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

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tude of Taddhṛti namely (R H sin δ) / (Sin ϕ cos ϕ) only H sin δ is
variable and Taddhṛti is directly proportional to H sin δ.
But in the present example both H sin δ and H sin ϕ are
both variables so that, that kind of rule of three does not
work.
Verse 89. Oh! Mathematician! At a place where
s = 5'', there 10 nādikas after Sun-rise the shadow S is
observed to be 9''. Tell me what the longitude of the Sun
would be, if you are an adept in computing as well as
understanding the geometry of the sphere.
Verses 90, 91. Answer to the question posed.
Assume H sine (Unnatakāla) to be Iṣṭāntyakā. Then
(K × H cos z × R) / (12 × I. A.) = H cos δ where I. A. = Iṣṭāntyakā.
R² − H cos² δ = H sin² δ ; from this approximate H sin δ
and the given s compute a more approximate I. A. Repeat
the process till an invariable quantity is obtained for
H sin δ, which will be its correct value. From this, using
the formula H sin δ = (H sin λ H sin ω) / R , λ could be had.
Comm. We know the formula for I. A. as
(R³ H cos z) / (H cos φ H cos δ) Assuming Unnatakālajyā as I. A.
H sin (Unnatakāla) = (R³ H cos z) / (H cos φ H cos δ)
∴ H cos δ = (R³ H cos z) / ((12 R. I.A.) / k) = (k × R × H cos z) / (12 × I. A.) ; from
which obtaining H sin δ and proceeding as indicated we
have λ. In the above proof we have used our formula.
But Hindu Astronomers proceed from first principles. Let
us hear Bhāskara. Since S = 9''; K = √(9² + 12²) = 15''