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Example 18.—The reverse process.

Rule. The time according to the Sūrya Siddhānta being given, apply to it Correction I. (Table XVILI., with signs reversed) and the time-difference between the śōdhya intervals according to the two authorities, viz. − 33 m. 18 s.

Wanted the time of mean Tulā saṁkrānti by the Ārya Siddhānta in Saka 1153 expired, A.D. 1231–32, the time of the same according to the Sūrya Siddhānta being given as Friday, September 26th, A.D. 1231, at 7 h. 30 m.

		h.	m.
Sū. Sid. hours and minutes		7	30
Corr. I (Table XVII.)	−	1	21
Śōdhya difference	−		33
		5	36

Answer. Friday, September 26th, A.D. 1231, at 5 h. 36 m. Laṅkā time.

Example 18a.—To find the moment of any true tropical saṁkrānti according to the Ārya Siddhānta.

Rule. (This is fully given above, under "Tropical saṁkrāntis," §§ 36 to 39. See also pp. 152–3 of the Indian Calendar, and the working for tropical Mēsha saṁkrānti shown in Example 9.) Find the moment of the corresponding true sidereal saṁkrānti as in Example 11.

Wanted the time of tropical Makara saṁkrānti in the year Śaka 1000 current, A.D. 1077–78:—

	d.	w.	h.	m.
Time of sidereal Mēsha saṁk. (Table I.)	23 March (82)	5	14	52
Coll. Dur. to Makara saṁk. (Table XVIII.A)	(275)	2	15	43
Time of sidereal Makara saṁk.	(358)	1	6	35

The Makara saṁkrānti therefore occurred on Sunday, December 24th (358, Table IX.), at 6 h. 35 m. Laṅkā time.

(a) Rough calculation. (The given year, Śaka) 1000 − 445 (the base year) = 555. 555 ghaṭikās = 9 days 15 gh. = 9 d. 6 h. Deducting 9 d. 6 h. from December 24th, 6 h. 35 m., we arrive at Friday, December 15th, at 0 h. 35 m. Laṅkā time, or 35 m. after sunrise, as the required moment of tropical Makara saṁkrānti in the given year.

(b, i.) Accurate calculation. The 9 days in question belong to the sidereal month Dhanus, the length of which is, by the Ārya Siddhānta (Table on p. 10, Ind. Cal., or Table XVIII.A), 29 d. 8 h. 24 m. 48 s. The difference of years, 555, multiplied by ⁠1/60⁠ = ⁠9+15/60⁠, or, in ayanāṁśas (degrees of precession), 9° 15′.

${\frac {{\text{29 d. 8 h. 24 m. 48 s.}}\times 9^{\circ }15'}{30}}={\text{9 d. 1 h. 11 m. 39 s.}}$

Call this 9 d. 1 h. 12 m., and deduct it from the time of the true sidereal Makara saṁkrānti already found.

d.	w.	h.	m.
24 Dec. (358)	1	6	35
9 Dec. (358)(9)	2	1	12
15 Dec. (358)349	6	5	23

Answer. Friday, December 15th, A.D. 1077, at 5 h. 23 m. Laṅkā time.

(b, ii.) The sum above being troublesome, we may substitute the hours, &c., for one degree of Dhanus, and carry out the calculation by col. 2 of Table XX.A. The sum is then simplified into—

0 d. 23 h. 28 m. 49.6 s. x 9° 15′

which yields the same result.