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

CUBEROOT 167
| - | - |
1 8 2 2 5
Subtract square 1 root = 1
Divide by twice ———
the root 2) 8 (3 placing quotient at the
6 next place, the root=13
———
22
Subtract square of
quotient 9
Divide by twice 26)132(5 placing quotient at the
the root 130 next place, the root=135
Subtract square 25
of the quotient 25
The process ends. The square-root of 18225 is thus 135.
It has been stated by Kaye, that Āryabhaṭa’s method of
finding out the square-root is algebraic in character, and that it
resembles the method given by Theon of Alexandria. Ārya-
bhaṭa’s method is purely arithmetic and not algebraic is the
view of Datta and Singh who do not agree with Kaye on this
point.
Cube Root
The Sanskrit term for cube-root is ghanamūla or ghanapada.
The first mention of the operation of cube-root is found in the
Āryabhaṭīya of Āryabhaṭa I (499 A.D.), though the operation
is given in only a concise form :
Divide the second aghana place by thrice the square
of the cube-root; subtract from the first aghana place
the square of the quotient multiplied by thrice the
preceding (cube-root); and (subtract) the cube (of the
quotient) from the ghana place; (the quotient put
down) at the next place (in the line of the root) gives
(the root).¹
As has been explained by all the commentators on the
Āryabhaṭīya, the units place is ghana; the tens place is first
aghana, the hundreds place is the second aghana, the thousands
place is ghana, the ten thousands place is first aghana, the hun-

1. अघनाद् भजेद् द्वितीयात् त्रिगुणेन घनस्य मूलवर्गेण।
   वर्गस्त्रिपूर्व गुणितश्शोध्यः प्रथमाद् घनश्च घनात् ॥ —Ārya. II. 5