पञ्चसिद्धान्तिका (आचार्य वराहमिहिर - सूर्य, रोमक, पौलिश, वासिष्ठ एवं पितामह सिद्धान्त)

120 PAÑCASIDDHĀNTIKĀ IV. 47 (v) Arc I = 38° 3′. Arc II = 9° 59′. The time from sunrise in nāḍis = 38° 3′ / 6 + 9° 59′ / 6 = 8 nāḍis. (Note that this work is the inverse of example 15 (a). There, 8 nāḍis were given, and the shadow 13 aṅg 56 vyaṅg was computed. Here, for the shadow 13 aṅg 56 vyaṅg, the nāḍis amounting to 8 have been computed). Example 17 (b) For the same place, on the same day, find the time when the shadow is 137 aṅg 57 vyaṅg. (i) ‘First sine’ = 1,72,800 ÷ (103′ 55″ × √(144 + 137 57/60² = 12′ 1″. (ii) Earth-sine = 60′ × 34′ 30″ ÷ 103′ 55″ = 19′ 55″. (iii) Sine I = (12′ 1″ − 19′ 55″) × 240 ÷ 229′ 51″ = − 8′ 15″ (iv) Sine II = 19′ 55″ × 240 ÷ 229′ 51″ = 20′ 48″. (v) Arc I = − 3° 57′, Arc II = 9° 59′. The time from sunrise = − 3° 57′/6 + 9° 59′/6 = 1 nāḍi. (Note that is the inverse of Example 15 (b). There the shadow 137 aṅg 57 vyaṅg was computed for one nāḍi from sunrise. Here for the same shadow the time one nāḍi is computed.) We shall do the same by the inverse operation of the work previously given by the author: The ‘First sine’, computed is 12′ 1″. The Earth-sine computed is 19′ 55″, and greater than the ‘First sine’. Therefore taking the ‘First sine’ alone, 12′ 1″ × 240 ÷ 229′ 51″ = 12′ 33″. The arc of this = 6° 2′. Dividing by 6, the time obtained is one nāḍi and 1/3 vinādi, and neglecting the negligible 1/3 viṇāḍi, we see the same time is got. For proof of the rules here given, we shall derive these from the rules for the shadow given the time, as the operation is practically the inverse of the operation given there. In the previous work, rule (ii) gives: 12 × √(14,400 − sin² altitude) ÷ sin altitude = shadow. ∴ 144 × (14,400 − sin² alt.) = sin² alt. = shadow². ∴ 144 × 14,400 = sin² alt. × shadow² ± 144 sin² alt. = sin² alt. (shadow² + 144). ∴ 12 × 120 = sin alt. × √(shadow² + 12²). ∴ 12 × 120 ÷ √(shadow² + 12²) = sin alt. = 12 × 120 × 120 × sin colat. ÷ (120 × sin colat. × √(shadow² + 12²) = ‘First sine’ × sin colat. ÷ 120, (because, 12 × 120 × 120 ÷ (sin colat × √(shadow² + 12²)) = 1,71,800 ÷ (sin colat × √(shadow² + 12²)) = ‘First sine’ as given). Similarly, in the previous rule (i), sin alt. = {sine (degrees of time ∓ degrees of half-cara) ± sin half-cara} × sin colat. × day-diameter ÷ 28,800, = ‘First sine’ × sin colat. ÷ 120. ∴ ‘First sine’ × 240 ÷ day-diameter = {sin (degrees of time ∓ degrees of half-cara) ± sin half- cara}. ∴ ‘First sine’ × 240 ÷ day-diameter ∓ sin half-cara = sin (degrees of time ∓ degrees of half-cara). ∴ ‘First sine’ × 240 ÷ day-diameter ∓ Earth-sine × 240 ÷ day-diameter = sin (degrees of time ∓ degrees of half-cara). ∴ (‘First sine’ ∓ earth-sine) × 240 ÷ day-diameter = sin (degrees of time ∓ degrees of half-cara) = sin (degrees of time after the Sun has touched the unmaṇḍala)