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

116 This is a crude approximation, the actual value being 48' — 48" — 14"'. Application of rule of three is justified if the H sine—differences are uniform, but they are not so, being in a decreasing order. So, the following H sine— difference namely 19' is to be rectified so as to be applicable at 24°. In other words we have to take such a H sine— difference which will hold good at 24°, not at 20° or 30° — from 10° to 20°, the H. S. d. (H Sine—difference) is 20', and from 20° to 30° it is 19'. If that be so what will be exactly at 24°? It should be less than 20' and greater than 19'. Now the argument advanced by Bhāskara is that the H Sine—difference at the mid-point of the 2nd and 3rd differences namely 20 and 19 should be (20 + 19)/2 = 19·5. The H sine—difference at the end of the third interval is 19. Then by the rule of three ‘If there is a decrease of 19·5 — 19 = ·5, during the course of the 10° of the third interval, what should the difference be for 4°? (where we have to find the H sine at 24°) The result is 4/10 × ·5 = (4 × 1)/20 which is given by the words “यातैष्ययोः खण्डकयोर्विशेषः शेषांश निघ्नो नखहृत् ”. This decrease makes the H sine— difference at 24°, 19·5 — 4/20 = 19·3 at 24°—Now the argu- ment to find the H sine at 24° is ‘If for an interval of 10°, the H sine—difference is 19·3, what should it be for 4°?’ The answer is 4/10 × 19·3 = 7·72. Hence the H sine of 24° is 21 + 20 + 7·72 = 48·72 = 48' – 43" – 12"' which is nearer the truth 48' – 48" – 14"' than what was obtained by the crude rule of three namely 48' – 36". Here we have to explain Bhaskara’s words more elabo- rately, because, Kamalākara happened to criticise Bhas- kara’s words in this context. Bhaskara Says “The H sine difference at the end of an interval is the arithmetic mean of the preceding and succeeding differences, whereas the succeeding one is that which holds good at end of the suc- ceeding. In between, we have to apply the rule of three to