________________
vi
both with respect to integers and fractions. The theory of indices as described in the Dhavalā is somewhat different from what is found in the mathematical works. This theory is certainly primitive and is earlier than 500 A. D. The fundamental 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 squareroot (vii) the oube:root (viii) the successive square-root, (ix) the successive cuberoot, etc. All other powers are expressed in terms of the above. For example, 83/3 is expressed as the first square-root of the cube of a; 89 is expressed as the cube of the cube of a; a6 is expressed as the square of the cube or the cube of the square of a; ebo. The successive squares and square-foots are as below
lat square of a means 2nd equare, of & means ord square of a means
(al )2 = af = 22
nth square of a means
Similarly
a 1/2
1st square-roof of a means 2nd square-root of a means 3rd square-root of a means
a 1/2
alla
........
nth square-root of a means
al
Vargita-samvargita-The technical term vargita-samvargita has been used for the raising of a number to its own power. For instance, ma is the vargitasamvargita of n. In connection with this the Dhavata mentions an operation called Viralana-deya-" spread and give ". The Viralana (spreading) of a number means the separating of the number into its unities, i. e, the viralana of n is
11111 ......n times. Deya ( giving ) means the substitution of n in the place of 1 everywhere in the above. The vargita-samvargita of n is obtained by multiplying together the n’s obtained by the viralana-deya. The result is the first vargita-samvargita of n, i.e.,
lat wargita-samvargita af n is
na The application of the process of viralana-deya once again, i. e., to ng gives the
Ind vargita-samvargita of n (n ) . A further application of the same procedure gives the
1. Dhavala Vol. III, p. 53.
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