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

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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