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

134 GREEK AND INDIAN METHODS
¹Problem IV :—
The Altazimuth Equation
Indiad Method
Let α denote the azimuth of the Sun from the south. In
the same triangle SKL in the same figure, we have,
LK : SL = R sin ϕ : R cos ϕ
or, ‘Śaṅkutala’ : ‘Śaṅku’ = R sin ϕ : R cos ϕ
∴ ‘Śaṅkutala’ = (R cos Z × R sin ϕ) / (R cos ϕ)
Now ‘Śaṅkutala’ is made up of two parts, namely, ‘Bāhu’
and ‘Agrā’, of which the former is the distance of L from the
observer’s East-West line; the ‘Agrā’ has been already found.
Here ‘Bāhu’ = (R sin Z × R cos α) / R and ‘Agrā’ = (R sin δ × R) / (R cos ϕ)
∴ ‘Śaṅkutala’ = ‘Bāhu’ + ‘Agrā’
or (R cos Z × R sin ϕ) / (R cos ϕ) = (R sin Z × R cos α) / R + (R sin δ × R) / (R cos ϕ)
or R sinδ = (R cos ϕ / R) ((R cos Z × R sin ϕ / R cos ϕ) - (R sin Z × R cos α / ϕ))
which is easily seen to be equivalent to
sin δ = cos Z sin ϕ - sin Z cos ϕ, cos α
Greek Method
Ptolemy² has also a method of finding the Sun’s altitude at
any hour of the day. His method is as follows :—
(i) He would find by
means of his tables for the
times of risings of the signs of
the zodiac, the orient ecliptic
point. (ii) He would then
find the culminating point of
the ecliptic. (iii) He would
finally apply Menelaus’s
theorem in spherics thus :—
Fig. 9
Let ASC be any position
of the ecliptic, (Fig. 9) NZC the

1. The equivalent of this, in a particular case, is first found in Brāhmasphuṭasiddhānta, Ch. III, 54-56 Cf. Sūryasiddhānta, III, 28-31, also Bhāskara Grahaganita, IX, 50-52.
2. Manitius, ibid, pp. 118, 19.