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

IV. 39 IV. THREE PROBLEMS 113 [शङ्कुच्छाया सममण्डलं च] कृतदि(ग्ग्र)हणे वृत्ते रेखां पूर्वापरां यदा छाया । प्रविशति सम्यक्छङ्कोस्सममण्डलगस्तदा सूर्यः ॥ ३८ ॥ Gnomonic shadow and the prime vertical 38. On a circle with the east-west line drawn, and the directions marked, (according to IV.19), the time when the gnomonic shadow perfectly coin- cides with the east-west line is the time of the Sun crossing the prime vertical. The idea is that if this time is found by measuring instruments, compared with the computed time and the agreement shown, people will acquire faith in the method. It can be shown that when the Sun is on the prime vertical, the gnomonic shadow must be along the east-west line. The prime vertical is the vertical great circle of the sky-sphere, passing through the east-west points and the zenith, and therefore the east-west line forms the intersection of this vertical plane and the plane of the horizon which is horizontal. As the gnomon standing vertical and also the Sun on the prime vertical lie in the vertical plane, the shadow (intercepted by the hori- zontal plane) must also lie on the vertical plane, and therefore must fall on the east-west line, which is the intersection of the two planes. [अग्रा-दिग्ज्या] इष्टक्रान्तिज्या(घ्न)व्यासशकललम्बकांशमुष्णांशुः । समपूर्वापररेखामतीत्य यात्यस्तमुदयं वा ॥ ३९ ॥ Agrā : Sine amplitude 39. Multiply the Sun's declination by the radius and divide by the sine of co- latitude, and find the sine (of the amplitude of the rising or setting point, called Agrā). At a point distant by this amount from the east-west line (accord- ing to the declination, north or south) the Sun rises or sets. Agrā, (i.e. sine amplitude) = sin dec × radius ÷ sin colat. Find the arc of this sine. By an angle equal to this from the east to west point does the Sun rise or set on the horizon. Example 14. The latitude of a place is 60°. The longitude of the Sun is rāśi 4-0. Find the direction of rising or setting of the Sun. First, sin declination is to be found. As the Sun is in the second quadrant, Sin rāśi 4-0 = Sin rāśi 2-0 = 103' 55". Sin dec = 103' 55" × 61/150 = 42' 15", and this declination is north. Sin amplitude = 42' 15" × 120' ÷ sin (90° - 60°) = 42' 15" × 120' ÷ 60' = 84' 30".

38. Quoted by Utpala on BS 2, p.41. 38a. A. कृतिदिग्रहणे c. A. शङ्कुः 39a. A. ज्याघ्ना; D. ज्याघ्नं b. A. व्यासकल; D. व्यासशक (लं) लम्बभक्तमुष्णांशुः d. A1. मतीत्या; A2. मलिप्त corrected to मलीप्त ।