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JANUARY, 1979
93
bānajudarumdavagge rumdakadi sodhidūņa duguņakado / jam laddham tam hodi hu karaņicāvassa parimāņam || 181 //
'From the square of the sum of the bāņa (height of the segment) and diameter, subtract the square of diameter and multiply by two. The square-root of the result is a measure of the arc (of the segment).'
If we take the approximation 22/7 for ī, then (8) will give k equal to 288/49 and in this case (7) will become s =Vc2+(288/49)h2
... (13)
A rule giving this formula is found in MS, XV, 94 (p. 173) as an accurate method for finding the circular arc. Thus we see that the authoof the MS, a non-Jaina work, tried to improve the formula (10) by eml ploying a better value of 7. However, the basic principle behind alr these formulas is same as stated above to derive (8).
Since the formulas (10) to (13) were derived by using analogous and empirical comparision with a semi-circle, it will not be fair to check their accuracy for small arcs as suggested sometimes. (e.g. see JPS, Introduction, p. 53).
For accurate rectification of small circular arcs the following formula is found in the works of Nilakantha Somayaji (c.1500)1) s =Vc2+(16/3)h 2
... (14)
However, this was based on a different principle and is the best formula of the type (7) for small arcs, because from
20r = V(2r sin 6)2+-k(r-r cos 6)2 we get
k =4(09-sin20)/(1-сos 0)2 which tends to 16/3 as 0 tends to 0.
The formula (10) seems to be used by Mahaviracarya even to find the (accurate) perimeter of an ellipse. In this connection his GSS, VII, 63 (p. 196) states
vyāsakrtișadguṇita dvi samgunāyāma krtiyutā (padam) paridhih
10 See R. C. Gupta, 'Neelakantha's Rectification Formula' (GAIM No. 1). The
Mathematics Education, Vol. VI, No. 1 (March, 1972), Sec. B, pp. 1-2.
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