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

III.10 III. PAULIŚA-SIDDHĀNTA 49 unit or radius being 120′, and the diameter 240′. The day-diameter is twice the tabular cosine of the Sun’s declination. Thus this formula reduces to the modern form: sin cara = k tan latitude × tan declination. The degrees of cara got from the formula, converted into minutes and divided by 3 gives the whole cara. The cara-khaṇḍas or differences are got deducting the whole cara of the beginning of a rāśi from that of the end of the rāśi. We shall not derive this formula now. We shall restrict ourselves to deriving the given cara- khaṇḍas from the formula. The ‘diameter’, as we have said, is 240′ (vide IV.1). The tabular sines of declination at the ends of the three rāśis Meṣa, etc. is 24′ 24″, 42′ 15″, 48′ 48″ (from IV.24). The day- diameters for the same are 235′, 224′ 40″, 219′ 15″ (from IV.25). Sin latitude ÷ sin (90° − latitude) = equinoctial shadow ÷ 12. (This will be shown when dealing with IV). Therefore; successively, sine cara = equinoctial shadow × {(240 × 24 2/5) ÷ (235 × 12), (240 × 42¼) ÷ (224 2/3 × 12), (240 × 48 4/5) ÷ (219¼ × 12)} = Eq. shadow × (2′.08, 3′.78, 4′.45). The arcs of these cannot be found in terms of the Eq. Shadow unless it is known. But as the author intends this rule only for North India, where the eq. shadow may be taken as 6 digits on the average, we shall frame the rule for 6 and then use it for other places by the rule of proportion. So, multiplying the numbers 2′.08, 3′.78, 4′.45 by 6, we get sine cara = 12′ 29″, 22′ 41″, 26′ 42″. Using the tabular sines, the arcs of these = 5° 58′.3, 10° 53′.6, 12° 51′.6 = 358′.3, 653′.6, 771′.6. Dividing by 3, we get the whole cara in vināḍīs, 119.4, 217.9, 257.2. Dividing these by 6 and multiplying by Eq. shadow, we get the whole cara vināḍīs for any Eq. shadow, viz., Eq. shadow × (19.9, 36.3, 42.9). Deducting the next from the previous, and because the cara is zero for the beginning of Meṣa, as declination is zero, the cara differences (khaṇḍas) got are, Eq. shadow × (19.9, 16.4, 6.6). This is practically the same as Eq. shadow × (20, 16½, 6¾) given by the author. In the Vākyakaraṇa, Mahābhāskarīya and Siddhānta Śiromaṇi, the same method is given. As we have said, the cara-vināḍīs are zero for the beginning of Meṣa when day-light is 30 nāḍis. Then at the end of each rāśi, they increase by a quantity equal to the differences, reaching a maximum at the end of Mithuna, the day-light then being a maximum also, as the declination has reached a maximum. After that the declination decreases in the same manner in which it has increased, and the day-light decreases from maximum and the cara also decreases. At the end of Kanyā the declination becomes zero again, the day-light equals 30 nāḍikās again, and the cara becomes zero again. So the differences have to be used in the reverse order for Karkaṭaka, Siṃha, and Kanyā. After this the South declination increases and decreases just like the North declination and corresponding to this the night-time increases from 30 nāḍikās to a maximum and decreases again to 30 nāḍikās at the end of the next six months; and the cara also increases from zero to a maximum and then decreases to zero, repeating what it is for North declination. Hence the instruc- tion to repeat for the next six months of the solar year. If the cara is required for any day within the month, it is to be got by interpolation, which is well known and therefore not mentioned: Example 4. (a). In a place the Eq. shadow is 5 digits. Find the cara-vināḍīs when the Sun is rā.2-10-0. (b) The Eq. shadow is 7. the Sun is at the end of Tulā, find the cara-vināḍīs. (a) The constant for Meṣa is 20, for Vṛṣabha 16½ and for 10° of Mithuna, 6¾ × 10°/30° = 2¼. Adding, the total cara is 20 + 16½ + 2¼ = 38¾. Multiplied by Eq. shadow, the vināḍīs are 5 × 38¾ = 193¾. (b) The Sun is rā.7-0-0 and therefore 1 rāśi has gone in the second half of the zodiac. Therefore the cara-vināḍīs = 7 × 20 = 140.