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motion, touches the first point of each fixed sidereal sign; a tropical (sāyana) saṁkrānti takes place when the sun touches the first point of any one of the tropical signs, which signs are, like the others, at intervals of exactly 30° from one another, but the first point of the first of which exactly coincides with the vernal equinox as calculated for the year. That is to say, in this last case the sun's longitude is calculated not from the sidereal fixed point, but from the vernal equinoctial point, the precession of the equinoxes being taken into account.

17. Though it is highly improbable that any Hindū pañchāng, or almanack for the year, was ever based on the tropical saṁkrāntis, it is possible that these were sometimes taken into account and were ascertained and stated as amongst the year's phenomena. Therefore, for computation of Hindū dates we have to calculate generally the moment of occurrence of apparent saṁkrāntis; sometimes, especially in earlier years, the mean saṁkrānti; and occasionally the tropical saṁkrānti. It is possible that the last might be calculated either by the sun's apparent or mean motion, but I think it unlikely that the Hindus would have used mean motion for calculating a tropical saṁkrānti, and such a process will not, therefore, be considered in this volume.

18. A purely lunar year, such as that of the Muhammadans, takes no account of the sun's place in the heavens, but, as explained above, the Hindū luni-solar year does so, each such year beginning with the new-moon next following the time of one of the solar saṁkrāntis. The difference between the lengths of the lunar year and the solar year necessitates the intercalation of a lunar month roughly every three years and the suppression of one at intervals ranging from nineteen years upwards.[1] This subject is fully considered in its place. Each lunation, or lunar month—new-moon to new-moon (the amānta system) or full-moon to full-moon (the pūrṇimānta system)—is divided into two pakshas or fortnights, and thirty tithis, fifteen to each fortnight, a tithi being the time occupied by the moon in travelling 12° from the sun. (§ 7, p. 3, Ind. Cal.) The tithi therefore may begin at any moment, day or night, of a solar day; but as a general rule a civil day is coupled in Hindū calendars with the name of the tithi current at sunrise. Our Tables are prepared so as to establish this concurrence for mean, and not true, sunrise, since the latter varies with the latitude and longitude of each place. To prove a date we have to ascertain the tithi current at sunrise of a certain civil day, or at any given moment of such a day, in any year; that is to say, we have to establish the position of the moon at that moment.

19. It is practically more convenient, especially for the examination of records bearing date within the last few centuries, to work by the amānta system and by true, not mean, tithis, and the Indian Calendar Tables are founded on this principle. According to it a lunar month begins at the moment of new-moon, or at the moment when the longitudes of the true sun and the true moon are equal. The first tithi ends and the second begins when the moon has increased her distance from the sun by exactly 12°, and so with the rest. Hence to find the mean time of beginning and ending of any given tithi we have to find first the mean longitudinal distance of the moon from the sun, and then to correct this to its true longitudinal distance by applying to it the equation of the centre for the moon and for the sun.

19A. This may be accomplished in several ways, but the way we adopted in the Indian Calendar was to work by what we may call the "a, b, c system," following Professor Jacobi in his Tables in the Indian Antiquary, Vol. XVII. (q.v., especially for present purposes, Part II., pp. 147–48). This system must now be explained. As before stated, both the sun and moon are in Hindū astronomy considered to be planets. Now a planet's true angular distance from the perihelion, or in the case of the moon from the perigee, point of its orbit is called its "true anomaly." Its "mean anomaly" is what that angular distance would have been if the planet had moved with mean or unvarying velocity. The difference between the true and mean anomalies, or the quantity necessary to make the mean equal to the true anomaly, is called the equation of the centre. (See Ind. Cal., § 107, p. 60.) Therefore, to find the

  1. ↑ Ind. Cal., § 50A
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