60
direction, place and time
Brahmagupta is more precise. He says :
"The point where the lines passing through the two fish-figures, which are drawn by means of three shadow-ends (of the gnomon), intersect each other is, for places in the northern hemisphere, the south direction* (if the midday shadow falls to the north of the foot of the gnomon) If the midday shadow falls towards the south of the foot of the gnomon, it is the north direction". 2
The above rule is evidently based on the assumption that the locus of the end of the shadow of the gnomon is a circle. In fact for places whose latitudes are less than 90°- Q (where Q is the obliquity of the ecliptic), this locus is a hyperbola, so the above assumption is not a correct one. The above rule will, however, give an approximately correct result if the three shadow-ends chosen are not far removed from the vertex of the hyperbola.
The method of drawing a circle through three given points by. the aid of two fish^figures is called "trisarkara-vidhana" by Bhaskara I. 3
A rule . for getting the length of the hypotenuse of the shadow :
4. The square root. of the sum of the squares of the gno- mon and its shadow (is equal to the hypotenuse of the shadow : this), say the learned (astronomers), is always the semi-diameter of its own circle in the calculations with the shadow. 4
By "the semi-diameter of its own circle" is meant "the semi-diameter of the circle of shadow".
The circle of shadow is, as Bhaskara I has said 5 , useful in the appli- cation of proportion in connection with the problems involving the shadow of the gnomon. For example, in finding out the Rsine 6 of the Sun's zenith distance from the shadow of the gnomon, the proportion is :
1 The north „ direction being indicated by the end of the midday shadow of the gnomon.
2 BrSpSi, iii. 2.
3 See LBh, vi. 16.
4 This rule is found also in A, ii. 14. 6 In his commentary on A, ii. 14.
• Rsine stands for "radius x sine".
