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

122 PAÑCASIDDHĀNTIKĀ IV. 49 whether actually the constants are 6 and 12, as here or 5 and 10, or some other number and double that, depends upon the accuracy of the result we get. For the matter of that there is another rule, very popular and attributed to our author himself in the following form: time = 5 × daytime ÷ (shadow − mid-day shadow + 10), given by the popular verse: chāyā nijeṣṭā dinamadhyabhāgacchāyonitā diksahitā tayāpte | dine śaraghne gatagamyanāḍīḥ śrīmān Varāho vadati syayuktyā || Here too the shadow is that of the 12 digit gnomon. Note that the multiplier here is 5, and the additive constant double that, viz. 10. Actually, different constants for different places, and for different times, even if the place is the same, may have to be used if sufficient accuracy is to be secured. So the average for a particular region may be used for that region in the rough rule. Let us now compute the constants using the data of Example 16 (a), and examine the degree of accuracy of the constants 5 and 10 used in the above verse. In the example we find that the time is 8 nāḍīs for shadow aṅg 13-56. The daytime for the day is nā. 33-20 and mid-day shadow, aṅg. 2-50, as we have already given in Example 17. Using the assumed form, x × 33 1/3 ÷ (13 14/15 + 2x − 2 5/6) = 8. x × 33 1/3 = 8 (2 x + 11 1/10) = 16 x + 88 4/5. 17 1/3 x = 88 4/5. x = 88 4/5 ÷ 17 1/3 = 444 × 3 ÷ (5 × 52) = 5 8/65. As 8/65 is small, x, the multiplier, may be taken as 5, and y (i.e. 2x) may be taken as 10, with tolerable accuracy, as VM himself seems to have done in the popular verse. Let us examine the accuracy given by this by working Example 17 using this. The time = 5 × 33 1/3 ÷ (13 14/15 + 10 − 2 5/6) = 500 × 10 ÷ (3 × 211) =nā. 7-54. Note how near this is to the correct 8 nāḍīs, and contrast with the result of the rule given by the text, nā. 8-37. Let us once again examine the relative accuracy by computing the time sought in the example under IV. 41-44, from the shadow caused by the Sun on the prime vertical, at the place and time of Example 16 (a). The prime vertical shadow was given as aṅg. 17-4. The time got there was nā. 6-53. Using the rule of the text, time = 6 × 33 1/3 ÷ (17 1/15 − 2 5/6 + 12) = nā. 7-37, which is far from the correct nā. 6-53. Using the popular verse, time = 5 × 33 1/3 ÷ (17 1/15 − 2 5/6 + 10) = nā. 6-53, agreeing exactly with the correct time. What are we to conclude from this? [नाडीतः छाया] छायाऽऽर्की नाडीभिर्दिनमानं षड्घ्नमुद्धरेत्तत्र । लब्धं द्वादशहीनं मध्याह्नच्छायया सहितम् ॥ ४९ ॥ Shadow from time 49. Roughly again, the shadow can be got thus from the time: Multiply the daytime by 6, and divide by the time for which the shadow is sought. Add the