Page:Sewell Indian chronography.pdf/25
22. Another point requiring notice concerns the time-duration of the tithi. Having found by the above process the value of t or the tithi-index for the given moment, we may require to ascertain the times of the beginning and ending of the tithi in question. Its beginning and ending values are given in Table VIII., col. 3, and by subtracting the found value of t from the ending value of the tithi we find how many lunation-parts intervened between the time for which we have worked and the ending-point of the tithi. We convert these intervening lunation-parts into mean time by the use of Table X. (heading "Tithi-Index"). And we similarly find the mean time of the beginning of the tithi by subtracting the value given in Table VIII., col. 3, for the beginning from the found value of t, and converting the difference into mean time as above. Say that we find that at sunrise on a certain civil day t = 2362. By Table VIII., col. 3, we find that the 8th tithi śukla, or the 8th tithi of the first fortnight, was current. It began (2362-2333 =) 29 lunation-parts = (Table X.) 2 h. 3 m. (mean time) before sunrise. It ended (2667-2362 =) 305 lunation-parts = (Table X.) 21 h. 16 m. + 21 m. = 21 h. 37 m. (mean time) after sunrise.
23. But Table X. shows only the values of the mean tithi, under both the headings "Tithi-parts" and "Tithi-index"; and as every true tithi varies in time-length because of the varying velocity of the true moon in its orbit, it is incumbent on us, if we desire absolute accuracy, to work the case to a conclusion as explained in the Indian Calendar (§ 82, p. 50, and pp. 81, 82.) Having done this we get the final and correct beginning and ending times of the true tithi.
24. In the examples given below, the figures given under head d (generally given in brackets, as in Table I., cols. 13, 19 of the Indian Calendar) refer to Table IX. of that work, from which the European month and day of the month are obtained. In the numbers given for (w) days of the week, 0 is Saturday, 1 Sunday, &c.
25. In adding together the necessary figures for a or t all values over 10,000 are taken less 10,000, and in figures for b and c all values over 1000 are taken less 1000; since the figures represent longitude measured by 10,000ths and 1000ths of a circle, and when a full circle has been described and the point 0 is reached a new circle of course commences. 26. In actual work we often find that our result differs by one unit in a or b or c from the expected result, and it is well to remember that this difference is inherent in the system. Each unit in a, for instance, represents a time-interval of about 4.25 m., and we advance by whole units; but such advance, with decimals or fractions ignored, often causes a difference in the ultimate figure. An example of this can be gathered from Table IV. If I desire to state the a for 395 days I can either do it by adding the a of 385 days to the a of 10 days, or by adding the a of 300 days to the a of 95 days. In the former case the result is (373 + 3386 =) 3759; in the latter it is (1590 + 2170 =) 3760. See also the case stated under "Correction III." below (para. 57). Here we have two figures stated with decimals which have to be added together. One is 200.6 and another is 7.83. Stated separately these would be stated in whole figures as 201 and 8. Total, as so stated, 209. But added with the decimals the total is 208.43, which, stated as a whole figure, would be 208. Such differences are practi- cally unavoidable in every working system.
27. The twelve sidereal saṁkrāntis occur at the beginning-points of the twelve signs of the zodiac, separated from one another by 30° of celestial longitude. The first, called the Mēsha saṁkrānti,