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

116 PAÑCASIDDHĀNTIKĀ IV. 44 [तत्कृतिविना (कृ) तानां 'खखवेदसमुद्रशीतरश्मीनाम्' । पदमर्कघ्नं शङ्कङ्गङ्गुलाऽऽख्यलिप्तोद्धृतं छाया ॥ ४४ ॥] 44. Square this and deduct from 14,400. Take its square root, multiply this by twelve, and divide by sine altitude. The result is the length of the shadow of the twelve-digit gnomon. The following are the steps in the work: (i) Sine altitude = {sine (degrees of the ∓ degrees of half-cara) ± sine half-cara} × sin colat × day-diameter ÷ 28,800. (Here, of ∓ or ±, the upper sign should be taken for the 6 signs Meṣa etc., and the lower for the 6 signs Tulā etc.) (ii) The shadow = 12 × √14,400 − sin ² altitude ÷ sin altitude Example 16 (a). At a certain place where the sine of the co-latitude (i.e. cos. lat.) is 103′ 55″, when the Sun is in the 6 signs from Meṣa on a particular day, the cara is 200 vināḍis, and the day-diameter is 229′ 51′. Find the length of the shadow at 8 nāḍīs from Sunrise. (i) Degrees of half-cara = 200 ÷ 20 = 10°. Degrees of time = 8 × 6 = 48°. As the Sun is in the six signs from Meṣa, deducting 10° from 48°, we get 38°. Sine 38° = 73′ 35″. The sine of the half-cara, i.e. sin 10° = 20′ 50″. Adding the two signs, (since the Sun is from Meṣa), 73′ 35″ + 20′ 50″ = 94′ 25″. Sine altitude = 94′ 25″ × 229′ 51″ × 103′ 55″ ÷ 28,800 = 78′ 19″ (ii) The shadow = 12 × √14,400 − 78′ 19″² ÷ 78′ 19″ = 13 aṅg 56 vyaṅ. Example 16 (b). At the same place, on the same day, find the shadow at one nāḍī after sunrise. (i) The degrees of half-cara (already found) = 10°. The degrees of time = 1 × 6 = 6°. The half- cara degrees have to be deducted, but cannot be deducted, being greater. Therefore, taking the sine of the 6° alone, we have 12′ 32″. Sin altitude = 12′ 32″ × 229′ 51″ × 103′ 55″ ÷ 22,800 = 10′ 24″. (ii) shadow × 12 × √14,400 − 10′ 24″² ÷ 10′ 24″ = 137 aṅgulas 57 vyaṅgulas. But it should be mentioned here, that the author's instruction for the case when the degrees of half-cara cannot be deducted from the degrees of time, will give only a rough result. This will not matter much in places where the degrees of half-cara is small, as in India, and therefore given by the author. For correctness, the following instruction is to be followed. If the degrees of half-cara cannot be deducted from the degrees of time, deduct the degrees of time from the degrees of half-cara, find its sine, and deduct this from the sine of half-cara. This sine should be multiplied by sin colat. etc. and sine altitude is to be got. Because this will not produce much differences in our country, the author has not given this detail. (Even if the cara is 5 nāḍīs the difference in sin alt. will be only 15′.) Further, the measurement of long shadows cannot be accurate, and any inaccuracy caused by the author's rough work will be submerged in the inaccuracy of measurement. 44a. C.D.विनाशकृतानां