Page:Sewell Indian chronography.pdf/54

97. Intercalations and suppressions have since, it is believed, about A.D. 1100, been always regulated by the sun's and moon's true motions. Prior to the 10th century A.D. it is probable that the calculation was generally by mean motions. In the Vēdāṅga Jyōtisha (about 1400 B.C.) no account is taken of the true motions of the sun and moon. Mean intercalations seem not to have been known in the time of Bhāskarāchārya (A.D. 1150). But in some parts of India at least the true or "apparent" system was used as early as the beginning of the 10th century A.D. (Above, § 32.) The systems are known as the "true system" and the "mean system." For purposes of our calculations it is necessary, when working by the true system, to add to $a$ the equations of $b$ and $c$, the result, $t$, showing the moon's age at the moment of the saṁkrānti; but in the mean system no account is taken of $b$ and $c$, the $a$ alone, when increased by 201 (§§ 20, 21), showing the correct age of the moon at the saṁkrānti. When we find the moon's age at the saṁkrānti near to but less than 10,000 (the moment of new-moon) we know that the moment of new-moon was after the saṁkrānti (e.g., the unit of $t$ being 4.25 m., $t=9996$ at the saṁkrānti shows that the moment of new-moon was ($t=4=$) 17 m. after the saṁkrānti); and when we find the moon's age at the saṁkrānti to be near to but greater than 0 (the moment of new-moon) we know that new-moon took place before the saṁkrānti (e.g., $t=4$ shows that the new-moon took place 17 m. before the saṁkrānti). Consequently when, working the $t$ for two consecutive saṁkrāntis, we find $t$ approaching 10,000 at the first, and only slightly exceeding 0 at the last, we know that no saṁkrānti occurred in that lunar month which lay between the two, and therefore that that lunar month was intercalated (adhika).

98. And similarly when we find $t$ near to but greater than 0 at one saṁkrānti, and near to but less than 10,000 at the next, we know that two saṁkrāntis occurred during the lunar month, and therefore that it is suppressed (kshaya). (See especially § 50, pp. 29, 30, Ind. Cal., and "Hints," Nos. 8, 9, 12–14, 18 below.) The same is of course the case with the figures for $a$ in calculations by the mean system.

99. Tables XVIII.A and XVIII.B are for work by the true system, and show the collective duration and the increase of $a,b,c$ from Mēsha saṁkrānti^[1] for each of the sidereal saṁkrāntis, as well as the individual lengths of solar months between saṁkrāntis, by the Ārya Siddhānta and Sūrya Siddhānta respectively; Table XVIII.C gives the figures to be used when converting a result by one into a result by the other of the two Siddhāntas. And Tables XIX.A and XIX.B are for work by the mean system similarly. There is no appreciable difference between these last two.

100. In practical work values of vipalas and seconds may be ignored, fractions over half being taken as one pala or one second respectively. (See also "Hint" No. 20 below.)

101. The general principles governing mean intercalations are similar to those for true or apparent intercalary months; but mean saṁkrāntis have to be calculated from the moment of mean, not true, Mēsha saṁkrānti (for the method of finding the moment of mean Mēsha saṁkrānti, see "Correction III." § 57 above); and we have to work by mean, not true, length of solar months. This done, the moon's place is found in the usual way, the $a+201$ alone being taken into consideration; in other words, we deal with the moon's mean longitude alone, without reference to the mean anomalies of moon ($b$) and sun ($c$). For actual practice, see Examples 36 to 39 below.

102. A lunar month is, as above stated, suppressed whenever two saṁkrāntis occur in it. This requires the combination of a long lunar month with a short solar one. The short solar months are between the Vṛiśchika and Mīna saṁkrāntis (see Tables XVIII.A or XVIII.B, cols. 6 or 7), and these are connected with (col. 1) the lunar months Mārgaśīrsha, Pausha, Māgha and Phālguna. The first two of these are the months most commonly suppressed. (See Examples 29, 32 below.)

1. It makes no difference whether we begin calculation for intercalation and suppression of the lunar months at true Mīna saṁkrānti and thence find the moment of the following new-moon, or at true Mēsha saṁkrānti and thence find the moment of the preceding new-moon; except in years in which the first month, Chaitra, is itself intercalary. I recommend beginning with Mēsha saṁkrānti because the day and hour of its occurrence are given in Tables I. and XXXVIII.A and B; and this forms a convenient starting-point.