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

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obtain the rectified difference." What Bhāskara means is
this. At the end of an interval, to obtain the H sine, it is
enough to add the H sine difference belonging to that inter-
val to the preceding differences. But when it is required
to find the H sine in the interior of an interval, we have to
construe that the difference at the end of the previous inter-
val is the arithmetic mean of the previous and succeeding
differences. There is apparently a self-contradiction in
Bhāskara's words; for, at the end of the interval, according
to his own words, the difference is that belonging to the
previous interval and not the arithmetic mean as postula-
ted. The contradiction will not be there when we read
Bhaskara's mind that he means "When we require to find
the H sine in the interior of an interval only, the difference
at the beginning of that interval is to be taken as the arith-
metic mean of the previous and the current differences, and
that at the end of the interval the current difference holds
good."
The truth of Bhāskara's statement could be seen
analytically as follows. The arithmetic mean of the prece-
ding and succeeding H sine differences is (The context is to
rectify the third H sine difference namely 19, for, we were
finding the H sine of 24°) (AB + BC) / 2 (Ref fig. 8) where OA,
AB, BC etc are the successive differences.
(AB + BC) / 2 = (H sin 20 - H sin 10 + H sin 30 - H S 20) / 2
= (H sin 30 - H S 10) / 2 = R ((Sin 30 - S 10) / 2)
= R × Cos 20 Sin 10 = (H Cos 20 H Sin 10) / R
The numerical difference of the preceding and succeed-
ing H sine differences is (i.e. यातैष्यखण्डविशेष). AB - BC
= (H sin 20 - H S 10) - (H sin 30 - H sin 20) =