Page:Sewell Indian chronography.pdf/93

10. In the Indian Calendar, p. 154, § 160 (b), we gave a list of points to be remembered in verifying dates, remarking that that list was not exhaustive. Let Addition to Ind. Cal., § 160 (b). the following be added as A (I.) (c): "The original date may have been calculated by mean months. " The present work shows how to calculate them.

12. When a saṁkrānti is found to take place within about (${\displaystyle t=}$) 33 lunation-parts of the moment of new-moon, the use of a different Siddhānta may cause the intercalation of a month Saṁkrāntis. Limit
of differences. next preceding or next following the one found to have been intercalated by the calculation. When, therefore, the results for the moments of two consecutive saṁkrāntis connected with a lunar month are both less than (${\displaystyle t=}$) 9967 or both greater than (${\displaystyle t=}$) 33 it may be assumed that the same month was intercalated by all Siddhāntas. (Ind. Cal., p. 29, note 1, § 88, p. 51). Where, however, a Siddhānta has been used with a bīja-correction, the limits must be enlarged to 113 lunation-parts, or ${\displaystyle t=9887}$ and 113.

13. (i.) When working for "true" intercalations and suppressions of lunar months the equations of (${\displaystyle b}$), the moon's mean anomaly, and (${\displaystyle c}$), the sun's mean anomaly, are added to the ${\displaystyle a}$ found by the calculation. It is convenient to note that no ${\displaystyle a}$ can possibly increase by these Intercalations and
suppressions of lunar
months. Limitations. additions to as much as 10,000 or 0, the moment of new-moon, if it (${\displaystyle a}$) is less than 9599, since the greatest possible increase caused by the addition of these equations is (280 + 121 =) 401. (See Tables VI., VII., col. 2.) If the ${\displaystyle a}$, therefore, of both the saṁkrāntis concerned is near to but less than 9599, the month must be a common one.

(ii) Similarly there can be no intercalation or suppression of a lunar month in a year the "${\displaystyle a}$" of whose Mēsha saṁkrānti is less than 5910, since the greatest increase of ${\displaystyle a}$ in a year (see Tables XVIII., XIX.A and XIX.B) is 3689, which, + 401 (above 13 (i.)), = 4090; and 10,000 − 4090 = 5910.^[1]

14. No mean intercalation is possible in a year in which the ${\displaystyle a}$ of mean Mēsha saṁkrānti is less than 6311, since the greatest increase of ${\displaystyle a}$ for any mean saṁkrānti of a year is 3689. 10,000 − 3689 = 6311.

15. In connecting tithis with solar days note that—in consequence of (i.) the chance that a Siddhānta has been used by the almanac-maker different from that by which the calculation is being made, (ii.) the difference between the mean time of Table X. and the possible Tithis, nakshatras and
karaṇas. Limitations. true time of the date, (iii.) the difference at the principal place in the neighbourhood of the inscription where probably the almanac makers resided, of the time of true and mean sunrise (which last is the basis in our system)—a result which shows a certain tithi, nakshatra, or karaṇa as beginning or ending within about 5 ghaṭikās, or 2 hours, of sunrise may be wrong by one day. Consequently, whenever, in a calculation for mean sunrise made by our Tables, the tithi-index (${\displaystyle t}$) or the nakshatra-index (${\displaystyle n}$) or the karaṇa index (Table VIII., cols. 3, 8, 9, 10) is within 30^[2] of the beginning index or ending index of the tithi, nakshatra, or karaṇa, it is possible that the result may be one day wrong. In the case of the yōga the limit may extend to 7 ghaṭikās, or 2 h. 48 m., or, in terms of the yōga index, 50. These limits are not fixed; but when the result ascertained is not within, or almost within, those limits it is safe to conclude that the result would be the same whatever Siddhānta might have been used, and whether the original calculators had worked by true or mean sunrise, and by true or mean time. When a doubt exists the date should be tested by Jacobi's Epig. Ind., Vol. I., Special Tables. (See Ind. Cal., § 37, pp. 20, 21.)

1. ↑ If, however, this "${\displaystyle a}$" is less than about 330 intercalation of Chaitra is possible.
2. ↑ In mean-time 2 h. 8 m. (Table X.)