Kato Fukutaro's Explanation

 

 

Let the largest square with sides (a + b) represents the radicand, or a number N, from which we want to calculate the square root. So, we want to find a two digit (a and b) number, such that (a + b) x (a + b) = N or (a + b)2 = N. If so, = . Subtract from the largest area (a + b)2 the hatched area (a2) . The figure that remains is shown on the next figure. It equals a2 + 2ab + b2 - a2 = 2ab + b2.
 

This fig. equals the areas of 2 rectangles each or area a x b plus the area of a small square of area b x b = b2).

 
 

If we divide 2 into the above area , this area remains, equal to the area or a rectangle (hatched) plus the area of a triangle, (half b-squared). Subtracting the hatched area, ab, the non-hatched area remains.

 

 

Subtracting from the remaining triangle its area = (half b-squared), we have no more area to subtract, provided that N = (a + b)2 is a perfect square.

 

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When the square root has 2 figures only (say a and b)

From high school, we learned that;

(a + b)² = a² + 2ab + b²       (Note ab = a × b, as we know.)

Let N (the number we want to extract the square root from) equals (a + b)². Or, in other words, given N, we must find 2 numbers, a and b, whose sum-squared equals N. Let a be a number whose square a² is less or at most equal to the last digit group from N, whose square root we want to calculate.

(a² +2ab + b²) - a² i) Square a and subtract a² from the last radicand group already set on Soroban. Call A this remainder.
 

(a² +2ab + b²) ÷ 2 ii) Simplify A by dividing 2 into it (or multiply A by 0.5; it is quicker), without displacing the orders. Call this quotient B.
 

(ab + b²/2) - ab iii) Evaluate the division of a into B and choose only the first figure; this partial quotient is "the best" approximation to the second figure of b. See if you do not need to "revise" b  (Example #4, on the main page illustrates an instance of revising an answer.)  Multiply a × b and subtract this product from B. The remainder on the Soroban right side is then b²/2. Let's call it C.
 

b²/2 - b²/2 = 0 iv) From C, subtract half of b squared. (see table) If the given radicand N that was supposed be equal to (a² + b²) is a perfect square, the remainder must be 0 (zero), as shown by the following equivalent equation, that summarizes the whole process:

((a + b²) - a²)/2 - ab - b²/2 = 0 (zero)