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EXAMPLES.—TRUE AND MEAN MĒSHA SAṀKRĀNTIS..

85

If we take the full figures as given, viz., 1.01128935̇18̇, this, multiplied by 9.25, gives exactly 9 d. 8 h. 30 m. 22 s.

Example 10. To find, according to the Sūrya Siddhānta, the time of tropical Mēsha saṁkrānti in any year. Rule. (See pp. 152–3, Ind. Cal., and above "Tropical saṁkrāntis," § 33, &c.)

Wanted the time of occurrence according to the Sūrya Siddhānta of tropical (sāyana) Mēsha saṁkrānti of Śaka 1000 current, K.Y. 4179 current, A.D. 1077–78, the same year as in the last example.

The remarks at the beginning of Example 9 apply equally to this problem. We start with true nirayaṇa (or sidereal) Mēsha saṁkrānti-time, as given according to the Ārya Siddhānta for the year in question, in Table I., viz., March 23rd, 14 h. 52 m. We next convert this to the time according to the Sūrya Siddhānta by Table XVII., adding 64 m. This gives us the same moment by the Sūrya Siddhānta as March 23rd, 15 h. 56 m.

(a) Rough calculation. (The given year Śaka) 1000 − 422 (the base-year, both current) = 578, and 578 ghaṭikās = 9 d. 38 gh. = (Table XXV.) 9 d. 15 h. 12 m. Deducting this period from March 23rd 15 h. 56 m., we have March 14th h. 44 m., and this was roughly the time of tropical Mēsha saṁkrānti in the given year.

(b, i.) Accurate calculation. Noting that the intervening days belonged to the sidereal month Mīna,^[1] and that the length of Mīna was, according to the Sūrya Siddhānta (Table XVIII.B), 30 d. 8 h. 29 m. s., while 578 × ⁠3/200⁠ = 8.67 = (in ayanāṁśas) 8° 40′ 12″, we multiply ⁠1/30⁠th of the length of Mīna by the ayanāṁśas.

${\frac {{\text{30 d. 8 h. 29 m. 1 s.}}\times 8^{\circ }40'12''}{30}}={\text{8 d. 18 h. 31 m. 52 s.}}$

		h.	m.
Sidereal Mēsha saṁkrānti	March 23	15	56
Less	8	18	32
Tropical Mēsha saṁkrānti	March 14	21	24

This was the required day and time.

(b, ii.) If, as in the last example (b, ii.) we use Table XX.A the sum to be done is―

${\text{1 d. 0 h. 16 m. 58 s.}}\times 8^{\circ }40'12''$

which comes to the same result.

(b, iii.) And if, as in the last example (b, iii.) we use the decimals for each element of the sum, and this is by far the easiest and shortest method, we have

1.01178 x 8.67

which = 8.7721326, and this, by Table XXXVI., = 8 d. 18 h. 31 m. 52 s., and gives us the same result.

[Comparing the results of Examples 9 and 10, it will be noticed that whereas the difference in the case of the former (according to the Ārya Siddhānta) between the rough and the accurate results, viz., 8 h. 52 m. and 6 h. 22 m. is only 2 h. 30 m.; in the second case Note. (by the Sūrya Siddhānta) the difference between the rough and the accurate results amounts to as much as 20 h. 40 m. When working by the Sūrya Siddhānta one must always expect considerable divergence in this respect, increasing with the distance of the given year from the base year; because the first rough calculation is made on the basis of the degrees of precession, or ayanāṁśas, amounting to 1′ per annum, which is taken as = 1 ghațikā per annum in time, while the second, accurate, calculation is based on the actual amount of precession according to that authority, viz., 54″ per annum, which = 0.9 gh. per annum in time. In 578 years this difference is considerable.]

↑ See note following Example 18B.

[1] See note following Example 18B.

[1]