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

36 PAÑCASIDDHĀNTIKĀ II.10 Example 8. (a) On a certain day the Sun’s longitude is 5 rāśis. (b) On another day it is rāśis 11-15. In both cases find the mid-day shadow. (a) The Sun is 5 rāśis and is within the six rāśis from Karkaṭaka, being 2 rāśis from the beginning of the first point of Karkaṭaka (i.e. 3 rāśis). So, 2 × 2 = 4 digits is the shadow. (b) The Sun’s longitude is rāśis 11-15. This is within the six rāśis from the first point of Makara (9 rāśis), the Sun’s position being 11ʳ 15° − 9ʳ 0° = 2ʳ 15° = 2½ rāśis from that point. 12 − (2½ × 2) = 7 digits is the noon-shadow, Example 9. (a) The Sun is in its southward course and the shadow is 4 digits. Find the longitude of the Sun. (b) The Sun is in its northward course and the noon-shadow is 7 digits. Find the Sun. (a) Half the shadow = 4/2 = 2. As the course is southward add 3 rāśis; the Sun’s longitude is 5 rāśis. (b) Half the shadow = 7/2 = 3½. As the Sun’s course is northward, deduct from 15. 15 − 3½ = 11½ rāśis. This is the longitude of the Sun. From the two sets of examples it can be seen that the two formulae are the inverse of each other. The formulae are explained thus: This Siddhānta assumes that the noon-shadow is zero when, at the end of its northward course, it reaches the first point of Cancer. Then as it moves southward, the shadow increases to 12 digits at the end of the course, i.e. after 6 months, when the Sun reaches the first point of Capricorn. Assuming the increase to be uniform, there is an increase of 2 digits per rāśi. As the shadow is zero for the first point of Cancer, the longitude in rāśis measured from this point, multiplied by 2 gives the shadow. Thus, if c is the Sun in rāśis measured from the first point of Cancer and s is the shadow in digits, s = 2c for the 6 months till the Sun reaches Capricorn, where the shadow is 6 × 2 = 12 digits. Then the shadow decreases at the same rate to zero at the first point of Cancer, in the course of 6 months. Therefore if c is the Sun measured from the first point of Capricorn, when the shadow is 12, and s the shadow, then s = 12 − 2c. Now for the longitude of the Sun from the noon-shadow. We have seen that for the six rāśis from Cancer, s = 2c. Therefore c = s/2. But c is counted from the first point of Cancer, i.e. 3 rāśis. There- fore the Sun’s longitude in rāśis is 3 + c = 3 + s/2, which is the same as the instruction to divide the shadow by 2 and add three rāśis. For the six rāśis from Capricorn, we have seen that s = 12 − 2c. Therefore c = (12 − s)/2. But c is counted from the first point of Capricorn. i.e. 9 rāśis. Therefore the Sun = 9 + c = 9 +(12 − s)/2 = 15 − s/2, which is the instruction given. It must be noted here that both the formulae are very rough. At summer solstice, when the Sun is at the first point of Cancer, its north declination is maximum and given by Hindu astronomy as 24°. At that time, the mid-day Sun is at the zenith at places on 24° north latitude (like the region of Ujjain) and so it is only in this region that the shadow is zero at this time. When the Sun reaches its southernmost point at winter solstice, i.e. the first point of Capricorn, its south declination is 24°. Therefore the zenith distance of the noon-Sun as seen from latitude 24° North at that time must be 48° towards the South, and the shadow at that time must be greater than 13 digits and not 12. (All this will be shown in Chapter IV). If the shadow is to be 12 digits, the Sun’s zenith-distance must be 45° and the region where the Sun is seen at this zenith-distance is 21° North latitude. Thus there is contradiction even here. In verse 8, we showed that the rule is intended for a region having about 36° North latitude, neither 24° nor 21°. Thus, so far as these things are concerned, the Siddhānta seems to be a hotch-potch.