पाटीगणित (श्रीधराचार्य - प्राचीन संस्कृत टीका एवं प्रो. कृपाशंकर शुक्ल सविस्तार व्याख्या)

SQUARE-ROOT 9
This rule is based on the formula
(a+b)² = a² + 2ab + b².
For details of the process, see B. Datta and A. N. Singh, History of
Hindu Mathematics, Part I, pp. 157-160
Other rules for squaring :
24. The square (of a given number) is also equal to
the product of two equal numbers (each equal to the given
number), or the sum of as many terms of the series whose first
term is 1 and common difference 2, or the product of the
difference and the sum of the given number and an assumed
number plus the square of the assumed number.¹
That is, (i) n² = n × n ;
(ii) n² = 1 + 3 + 5 +......to n terms ;
(iii) n² = (n - a) (n + a) + a².
Ex. 4. Tell (me) the squares of 1 to 9, 25, 36, 63, 432,
and 7802.
(6) Square root (varga mūla)
Rule for finding the square root :
25-26. Having subtracted the (greatest possible) square
from the (last) odd place (set down double the square root
underneath the next place). By that double square root, that
has left its place (i.e., which has been set down underneath the
next place), divide the remainder ; set down the quotient in
the line (of double the square root), and, having subtracted
the square of that (from the number above), make double of
that (quotient). Then having moved the resulting quantity
(in the line of double the square root) one place forward,
divide by it as before. (Continue this process till all places
are exhausted, and then) halve the doubled quantity (to get
the square root).²
This rule will be clear by the following example :

¹ Cf. GSS, ii. 29 ; GK, I, p. 6, vs. 18. Also see Â. ii. 3(i) ; BrSpSi,
xii 63(ii); MSi, xv. 6(i); GT, p. 8, vs. 21(ii); SiŚe, xiii 4(i); L (ASS), 20(ii).
² Cf. GSS, ii. 36 ; GT, p. 9, vs. 23 ; MSi, xv. 6(ii)-7 ; SiŚe, xiii. 5 ;
L (ASS), p. 21, vs. 22 ; GK, I, p. 7, lines 2-9. Also see Â, ii. 4.