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4. DERIVATIONS AND RATIONALES
For the area of closed round figure a general ancient prototype rule was Area = (perimeter).(width)/4
............. (22) This was usually used for a circle for which it gives exact result. Surprisingly it is also true for a square. Mahavira used it for āyatavrtta (elongated circle or ellipse) in his GSS, (VII, 21 and 63, pp. 185 and 196). 11
KKHN
C2h
FIG. 3
FIG. 4 If we apply (22) to the figure (see FIG. 3) formed by double segment, we will get
2A = (28).(2h)/4 A = (s.h)/2
............ (23) Directly as such, this rule (23) for the area of a circular segment (FIG. 2) is found in Karvinda's commentary on the Apastamba Sulbasūtra. 12 Now the following ancient empirical relation has been found S = c + h ...
............ (24) Putting this in (23), we get the practical and popular classical formula (13)
The author of the present paper has discovered the presence of the simple rule (24) in an old Babylonian text (BM 85194). 13 Mahavira used this simple method for finding the vyāvahārika (approximate) perimeter of an ellipse (GSS, VII, 21), but not for its sūksma (accurate) perimeter (GSS, VII, 63). For arc of a circular segment his rules were different namely (9) and (10).
On the other hand Narayana (1356 A.D.) in his Ganita Kaumudi (IV, 12) used (24) for circular segment which was not greater than a semi-circle." His example on yavākāra-ksetra (barley-shaped figure)
Arhat Vacana, 14(1), 2002
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