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

544 In this example the declinations of the Sun and Moon are respectively 1416′, 1324′, when their tropical longitudes are 80° and 100°. The Moon is in an even quadrant, and his declination is less than that of the Sun. As per the criterion of Lalla given in the verse “सूर्यापमात् etc.” the pāta is not there at all. But Bhāskara computes and shows that the pāta has occured 70 nādīs before (Rāhu having a reverse tropical longitude of 100°). We shall show here, how using differential calculus we could cut short the process. We have sin δ = sin λ sin ω ; differentiating cos δ Δδ = cos λ δλ sin ω or Δδ = (sin ω cos λ / cos δ) Δλ = Āyanavalana × Variation in Bhuja. This could be seen graphically also from fig. 129. Let P and Q be two contiguous positions of the Sun (say) on the ecliptic when arc pQ = Δλ. Let the increment in his Fig. 129 declination in going from P to Q be NQ equal to Δδ. Taking PNQ to be a plane Δ Δδ = Δλ sin v where v̂ = QP̂N. We have named this angle to be v, because it is no other than the Āyanavalana which was formulated to be equal to (sin ω cos λ / cos δ) . Thus Δδ = Δλ × Āyana- valanajyā.