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

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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.