CHAPTER VII
SQUARE AND CUBE ROOT

The process of extracting the square and cube roots of numbers by the Bead Arithmetic, as has been stated before (page 4) is merely a matter of repeated subtraction. Howver, as the reader will find, it is rather cumbersome to use. It is given here only to show how it can be done on the abacus.

TO EXTRACT SQUARE ROOT

Method of Procedure.

1. Place the number on the right side of the abacus (for convenience call it the “square number,”) and separate it into periods of two figures or columns each beginning from the decimal point, the same as we do in pen arithmetic.

2. Mark one (1) on the left side of the frame call it the “root number,” and subtract it from the left-hand period of the square number.

3. Add two (2) to the root number and subtract the sum again from the left-hand period of the square number. Add two again to the root number and again subtract the sum from the same period, repeating this process until the root number (which increases at every such operation) is greater than the number in that period.

4. Now, bringing down the next period annex one cipher to the root number and add eleven (11), and from the new period subtract the sum. Repeat the process described in (3) until the root number is again too large to be subtracted from that period. Then repeat (4), thus continuing adding and subtracting until the whole number is finished.

5. If, as is sometimes the case, after bringing down the next period, the root number is still too large to be subtracted, bring down another period, but instead of annexing one cipher to the root number and adding 11; annex two ciphers and add 101, and then proceed as in (3) and (4).

6. After the whole square number is thus used up, add one (1) to the final root number and divide this sum by two (2). The result is the square root of the number sought.

Example 1. Find the square root of 625.

In accordance with the method of procedure just given, after placing the square number, 625, on the right side of the frame, we separate it into two periods, the first period, containing one figure, 6, and the second period two figures, 25. These two periods indicate that there will be two figures in the square root.

We mark one on the left-hand side of the frame as the root number and subtract it from the left-hand period, 6, thus leaving 5.

We then add 2 to the root number making it 3, and then subtract it from the left-hand period, 5, thus leaving 2.

Now the root number is greater than the square number up to that period. Therefore, we annex one cipher to the root number, 3, making it 30, and add 11, thus obtaining 41. We now subtract 41 (the sum) from the next period of the square number 225, leaving 184. Proceeding as the method indicates, 41, the root number plus 2 equals 43 and 184 minus 43 equals 141. Root number 43 plus 2 equals 45. Square number 141 minus 45 equals 96. Root number 45 plus 2 equals 47. Square number 96 minus 47 equals 49. Root number 47 plus 2 equals 49. Square number 49 minus 49 equals 0. Therefore the number is a perfect square.

The final root number plus 1 equals 50 (that is 49 plus 1), 50 divided by 2 equals 25 which is the square root of the number 625.

Example 2. Find the square root of 363,609.

Proceeding as before, after the first period is finished, we have 11 in the root number and 36 in the next period. If we were to add 11 to 110 (root number plus 0) we would find that the sum is greater than 36. Therefore we know that there is zero in the square root. Therefore we bring down another period, making the square number 3609. However, we do not subtract 121 but, instead, 1201 which is the sum of 1100 and 101 (see the 5th step of method of procedure). Now proceed as usual. 3609 minus 1201 equals 2408, etc. When the square number is finished we have 1205 for the final root number. Adding 1 to this and dividing by 2 equals 603 which is the square root of 363,609.

TO EXTRACT CUBE ROOT

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