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

VI.4 VI. VĀS.-PAUL. SIDDHĀNTA – LUNAR ECLIPSE 159 We can show this by theoretical considerations as well, thus: Rāhu lessened by 1° 36′, is equiva- lent to Moon increased by 1° 36′, in its effect on Moon – Rāhu. The latitude is proportionate to the distance of the Moon from Rāhu in each quadrant. In the first and third quadrants, the Moon is greater than Rāhu, and the increase in the Moon by the correction increases the latitude. In the second and fourth quadrants, the Moon is less than Rāhu, and the increase in the Moon by the correction lessens its distance from Rāhu, with the result that the corrected latitude is less. Thus for Moon greater than Rāhu the correct latitude is greater, and for Moon less, it is less. The rest is as we have shown already. Now we shall find the quantity of correction: As the angles at R and R′ are equal, the corrected and the original orbits are parallel. (see the Fig.) Therefore the differences in the latitudes at any position like M′₁, M₁, M′M, and C R are equal. But C R, being the latitude caused by 1° 36′ of longitude, is equal to 380′ × 1° 36′ ÷ 90° = 6¾ minutes of arc. Therefore the difference in any position is 6³/4′. We shall first see how much difference this will produce in the half duration measured in minutes of arc. Clearly it is = √(55² – (lat ± 6³/4)²) – √(55² – lat²) = √(55² – lat² ∓ 13½ lat) – √(55² – lat²), (if 6³/4² is neglected, being small in comparison with 55².) = √(55² – lat²) √{1 ∓ 13½ lat/(55² – lat²)} – √(55² – lat²) = √(55² – lat²) {1 ∓ 13½ lat/2(55² – lat²)} – √(55² – lat²) (if higher powers of 13½ lat/(55² – lat²) are neglected. = ∓ 13½ lat/2√(55² – lat²), and the author has neglected lat² and taken this as ∓ 13½ lat/2 × 55. Using the difference of the mean daily motions of the Sun and the Moon, because this will not matter in the already rough result, and doubling for the whole duration, the vināḍis of correction are, 13½ lat × 2 × 60 × 60 ÷ (2 × 55 × 720) = 13½ × lat × 5/55 = 13½ × {(Moon ∼ R)° × 55/13°} × 5/55 (since lat = Moon ∼ Rāhu)° × 55/13°,) = 5 × (Moon ∼ Rāhu), roughly, as given here. The greater the latitude, the greater the roughness, but this will be submerged in the roughness caused by several other things like the incorrect semi-diameters etc., but the method has the advan- tage of being easy to apply. We shall consider the readings now. In verse 3, the need for correcting mūlaḥ into mūlam and kālasthiteḥ into kālaḥ sthiteḥ will be clear, as also for sthityā into sthityām in verse 4. As for correcting bhāgaiḥ into bhāgāḥ, this is justified by what we have shown in the explanation, viz. that it is Moon ∼ Rāhu that is to be multiplied by 5 to give the vināḍis of correction. If the word bhāgaiḥ, is taken as it is, the instruction should be taken to mean “five multiplied by the difference of Moon ∼ Rāhu and 13°. By this, the correction, instead of being zero, as it should be for zero latitude, is the maximum of 65 vināḍis. Instead of being the maximum for maximum latitude, the correction becomes zero. Further, on both sides of zero latitude, where there is a transition from the Moon being greater, to Rāhu being greater, there is a jump from – 65 vināḍis to + 65 vināḍis, which itself is an indication that the formula is incorrect. But it may be objected that if our correction into bhāgāḥ is accepted the word trayodaśonāḥ serves no purpose, for the computation will be begun only if the difference is less than 13°, and therefore this need not be mentioned. The answer is this: From our explanation of the formula for correction it may be seen that it is applicable if the difference is