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
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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.