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

C U B E 163
Multiplying the remaining digit 1 by 4, and setting the
product above it, we have
5625
1
Then moving the remaining digit 1, we obtain
5625
1
Thus the second round of operations is completed.
Next setting the square of 1 above it the process is comple-
ted for there are no remaining figures, and the result stands thus :
15625
Algebraic Method of Squaring
Brahmagupta in his Brāhmasphuṭasiddhānta gives a minor
method of squaring thus :
The product of the sum and the difference of the
number (to be squared) and an assumed number plus
the square of the assumed number give square¹.
This may be represented by the following identity :
n²=(n—a) (n+a)+a²
This identity has been used for squaring by most of the
Indian mathematicians. Thus
15²=(15—5) (15+5)+5²=225
We are not giving here other identities which have been
used by latter mathematicians of India in getting the squares
of numbers; for example, when Mahāvīra says :
The sum of the squares of the two or more portions of
the number together with their products each with the
others multiplied by two gives the square² :
he obviously refers to the identity
(a+b+c............)²=a²+b²+c²+......+2ab+......
Cube
The Sanskrit term for cube is ghana. It when used in
the geometrical sense also means the solid cube. In the arithmeti-
cal sense, it means the continued product of the same number
taken three times. Thus we have the definition in the Ārya-

1. राशेरिष्टयुतोनाद्वधः कृतिर्वेष्टकृतियुक्तः । — BrSpSi. XII. 63
2. GSS. p. 13.