ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

168 BRAHMAGUPTA AND ARITHMETIC
dred-thousands place is second aghana, and so on. Thus to find
out the cube-root, one has to mark out the ghana, first aghana
and second aghana places, then the process of finding out the
cube-root begins with the subtraction of the greatest cube num-
ber from the figures up to the last ghana place. Though this has
not been explicitly mentioned in the rule, the commentators
say that it is implied in the expression ghanasya mūla-vargeṇa
etc. ("by the square of the cube-root etc.")
We are reproducing here an illustration given by Datta
and Singh.
Example. Find the cube-root of 1953125.
The places are divided into groups of three by marking
them as below [ghana ( | ) first aghana (—) and second aghana
(--)]:
| — — | — — |
1 9 5 3 1 2 5
Subtract cube 1 ... ... ... ... ... (c) Root=1
Divide by thrice ——————————————
square of root,
i.e. 3.1² 3)9(2 ... (a) Placing quotient
Subtract square 6 after the root 1
of quotient mul- —— gives the root 12
tiplied by thrice 35
the previous root, 12 ... (b)
i.e. 2².3.1 ——
Subtract cube of 233
quotient, i.e. 2³ 8 ... (c)
Divide by thrice
square of the root,
i.e. 3.12² 432)2251(5 ... (a) Placing quotient
Subtract square 2160 after the root
quotient multiplied ———— 12 gives the
by thrice the pre- 912 root 125
vious root, i. e.
5².3.12 900
————
Subtract cube of 125 ... (b)
quotient, i.e. 5³ 125 ... (c)
————
Thus the cube-root=125.
From the details given, it would be clear that the present