ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

SIDEREAL TIME-INTERVALS 131 circle of a heavenly body with declination δ, NEHW the horizon, PEP′ W the six O′ clock circle. Here AA′ the line of intersection of the diurnal circle with the horizon is called the “udayāsta-sūtra”¹ (or the thread joining the rising and setting points). SS′ the line of intersection of the diurnal circle and the six o′ clock circle, is the horizontal diameter of the diurnal circle. From S draw SK and SL perpendiculars respectively to AA′ and EW. Join KL. Now since PN=ϕ, the latitude of the station, in the small right-angled triangle KLS, the ∠ KLS is also=ϕ. ∴ SK : SL=QM : MO or SK = (SL × QM) / MO = (R sin δ × R sin ϕ) / (R cos ϕ) Now SK² is a “sine” in the small circle AB A′B′ of which the radius is R cos δ; this “sine” reduced to the equator (radius R) is the ‘sine’ of cara. ∴ R sin ch = R sin EPA = (R sin δ × R sin ϕ × R) / (R cos ϕ × cos δ) Greek Method Let³ the arc PA be produced to meet the equator at C. Take PCQ′ for the triangle and EAN for the transversal. Then we get, (sin PA / sin AC) × (sin CE / sin EQ′) × (sin Q′N / sin NP) = 1 or (cos δ / sin δ) × (sin CE / 1) × (cos ϕ / sin ϕ) = 1 ∴ sin CE = sin ch = (sin ϕ × sin δ) / (cos ϕ × cos δ) Note—The perpendicular distance between AA′ and Ew is called the ‘sine’ of the amplitude or the ‘Agrā’ which is thus calculated :— KL : LS = QO : OM ∴ ⁴R sin amplitude = ‘Agrā’ = KL = (LS × QO) / OM = (R sin δ × R) / (R cos ϕ) It is now evident that the ancient Indian method is different ────────────────────────────────────────

1. Bhāskara, Gola, VII, 39.
2. This is called by the name ‘kujyā’ or ‘kṣitijyā’. i, e.. earth-sine. Āryabhaṭa, Gola, 26, Brahmagupta, II, 57, Sūrya-siddhānta. II, 61 etc.
3. Manitius, ibid, p. 84.
4. Āryabhaṭa, Gola. 30, etc.