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

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