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

128 GREEK AND HINDU METHODS *R sin R.A. = (R sin l × R sin ω) / (R cos δ) , where R is the radius of the sphere. Note:— If R be the radius of the circle of reference, the Indian trigonometrical functions for the arc θ, are (1) the 'sine,' (2) the 'cosine' and (3) the versed sine. They are respectively equal to R sin θ, R cos θ and R vers θ. In the adjoining figure, O is the centre of the armillary sphere, YQ, YC are quadrants of the equator and the ecliptic, respectively. P is the celestial pole, PCQ the summer solsti- tial colure. Join OY, CQ, OP and OC. Let YS be=l, YM=R.A., CQ=∠SYM=ω, SM=δ. Join OS, OM. PSM is the secondary to the equator. [Fig. 6] From C draw CK perpendicular to OQ. From S draw Sm and Sn perpendicular to OM and OY, respectively. Join MN and from M draw MN perpendicular to OY. Then the triangles Smn and CKO are similar. They are called 'Krānti-kṣetras'¹ or declination triangles,—similar right- angled triangles having one acute angle=ω. ∴ Sm : Sn = CK : OC or R sin δ : R sin l = R sin ω : R R sin δ = (R sin l × R sin ω) / R ......(I)

---

The Āryabhaṭīya, Gola, 25. Varāhamihira, in the Pañcasiddhāntikā (IV. 92) states it in the form 2R √((R² Sin ²l)—R² Sin ²δ) / (2R cos δ) = R Sin R.A., which is evident from figure. Brahmagupta's equation is identical with that of Āryabhaṭa, (BrSpSi. III. 15, Sūrya-siddhānta III. 40-41. Also Bhāskara II, Grahagaṇita cap. VIII, stanzas 54-55, is in agreement with Varāhamihira's forms.