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.