annual precession assumed for that authority. In the case of the Sūrya Siddhānta the fraction ⁠3/200⁠ represents ⁠3/200⁠° = 54″, which is its amount of precession per annum. The error inherent in this more accurate method of calculation is estimated by Sh. B. Dikshit as amounting to only about 1 hour in every 240 years.

39. To simplify the second rather complicated process of multiplication I have prepared Tables XX.A and XX.B. Table XX.A shows in time and in decimals of a day the time occupied by the sun in travelling 1° during each sidereal month, according to either the Ārya or the Sūrya Siddhānta; and Table XX.B shows in degrees and in decimals of a degree the amount of ayanāmśas we require when using the Sūrya Siddhānta, those required when using the Ārya Siddhānta being too easy to require a Table. Thus, referring to Examples 9 and 10, instead of the rather troublesome process of calculating ⁠30d. 8 h. 7 m. 42 s. × 9¼°/30⁠ in Example 9, or the still more intricate ⁠30d. 8 h. 29 m. 1 s. × 8° 40′ 12″/30⁠ in Example 10, we take the former sum as 1.0113 × 9.25, and the latter as 1.0118 × 8.67.^[2]

1. ↑ Precession per annum being 1′ (Ārya), or 54″, i.e. nearly 1′ (Sūrya), and the earth's motion being about 1° per day, or 1′ in a ghaṭikā approximately.
2. ↑ In Example 10 the difference of years was found to be 578, and col. 5 of Table XX.B gives us for 500 years 7° 5′, for 70 years 1° .05′, and for 8 years 0° 12′, total 8.67 in ayanāmśas; or the same amount may be gathered for A.D. 1077 from cols. 2, 3, viz. for A.D. 1000 7.515°, and for 77 odd years 1.05° + 0.105°, total 8.67°.