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IV. 56 IV. THREE PROBLEMS 133
Sūryāgrā is greater, then the Sun is in the northern hemisphere, i.e. in the six
signs from Aries).
56. The Agrā thus got multiplied by the sine of colatitude, and divided by 48′
48″. is the sine of the Sun’s longitude and from that the sun is obtained. (From
this sine, first the degrees of Bhuja, D, is got. If the Sun is in the northern
hemisphere, then the Sun’s longitude is D, or (six signs − D). If the Sun is in
southern hemisphere, the Sun’s longitude is six signs + D, or (twelve signs −
D). What exactly it is of the diad must be determined by the Sun’s ayana).
(Following the method for the Sun, the other grahas also can be got).
The following are the steps in the work:–
(i) As already seen, shadow-hypotenuse = √(shadow² + 144).
(ii) As already seen, Sūryāgrā = 12 × 120 × sin lat ÷ (shadow-hypotenuse × sin colat).
(iii) ‘The sine’ = the distance in aṅgulas from the east-west line to the tip of the shadow × 120′
÷ shadow-hypotenuse. (If the shadow is north of the east-west line, ‘the sine’ is north, if the shadow
is south of the east-west line, ‘the sine’ is south).
(iv) (a) If ‘the sine’ is north, and greater than the Sūryāgrā, Agrā = ‘the sine’ − Sūryagrā, and the Sun
is in the southern hemisphere.
(b) If ‘the sine’ is north and less than the Sūryāgrā, Agrā = Sūryāgrā − ‘the sine’, and the Sun
is in the northern hemisphere.
(c) If ‘the sine’ is south, Agrā = Sūryāgrā + ‘the sine’, and the Sun is in the northern hemis-
phere.
(v) Sine longitude of the Sun = Agrā × sin colat ÷ 48′ 48″ = Agrā × sin colat × 5 ÷ 244.
(vi) From the sine of longitude, the Bhuja degrees D, and using that the longitude of the Sun by
examination, are to be obtained.
As in the previous work, (iii), (iv) and (v) can be simplified thus: Sine sun’s longitude = (12 ×sin
lat ± sin colat × the distance in aṅgulas between the tip of the shadow and the east-west line) × 150
÷ (61 × shadow hypotenuse). In ± if the shadow is south of the east-west line then the upper sign
is to be taken, and the Sun then is in the northern hemisphere. If the shadow is north, the lower
sign is to be taken. In this case, if 12 × sin lat is greater, the Sun is in the northern hemisphere, and
if sin colat × distance in aṅgulas, is greater, the sun is in the southern hemisphere.
Example 25. Of a certain place, sin lat = 60′, sin colat = 103′ 55″. There, on a day during Uttarāyaṇā,
when the length of the shadow is 5 aṅgulas, the distance of the shadow tip from the east-west line is measured to
be aṅg. 1-36.6, north of the east-west line. Find the longitude of the Sun.
(i) Shadow-hypotenuse = √(5² + 144) = 13 aṅg.
55a. A1. ॰न्वे तितेष्या; A2. ॰न्वे तिरतेष्या; A.D. सा ज्या
D. ॰न्वेज्जारितैष्या b. A. काष्टा. A. मनोर्कः; D. हतार्कः
d. A. सूर्याग्रा. A2. न्यवे c. A2. सूर्यो-वेन
56a. A. लम्बगुणिता d. A. ग्रहक्षतो