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वीर ज्ञानोदय ग्रंथमाला
JAINA COSMOGRAPHY AND PERFECT NUMBERS
By R.C. GUPTA, M.Sc. (Double Gold Medalist) Ph.D. (Hist. of Math), Hon. Doct. (Hist. Sci.), F.N.A.Sc., (Member, International Commission of History of Mathematics)
Professor of Mathematics, Birla Institute of Technology P.O. Mesra, RANCHI-835 215
According to Jaina cosmography, Jambūdvīpa ("Jambū Island") is situated in the Madhyaloka. It is circular in shape and has a diameter D equal to one lakh yojanas. It is surrounded by an unenumerable number of concentric circular rings of sea and land alternately (see Fig. 1).
Acara Islan
ka se
bak
16D
FIG. 1 (nor to scale) The width of the nth ring (whether sea or land) is given by W. = 21 D
..... (1) Thus the width (valaya-vyāsa or vistāra) of the Lavaņa Sea (n = 1) will be 2 lakh yojanas, that of the Dhātaki Land will be 4 lakh yojanas, etc. The width of the Jambu Island, should be taken as D, and may be denoted by Wo.
The diameters of the outer boundaries of the various rings will be as follows (see FIG. 2): For Lavana Sea, D = D + 2 W
= D + 4D = 5D. For Dhătaki Land, D2 = D + 2 W. + 2W2
= 5D + 8D = 13D. For Kálodaka Sea,
D3 = D + 2W + 2W2 + 2W
= 13D + 16D = 29D and so on
0010101010
FIG. 2 (net to scale) In general the diameter of the outer boundary suci-vyāsa or sūcī) of the nth ring, whether sea or land will be given by D. = D + 2 (W; + W2 + .............. + W)
= D + 2 (2 + 22 + 23 + .......... + 2n)D
D + 2 (2n+1 - 2) D = 2n+2 – 3) D = 4W. - 3D
.. . (3)
"
(2)
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