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

140 GREEK AND INDIAN METHODS point A. Then he subtracts 90° from the longitude of A. Thus having the longitude of N', he next finds the part of the day elapsed of N' ; from which by the time-altitude equation discussed above, he finds ZN'. This is of course more accurate than that of Āryabhaṭa. Bhāskara¹ here follows Brahmagupta. Greek Method : Let the ecliptic CN'A cut the lower half of the meridian at F. Ptolemy takes AK along the ecliptic=90° and AR along the horizon=90°; then the great circle passing through R and K passes through the nadir Z'. Now take Z'FK for the triangle and ANR for the transversal, then by Menelaus's theorem.² (sin FN / sin NZ') × (sin Z'R / sin RK) × (sin KA / sin AF) = 1 ∴ sin RK = (sin FN / sin AF) = (cos FZ' / sin AC) = (cos CZ / sin AC) = (sin CH / sin AC) or sin MN' = sin CH / sin AC. Here Ptolemy's equation is simpler than that of Āryabhaṭa; hence they must be independent of each other. [Figure 12] Problem VII:— To find the Angle made by the Vertical through any Point of the Ecliptic with the Latter This problem is considered by Ptolemy but it is not consider- ed separately in Indian Astronomy, but from the rule for parallax in longitude, the rule for its calcula- Fig. 12 tion can be deduced. Indian Method : In Fig. 12 S represents the true position of the Sun and S' the Sun's position as depressed by parallax. N'SA is the ecliptic. If from S', S'Q be drawn perpendicular to the ecli- ptic, then, if P is the horizontal parallax,

1. Grahaganita; XII, 3-4.
2. Manitius, ibid, pp. 110-111.