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.