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52 PAÑCASIDDHĀNTIKĀ III. 14 by adding or subtracting this, according as the place is East or West, to or from nāḍīs 7-20 (for Ujjain) or 9-0 (for Banaras), respectively, the time difference for longitude from Yavanapura can be got. Example 6. The latitude of a place west of Ujjain is 27° and the distance between them is 44 4/9 yojanas. Assuming the latitude of Ujjain to be 24°, find the time difference for longitude of the place from Yavanapura. The distance in degrees = 9 × 44 4/9 = 5°. The difference in latitude in degrees is 27° − 24° = 3°. √(5² − 3²) = 4 = the east-west difference in degrees. 4/6 nāḍikās = 40 vināḍikās is the time differ- ence. As the place is West of Ujjain, deducting this from 7-20, the time difference from Yavanapura is nāḍikās 6-40. The rule is explained thus: It is well known that in a plane right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the sides containing the right angle. Now, let the places be P₁ and P₂. Let the point where the line of longitude passing through one of the places, say P₁, cuts the latitude passing through the other say P₂, be C. Then C is practically a right angle of which CP₁ is one arm and CP₂ the other, and P₁P₂ is the hypotenuse of the right angled triangle P₁CP₂. If P₁P₂ is not too great, then this triangle, though on the surface of a sphere, may be treated as practically a plane triangle. P₁C is the difference in latitude of the places. P₂C is the difference in longitude, which is wanted. P₁P₂ is the distance between them. We are going to find the differ- ence in longitude P₂C in degrees. The difference in latitude P₁C is also in degrees. So P₁P₂ also must be found in degrees. So the distance in yojanas is converted into distance in degrees by multiplying the yojanas by 9 and dividing by 80, because according to this Siddhānta there are 9° for 80 yojanas on the Earth, i.e. the circumference of the Earth (360° if given in degrees) is 3200 yojanas. Thus we have the difference in longitude in degrees = CP₂² = √(P₁P₂² − CP₁²). The degrees are converted into time by the proportion, if there are 60 nāḍikās for 360° of longitude, how much for the degrees got. Therefore degrees got × 60/360 = degrees got/6, are the nāḍikās of difference in longitude. In view of the right angled triangle not being exactly plane, a better result will be got if the nāḍikās obtained are multiplied by the circumference of the earth and divided by the circumference of the line of latitude midway between the places. But the author has not mentioned this because this method is intended for India and in India the two circumference do not differ much and therefore the difference between the two methods will be negligible. Sūrya Siddhānta etc. give the correct method. The author’s method is given by the Mahābhāskarīya with the name Adhvā (‘Path’) and by the Vaṭeśvara Siddhānta with the name ‘Adhvavaha’ (‘Marching on the Path’), only to be condemned as inaccurate. In the Vākyakaraṇa, which is also satisfied with rough results, the distance along the latitude (CP₂ in the explanation) is taken as found and a rule given. The method of determining the latitude of a place is given in IV.20-21 and the author expects us to get it by using that method and use it in the formula. Another thing must be mentioned here. If the two places are distant from each other or inter- vened by a sea or mountain or some such obstacle, as for instance Yavanapura and Ujjain, Yavanapura and Banaras, or Ujjain and Banaras, then by observation, from the two places, of celes- tial phenomena that are visible everywhere at the same moment, like the circumstances of a lunar eclipse, the time difference for longitude can be obtained. (All Siddhāntas give methods based on this principle, and the Mahābhāskarīya in Chapter II, which is, devoted exclusively to Deśāntara, gives two methods.) The following is the method: Let us assume that the meridian of Ujjain as the prime meridian (generally given in all Siddhāntas do) and by computation it has been found that at 5 nāḍīs after midnight the total obscuration of the moon begins. (The beginning or end of the