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

C U B E 165 After this we take the next figure, 5, i.e., the number 135, and in this consider 13 as the last and 5 as the succeeding. Then the method proceeds thus : (i) The cube of the last (13³) as already obtained = 2197 (ii) Thrice the square of the last multiplied by the succeeding, i.e. 3.13².5 = 2535 (placing at the next place) (iii) Thrice the square of the succeeding multiplied by the last, i.e. 3.5².13 = 975 (placing at the next place) (iv) Cube of the succeeding, i.e. 5³ = 125 (placing at the next place) Thus 135³ is the sum 2460375 Now the remaining figure 7 is taken, so that the number is 1357, of which 135 is the last and 7 the succeeding. The method proceeds thus : (i) Cube of the last, i.e. (135³) as already obtained = 2460375 (ii) Thrice the square of the last into the succee- ding, i.e. 3. (135)². 7 = 382725 (placing at the next place) (iii) Thrice the square of the succeeding into the last, i.e. 3.7². 135 = 19845 (placing at the next place) (iv) Cube of the succeeding i.e. 7³ = 343 (placing at the next place) ———————— Thus (1357)³ is the sum 2498846293 Evidently these methods of cubing are based on the identity: (a+b)³=a³+3a²b+3ab²+b³ and keeping in mind the place values of numerals in a given