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

273 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 35