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