सिद्धान्तशिरोमणि: गणिताध्याय (भास्कराचार्य - ग्रहगणित, मध्यमाधिकार व स्पष्टाधिकार सटीक)

250 We saw before that Agrā = Śaṅku-tala + Śaṅku- bhuja. So, in the first place as a first approximation, Agrā is taken as Śaṅku-bhuja. Since, when z = 45° the perpendiculars from the celestial body on the planes of the prime-vertical as well as meridian are equal, and since the perpendicular on the plane of the prime-vertical is called Śaṅku-bhuja = b (say) 2b² = H sin² z. This is so because H sin² z = Sum of the squares of the perpendiculars on the planes of the meridian and prime-vertical. But H sin² z = R² − H ocs² z. ∴ R² − 2b'² = H cos² z. So, taking Agrā as the bhuja b as a first approximation, √R² − 2b² gives us H cos z. From this using formula III under latitudinal triangles namely H cos z × s/12 = Śaṅku-tala, obtain the approximate Śaṅku-tala, from the approximate H cos z got above. Now using the formula Agrā = Śaṅku-tala + Śaṅku-bhuja obtain Śaṅku-bhuja as Agrā ⩲ Śaṅku-tala, where the +ve sign is taken when the Sun has a southern declination, and the difference sign when the declination is north. Taking this Śaṅku-bhuja, b', Koṇa-Śaṅku is now √R² − 2b'². In the first place we took the Agrā itself as the bhuja; but here we have a better approximation for the Śaṅku-bhuja. From this Koṇa-Śaṅku again, obtain a still better approximation for Śaṅku-bhuja and proceeding thus till a constant value is obtained, we have the required Koṇa-Śaṅku. This is a beautiful example where the method of successive approxi- mation was used by the Hindu Astronomers to a good advantage. It will be noted here that the Śaṅku-tala is always treated as extending south ie. the Śaṅku-tala will be south of the Śaṅku since India's latitudes are all north. Also it is said that when the Sun has southern declination, A + S = B and when northern A − S = B when A = Śaṅku-Agrā or simply Agrā (in contradistinction to Karṇāgrā reduced to a circle of radius K the chayākarṇa) S = Śaṅku-tala and B = Śaṅku-bhuja. This convention