We thus learn that the moment of true Mēsha saṁkrānti as calculated by the Siddhānta Śirōmaṇi occurred in A.D. 1150 on March 23rd at 10 h. 30 m. after mean sunrise, differing by more than a day from the same as calculated by the Ārya Siddhānta; and that in A.D. 1250 it occurred by the Siddhānta Śirōmaṇi on March 24th at 6 h. 45 m. after mean sunrise, with a similar difference between it and the Ārya Siddhānta.
[Unfortunately, I have no means at hand for checking and comparing these results both by the Brāhma Siddhānta and Siddhānta Śirōmaṇi, but they are decidedly suggestive; for it would seem necessary, after observing these results, that a number of records hitherto examined only by the Ārya or Sūrya Siddhāntas should again be subjected to scrutiny. The result would teach us till how late and in what parts of India the other authorities were used, and would form a most useful element in the discovery of mediæval forgeries.]
In all these cases there may be an error in the result of one minute, if we begin our calculations N.B. by taking the time of true Mēsha saṁkranti according to the Ārya Siddhānta from Table I. of the Indian Calendar, which does not take account of seconds. But if greater accuracy is desired, it can be obtained by applying the rule given in "Hint" No. 20 below.
63. Before proceeding to consider this subject a few notes may not be amiss.
The meaning of our w, a, b, c and t is explained above in the Introduction.
A true lunation is the time actually taken by the moon to move through one synodical revolution, new-moon to new-moon. Its length varies constantly from 30 d. 1 h. 15 m. to 29 d. 0 h. 40 m., in consequence of her orbit being elliptical and because of the earth's unequal velocity at different parts of its orbit round the sun.
A mean lunation, or mean synodic lunar month, consists, according to modern science and the Sūrya Siddhānta, of 29 d. 12 h. 44 m. 2.87 s. Authorities differ only in the fraction of the second.
While the daily motion of the moon varies like that of the sun in consequence of her orbit being ellipse, being more rapid when near perigee than when she is near apogee, so that during part of her an revolution she is six or seven degrees behind, and in part six or seven degrees in advance of, her mean place, this retardation and acceleration does not, as in the case of the sun during the 5000 years of the Kaliyuga, always take place in the same region of the heavens; since the line of the moon's apsides makes a complete revolution once in a little less than nine years, while the movement of the sun's line of apsides is exceedingly slow.
A tithi is one-thirtieth of a synodic lunation, that is to say, it is the time occupied by the moon in increasing her distance from the sun in longitude by 12 degrees, the thirtieth part of the circle of 360 degrees.
A mean tithi is the exact one-thirtieth of a mean lunation, viz., 23 h. 37 m. 28.093 s.
A true tithi is the time in true longitude, and varies in length from 21 h. 34 m. 24 s. to 26 h. 6 m. 24 s.
Table I. of the Indian Calendar, cols. 21–25, gives, according to the Siddhāntas generally, but more particularly according to the Sūrya Siddhānta, the elements of luni-solar reckoning for mean sunrise at Laṅkā (Ujjain) on the first civil day of each luni-solar year, i.e., of the sunrise next following the moment of the new-moon next after apparent Mīna saṁkrānti.
64. Amānta and pūrṇimānta systems of lunar months. The amānta lunar month begins astronomically at the moment of new-moon; the pūrṇimānta lunar month begins astronomically at the moment of the previous full-moon. In each case the first tithi of the lunar month begins at that moment; and the civil day at whose sunrise this first tithi is current is the first civil day of the lunar month.
