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गणिनी आर्यिकारत्न श्री ज्ञानमती अभिवन्दन ग्रंथ
६७१
by using (1).
Actually three types of Suci have been defined for any ring: (i) Adima or Adya Sūci which is the diameter of the inner boundary of the ring. (ii) Madhyama Suci which is the diameter of the central circle that lies exactly in the middle of the inner
and outer boundaries of the ring. (iii) Bāhya Súci which is the diameter of the outer boundary of any ring. This is often called Suci only and has been denoted above by D.
Let the Adima Suci be denoted by I, and the Madhyama Suci by C, fo the nth ring. Then we will have
In = D + 2W, + 2W2 + ........ + 2W.-1
= 2n+1 - 3) D = 2 W. - 3D
.. ..(5)
by following a process similar to (2) and (3).
Also we have .CD + 2W, + 2W2 + ........ + 2W.-1 + W.
= I. + W. = (21+1 – 3) D + 2"D, by (1) and (4) = 3(21 - 1) D
(6) = 3W, - 3D
The Tiloyapannatti, Chapter IV, gätha 2601 (vol. II, p. 693) as well as chapter V, gäthä 34 (Vol. III, p. 9), states the formulas (3), (5) and (7) in a remarkably concise way as (taking D = 1 lakh) In, C, D = (2, 3, 4) W.-3 lakh yojanas
.. (8) It should be noted that for any ring the madhyama diameter is, as the name implies, the mean of its inner and outer diameters. That is, Co = (1, + D.)/2
.. (9) Now the area of the nth ring will be given by A (1/4). D? - 13)
= (1/4). (D, + 1) (D. - In) = (1/4). (6 W. - 6D) 2 W,, by (3) and (5) = 3 7 (W. - D) W
.. (10)
For quick calculation of the area approximately the value of 77 was often taken to be 3 in ancient times. With this value and D = 1, the area will be given by A, = 9 (W. - 1) W, units
.. (11) In fact this elegantly simple form of the formula is found stated in the Tiloyapannatti, chapter V, gāthā 244, in the following words (vol. III, P. 56):
लक्खेणूणं रुंदं णवहि गुणं इच्छियस्स आयामो।
तं रुंदेण य गुणिदं, खेत्तफलं दीव-उवहीणं ॥२४४ ॥ 'The width as diminished by a lakh (yojana) and then multiplied by nine, is the ayāma (effective length) of a desired ring. That (āyāma) multiplied by the width is the area of the Island or Sea.'
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