Basic Mathematics

Author(s): L C Jain
Publisher: Rajasthan Prakrit Bharti Sansthan Jaipur

Page 46

Actually, this number, occupying twenty-nine notational places is 79, 22, 81, 62, 51, 42, 64, 33, 75, 93, 54, 39, 50, 336, based on the calculated area of inhabitants of the world human beings, showing inconsistency of other views, as shown by the author.
The theory of indices as described in the Dhavalā is to some extent different from other mathematical works. This theory appears to be primitive
in use before + 500. The basic ideas seem to be those of (i) the square, (ii) the cube, (iii) the successive square, (iv) the successive cube, (v) the raising of a number to its own power, (vi) the square root, (vii) the cube root, (viii) the successive square root, (ix) the successive cube root, etc.
For example, the third vargita samvargita of 2 may be found to be (256)256. The operations of such duplation and mediation were recognized by the Egyptians and the Greeks but there is no trace of them in the Indian mathematical works :
2 2
2 2
2 2
(c) Logarithms :
Singh66 recognized the Ardhaccheda of a number as logarithm to the base two. Thus (Ardhaccheda of 291 ) = m. Similarly he recognized the function of a function in finding Vargaśalākā of Dhavalā as logarithm to the base two of the logarithm of a number to the base two. Thus Vargaśalākā of 22" could be noted to be m. Similarly Trakaccheda of a number was found to be equal to logarithm to the base three, and so on.67
The following table gives the results known to the author of the Dhavalā.68
66. Cf. Mathematics of Dhavalā, op. cit. In this connection. Kapadia,
H. R., appears to have recognized already that if x=2", then n is called the ardhaccheda of x. He reminds that if ax = n, x is called the logarithm of n to base a: Cf. Ganita Tilaka, Baroda, 1937,
p. xxv (intro.). 67. T. Heath writes, “The Neo-Pythagoreans improved the classification
thus. With them the 'even-times even' number is that which has its halves even, and so on till unity is reached," A History of Greek
Mathematics, Oxford, (Pts. I & II, 1921), Pt. I, p. 72. 68. Cf. Jain, L. C., On certain Mathematical Topics of the Dhavalā Texts,
I. J. H. S., Vol. 11. no. 2, 1976, 85-111.
29