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

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षष्ठात्पञ्चदशादपि, सप्तमात् द्वादशात्सप्तदशाज्ञाधोत्तरं मतम् ” quoted
from the Brahma Siddhānta by Ranganātha in his com-
mentary of Sūryasiddhānta were alluded to in the articles
cited but no satisfactory mathematical explanations were
given by them. We shall give hereunder a satisfactory
explanation of the matter discussed in the articles.
In the first place it may be noted that in the table of
those 24 H sines, the sixteenth as given by Bhāskara
namely 2977 is more correct than that given in the Sūrya-
siddhānta namely 2978 (Lakshmi Venkateswara press edi-
tion 1955 Bombay).
In the course of the Commentary under the verses 15,
16 of the Sūryasiddhānta, Ranganātha gives the hint which
must have been at the back of the mind of the author of
the Sūryasiddhānta when he gave the rule to construct the
table cited. Just as δ (H Sin θ) = (H Cos θδθ) / R , Similarly
the formula δ (H Cos θ) = - (H Sin θ δ θ) / R must have
been known to the author. The negative sign means that
the successive differences of the H sines namely 225, 224,
222, 219 etc. are decreasing and also that the successive
differences of these differences are increasing according to
the H sine. Just as Bhāskara could see that the H sines
were increasing and the successive differences of the
H sines were in Kotijyānupāta i.e. in direct ratio to the
H Cosine at their respective place, similarly, the author of
the Sūryasiddhānta could see that the second differences
cited above were in Kramajyānupāta as hinted by Ranga-
nātha.
From the formula δ ((H cos θ δ θ) / R) = - (H sin θ δ θ²) / R²
putting θ = 90°, we have the second difference numerically