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

170 BRAHMAGUPTA AND ARITHMETIC continue the process till all ghana-padas (cube-places) are exhausted). This will give the (cube) root (of the given number).¹ K.S. Shukla in his translation and commentary of this book has given the illustration of extracting cube - root as follows : Example :- To find the cube root of 277167808. Let us indicate ghana-padas or 'cube' places by "c" and aghana-padas or non-cube places as "n" : n n c n n c n n c 2 7 7 1 6 7 8 0 8 Subtract the greatest possible cube (i.e. 6³ or 216) from the last 'cube' place (i.e. from 277) and place the cube- root (i.e. 6) underneath the third place to the right of the last 'cube' place; thus we have n n c n n c n n c 6 1 1 6 7 8 0 8 (remainder) 6 (line of cube-root) Dividing out by thrice the square of the cube-root (i.e. by 3.6² or 108) the remainder up to one place less than that occupied by the cube-root (i.e. 611) and setting down the quotient in the line of the cube-root (to the right of the cube-root), we have n n c n n c n n c 7 1 6 7 8 0 8 (remainder) 6 5 (line of cube-root) Let now quotient 5 be called the 'first' (ādima) and the cube-root 6 the 'last' (antya). Then subtracting the square of the 'first' (ādima) as multiplied by thrice the 'last' (antya) (i.e. 3×6×5² or 450) from one place less than that occupied by the quotient (i.e. from 716), we get

1. घनपदमघनपदे द्वे घन (पद) तोऽपास्य घनमदो मूलम् । संयोज्य तृतीयपदस्याधस्तदनष्टवर्गेण ॥ २९ ॥ एकस्थानोनतया शेषं त्रिगुणेन (सं) भजेत्तस्मात् । लब्धं निवेश्य पङ्क्त्यां तद्वर्गं त्रिगुणमन्त्यहतम् ॥ ३० ॥ जह्यादूपरिमराशेः प्राग्वद् घनमादिमस्य (च) स्वपदात् । भूयस्तृतीय पदस्याध इत्यादिक विधिर्मूलम् ॥ ३१॥ —Śrīdhara, Pāṭīgaṇita, 29-31