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On the basis of the Fig. 1a, Hayashi has developed a Fig. 1b -
em
Fig. 1b : A conch - like plane figure developed by Hayashi
Let the diameters of the two semi circles, AB and BC, be 'd' and (d-m) respectively, then the sum of their circumferences (except for the diameters AB and BC) and that of their areas are the perimeter 'P' and area 'A' of the conch-like plane figure (cf. Fig. 1a and 1b).
P= d+p/d-m)
[1a]
[2a]
Mahavira gives two types of formulae for the perimeter and area of a conch-like plane figure, one for gross (sthula) or practical (vyāvahārika) results and the other for subtle (sūksma) ones. For gross results :
वदना?नो व्यासस्त्रिगुणः परिधिस्तु कम्बुकावृत्ते। वलयाकृतित्र्यंशो मुखार्धवर्गत्रिपादयुतः ।।
(GSS v. 7.23, p. 437) In the case of a conch-circle (kambukāurtta), the diameter (vyāsa) 'd' diminished by half the face (vadana) 'm' and then multiplied by three gives the perimeter (paridhi) 'P'. One-third of the square of half (this) perimeter (valaya), added by three-fourth of the square of half the face (mukha), (gives the area 'A').
32
Arhat Vacana, 14(1), 2002
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