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