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

112 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