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

164 BRAHMAGUPTA AND ARITHMETIC
bhaṭīya :
The continued product of three equals and also the
solid having twelve (equal) edges are called ghana.¹
A method of cubing applicable to numbers written in the
decimal place-value notation, has been in use in this country
from before the 5th century A.D. Āryabhaṭa I (499 A.D.) had
the familiarity with this method; he, however, does not give the
method of cubing in his treatise, though he describes the inverse
process of extracting the cube-root.
Brahmagupta gives the method of cubing in the following
verse :
Set down the cube of the last (antya); then place at the
next place from it, thrice the square of the last multi-
plied by the succeeding; then place at the next place
thrice the square of the succeeding multiplied by the
last, and (at the next place) the cube of the succeeding.
This gives the cube.²
The rule may be illustrated by an example.
Example : To cube 1357.
The given number has four places, i.e., four portions. First
we take the last digit 1 and the succeeding digit 3, i.e. 13 and
apply the method of cubing thus :
(i) Cube of the last (1³) = 1
(ii) Thrice the square of the
last (3.1²) multiplied by
the succeeding (3) gives
(3.3.1²) = 9 (placing at the
next place)
(iii) Thrice the square of the
succeeding, multiplied by
the last gives (3.3².1) = 27 (placing at the
next place)
(iv) Cube of the succeeding (3³) = 27 (placing at the
next place)
——————
Thus 13³ is the sum 2197

1. सदृशत्रयसंवर्गो घनस्तथा द्वादशाश्रस्स्यात् ॥ — Ārya, II. 3.
2. स्थाप्योऽन्त्यघनोऽन्त्यकृतिस्त्रिगुणोत्तरसंगुणा च तत्प्रथमात् ।
   उत्तरकृतिरन्त्यगुणा त्रिगुणा चोत्तर घनश्च घनः । — BrSPSi. XII, 6.