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No. 1.] THE TRUE LONGITUDE OF THE SUN IN HINDU ASTRONOMY.
A. The exact nakshatra (i) by the First Arya-Siddhanta.-Mesha-samkranti occurred in the year A.D. 1490 (Table I, cols. 13, 17) at 10 h. 55 m. after mean sunrise on Day 86. The day of the record is 106. 106-86 20. Turning to day 20 in Table XLVIIIA, col. 1, we see that the sun's true longitude, s, was, at 10 h. 55 m. after mean sunrise, 540-6811, in 10,000ths of the circle. Deduct the sun's true motion on Day 20 for 10 hours and 55 minutes; viz. (Table XLIX) for 10 hours 11-2059, and (Table L) for 55 minutes 10457, total 12-2516. 540-6811-12-2516 528-4295. This is the value of s at mean sunrise on the given day. Add s to t, 528+8808 (in whole numbers) = 9336. This is the proper nakshatra-index, n, for mean sunrise, always supposing the tithi-index, t, to be correct. It differs by 8 units from the Indian Calendar valuation 9328, which is found therefore so much in error. This, in time (Table X), is over half an hour; but we find that the moon was certainly in Uttara Bhadrapada at mean sunrise.
(ii) By the Surya-Siddhanta.-The process, mutatis mutandis, is similar. Mesha-samkranti occurred at 12 h. 44 m. after mean sunrise. On Day 20 (Table XLVIIIB) s, at 12 h. 44 m. after mean sunrise, was 540-5000. Deduct for the sun's true motion on Day 20, for 12 h. (Table XLIX) 13-4471, and for 44 m. (Table L) 0-8365. Total 14-2836. 540-5000 -14-2836: 526-2164. This was the s at mean sunrise. 8+t=526+8808 9334. This, if the t is correct, is the correct value of the nakshatra-index, n, at mean sunrise; and we see that the Indian Calendar n, 9328, was in error by 6 units, or 24 minutes of time.
=
B. The exact tithi (t) by the First Arya-Siddhanta.-We first check the mean anomaly of the sun, c. Our preliminary work shewed this to be, in thousandths of the circle, 332. Table XLVIIIA, col. 3, shews that at 10 h. 55 m. after mean sunrise the mean anomaly was, in tenthousandths, 3322-1148. Deduct by Table XIV the sun's mean motion during 10 h. 11-4074, and during 55 m. 10457, total 12-4531. 3322 114812-45313309-6617. This is the exact mean anomaly of the sun, c, at mean sunrise. If we work by the simple Table VII of the Indian Calendar and take the c in thousandths as 331, we see that equation c would remain unchanged at the value 7. Hence, accepting the values assumed for the moon's mean place and equation, we find that the tithi-index was actually, as found by us, 8808.
We try this again by the more elaborate, but more, accurate process recommended above (§ 247, ii, last para.). By Table XLVIIIA, col. 7, we have the equation of the centre at 10 h. 55 m. after mean sunrise on Day 20 given as 51-8996 in 10,000ths of the circle. The difference in 24 hours had been (day 19) 52-3833-(day 20) 51-8996, or 0-4837. A 24th part of this is '02015, the average difference per hour. Accepting 10 h. 55 m. as 11 h., we have for 11 h. the difference (02015 x 11) 2216. Call this 222 and add it to 51-8996. Result gives the equation of the centre at mean sunrise 52-1216. Then 604 52-1216 8-2784, the value of equation c. This is over 8, and hence we find that the former valuation 7 is incorrect, and that the tithi-index at mean sunrise was really 8809, not 8808, by the Arya-Siddhanta. But we have also found out a little more regarding the nakshatra-index, n. s was found to be really 528-4295; and if we substitute the full value of equation c, viz. 8-2784, for the value found by the original calculation, viz. 7, we have for the tithi-index:
a
Equation b Equation c
t =
Add (8)
7
n=.
8541 260
8.2784
8809-2784 528.4295
9337-7079, or, say, 9338 in whole numbers.
