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

301
Bhāskara’s proof of I.A. = kR² / (H cos δ × I.K.) proceeds
from first principles as follows :—(i) If by k we have 12 as
the Koṭi what have we for R? The result is H cos z,
Mahāśaṅku. ∴ Mahā Śaṅku = 12 R / I.K. From Mahā Śaṅku
we pass on to Iṣṭa Hṛti with which it forms a latitudinal
triangle. If by the gnomon of 12 units we have k the
Viṣuvat Karṇa, what have we by Mahā Śaṅku? The
result is (12 R / I.K.) × (k / 12) = kR / I.K. Again from the Iṣṭa-Hṛti we
pass on to I.A. by multiplying by R / (H cos δ) so that I.A. =
kR² / (H cos δ × I.K.) as given.
Second half of verse 68. The inverse Hversine if a
quantity x greater than R, is 5400 + H sin ⁻¹ (θ) when
x - R = θ.
Comm. Since Hvers (90 + θ) = R + H sin θ = x (say)
90 + θ = Hvers ⁻¹ (R + H sin θ) =
But 90° are equal to 5400 asus and θ = H sin ⁻¹ (H sin θ)
= H sin ⁻¹ (x - R) = त्रिज्याधिकभागस्यक्रमचापम्
∴ 5400 + त्रिज्याधिकभागक्रमचापम् = Utkrama Cāpa of a
त्रिज्यादिक quantity.
Verse 69. Alternate method of obtaining the time
that has elapsed after Sunrise noting the shadow S.
Subtract Charajyā from or add it to Iṣṭāntyā according
as δ is north or south. Obtain inverse H sine of the
remainder and add the Caracāpa to this inverse H sine.
Then we have the unnatakāla by converting the result into
time.