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

350 The procedure originally called for a rectification is that taking K to be R, we have to compute r and again taking the resulting K to be R, we have to compute r and so on repeating the process till an invariable value for K is obtained. This means that we should go on substituting for r, r K / R . Instead of following this laborious process of ' Asakṛt-Karma ' ie. method of successive approxi- mations, Bhāskara gives an alternative in the verse, whieh is as follows. Let K be the value of the Karṇa, for a value r of r. Since we are directed to make this K as R, ie. we have to add R—K to K thus making it R, we add also R—K to R, to keep the relative position of R and K to be almost the same. In other words considering the fraction K / R , adding R—K to both the numerator and denominator we have R / (2 R—K) whieh means that for a radius 2 R—K, the Karṇa will be R; that being so for a Radius R what will be the Karṇa ? The result is R² / (2 R—K) as given. It will be noted that the above interpolative procedure is adopted as a short cut technique to the otherwise laborious process. The mathematical correctness of this procedure will be seen from the following analysis. The problem is to change the Mandaparidhi to a radius K of the deferent by the formula (as indicated by Bhāskara in the course of the commentary) r' = r K / R so that δr = r¹ — r = r K / R — r = r (K—R) / R (1) Now we have K² = R² + r³ + 2 R r cos m construing R and m as constants we have to find δ K for δ r