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

117 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) =