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

CUBE ROOT 169
method of extracting the cube-root is almost a contraction of the
method first given by Āryabhaṭa I (499 A.D.)
The method of Āryabhaṭa has been invariably followed by
Indian mathematicians. Brahmagupta in his Brāhmasphuṭa-
siddhānta repeats the method in the following words :
The divisor for the second aghana place is thrice the
square of the cube-root; the square of the quotient
multiplied by three and the preceeding (root) must be
subtracted from the next (aghana place to the right).
and the cube (of the quotient) from the ghana place
(the procedure repeated gives) the root.¹
Śrīdhara and Āryabhaṭa II have further improved on the
method of extracting cube-root proposed by Āryabhaṭa I and
followed by Brahmagupta. Rule for finding the cube-root as
given by Śrīdhara in his Pāṭīgaṇita is as follows :
(Divide the digits beginning with the units' place into
periods of) one ghana-pada (one "cube" place) and two
aghana-padas (two "non-cube" places). Then subtrac-
ting the (greatest possible) cube from the (last) ghana-
pada and placing the (cube) root underneath the
third place (to the right of the last ghana-pada),
divide out the remainder up to one place less (than
that occupied by the cube-root) by thrice the square
of the cube-root, which, is not destroyed. Setting
down the quotient (obtained from division) in the
line (of the cube-root), (and designating the quotient
as the 'first' (ādima) and the cube-root as the 'last'
(antya), subtract the square of that quotient, as multi-
plied by thrice the 'last' (antya) from one place less
than that occupied by the quotient (uparima-rāśi) as
before, and the cube of the 'first' (ādima) from its own
place.
(The number now standing in the line of cube-root is
the cube-root of the given number up to its last-but
one ghana-pada (cube place) from the left).
Again apply the rule, "(placing cube-root) under the
third place" etc. (provided there be more than two
ghana-padas (cube places) in the given number; and

1. छेदो घनाद् द्वितीयाद् घनमूलकृतेस्त्रिसंगुणाप्ताप्तकृतिः ।
   शोध्या त्रिपूर्वगुणिता प्रथमाद् घनतो घनो मूलम् ॥ —BrSpSi. XII. 7