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