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

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Note (1) When Bhāskara said 'If H sin a < Agrā'
he had in mind evidently the azimuth circle MZN which
cuts the diurnal path A Q₁ R₁ at S₁ and S₂. At S₁ the
azimuth EZM < EZA so that he stipulated that H sin a
should be less than the Agrā. But, let the diurnal path of
the Sun be Q₂ R₂ where Q₂ falls in between z and p. In
such a case we know that the azimuth does not take all
values but has a maximum where the vertical touches the
diurnal path at T. From PTZ where T is a right angle,
taking PZT = 90 - a, a being the Hindu azimuth we have
by Napier’s rule sin PT = sin ZP sin (90 - a) or
sin (90 - δ) = sin (90 - φ) sin 90 - a or cos δ = cos φ cos a.
If a has a lesser value than is given by this equation,
the diurnal path does not cut the azimuth circle
ie. if cos a > cos δ / cos φ, there will be no shadow in the given
direction even though the situation satisfies Bhāskara’s
condition namely H sin a should be less than Agrā.
Bhāskara has overlooked this case. This may be seen
analytically also as follows. We have from the spherical
triangle PZS, sin δ = sin φ cos z + cos φ sin z sin a
= A cos z + B sin z (say). We know, the maximum
value of A cos z + B sin z is √(A² + B²) which is here
√(sin² φ + cos² φ sin² a) = √(sin² φ + cos² φ - cos² φ cos² a)
= √(1 - cos² φ cos² a). Thus there will be no solution for
z if the quantity on the left hand side namely
sin δ > the above max. value ie. if
sin δ > √(1 - cos² φ cos² a) ie. if sin² δ > 1 - cos² φ cos² a
ie. if cos² δ > cos² φ cos² a ie. if cos δ > cos φ cos a
ie. if cos a > cos δ/cos p as derived above.
Hence even if H sin a > Agrā, there need not be a
shadow at all in the given direction. In other words when
the Hindu azimuth given is very small and when the decli-
nation is too great north or south, there may not be a
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