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

887 Note. A small circle parallel to the prime-vertical is called Upa-Vṛtta. Also secondaries drawn to the Ecliptic, Equator and the prime-vertical are called Kadamba-Sūtra, Dhruva-Sūtra and Sama-Sūtra. They are also called occasionally as Kadamba-prota-Vṛtta, Dhruva- prota-Vṛtta, and Sama-prota-Vṛtta. Bhāskara proceeds to find the Āyana Valana in a very ingenious way in verses 69-74. We shall first give it a modern treatment so that we may better appreciate his genius. Let (S) be the Sun's disc. (It does not matter whether we take the Sun or the Moon). EQ is its diameter along the diurnal circle, and CL along the Ecliptic. LM is the difference of the declination of L and S. Let SL = Δ λ and LM = Δ δ. We have sin δ = sin λ sin ω ; differentiating cos δ Δ δ = sin ω cos λ Δ λ ∴ Δ δ = (sin ω cos λ / cos δ) × Δ λ ; put Δ λ = b, the angular radius of the disc ∴ Δ δ = (b × sin ω cos λ / cos δ) = (b H sin ω × H cos λ / R × H cos δ) . This gives the Valana in the disc of radius b. If that be so, what will it be on the sphere of radius R? The result is (b H sin ω × H cos λ / R × H cos δ) × R / b = (H sin ω × H cos λ / H cos δ) as got before. Let us hear Bhāskara, " put the disc of the Sun at the point of intersection of the Ecliptic and the diurnal circle. The Valana (LM of fig. 84) at the periphery of the disc is the difference of the declinations of L and S. To get the value of this let us first get the value. SL in terms of λ, the longitude of S. It is (b × B / 225) ; so that LM