"Double the figures already found in the root for a new divisor, (or bring down your last divisor for a new one, doubling the right hand figure of it,) and from these, find the next figure in the root, as last directed, and continue the operation in the same manner till you have brought down all the periods."

It frequently happens that the left hand period is not a full period, but, notwithstanding, it must be considered the first period. When there are decimals in the given number, it must be pointed both ways from the place of units. If, when there are integers, the first decimal period be deficient, it may be completed by annexing so many ciphers as the power requires.

To extract the root of a vulgar fraction; reduce it to a decimal, and proceed as directed above.

What is the square root of 676 ?

676(26 Ans.

4

46)276

276

Pup. I understand how the work is done, but do not understand the reasons of it; I should like an explanation of the whole work.

Tut. "The superficial contents of any thing, that is, the number of square feet, yards or inches, &c. contained in the surface of a thing, as of a table, a floor, a picture, a field, &c. is found by multiplying the length into the breadth. If the length and breadth be equal, it is a square; then the measure of one of the sides, as of a room, is the root, of which the superficial content in the floor of the room is the second power." Hence to ex

tract the square root of any number, is so to arrange the parts of that number that they may be in a square form.

Suppose you should have 144 square pieces of board, and you wished to know how many of those pieces would be on a side, if the whole were arranged into a square form. To determine this, you must extract the square

root of 144; the first step of which is to point off the number into periods of two figures each. This shows how many figures the root will consist of, and is done on this principle, that the product of any two numbers can have, at most, but as many places of figures, as are in both the factors, and at least but one less.

144(1

1

44

Fig. 1.

The left hand period being 1, the square of it will be 1, and likewise the root will be 1. But as we have nothing to do at present with the right hand period, we will omit it, and consider only the left hand period, which being in the place of hundreds, must be called 100; hence the operation. at present, will be to find the square root of 100. root of 1, is 1, but as there are two periods in 100, there will be two figures in its root, and as the figure already obtained in the root is equal to its period, there is nothing remaining for the next period; and as the next period consists wholly of ciphers, the next figure of the root will be a cipher, so that the root of 100 is 10. By this process we have disposed of 100 of the pieces into the form represented by Fig. 1, viz. 10 pieces on a side.

The

The reason for placing the square number underneath the period, and subtracting it from the period, as directed in the rule, is as follows. When we have obtained the root of the left hand period, we have disposed of as many pieces as the greatest square of left hand period represents, and by subtracting the square of the root from its period, we make it smaller by as many as the square of the root represents; thus in the example given, 1 in the quotient represents 10, the square of which is 100, which 1, under the left hand period, represents. This, subtracted from the left hand period, leaves 44; so that 100

pieces have been disposed of as represented by Fig. 1, and 44 pieces are now to be added to it, in such manner that the square form will be preserved. To do this, the rule directs to "place the double of the root already found on the left hand of the dividend for a divisor."

144(12

a[

1

22)44

44

Fig. 2.

Now the first figure of the root shows the number of pieces there are on a side of Fig. 1, viz. 10. In order to A. preserve the square form the additions must be made on two adjoining sides of the square, as in Fig. 2. Now it is evident that if there were just 20 pieces left, after disposing of 100, there would be just enough to make a row on two sides of Fig. 1, and if there were 40 pieces left, they would make two rows, on two sides, as represented by the rows a, e, and o, n, Fig. 2. Hence the reason of placing the double of the root on the left of the dividend for a divisor. In making the additions a, e, and o, n, you will observe there is a deficiency, A. which is not filled. To fill this deficiency, the rule directs to "except the right hand figure," and likewise to "place the quotient figure on the right hand of the divisor.' Now the dificiency A. must be limited by the additions a, e, and o, n, consequently the figure, expressing the width of these additions, expresses the root of this deficiency, which, multiplied into itself, gives the superficial contents of the deficiency. Thus Fig. 2 shows the disposition of 144 pieces into a square

form.

10171

Pup. This appears much plainer than I at first imagined; there is nothing which is at all difficult.

Tut. After performing a few examples, we will proceed to examine the cube root.

What is the square root of 176248 ?

The extraction of the cube root, is the finding a number, which multiplied into its square, will produce the givnumber.

Rule.

"Separate the given number into periods of three figures each, by putting a point over the unit figure, and every third figure beyond the place of units.

"Find the greatest cube in the left hand period, and put its root in the quotient.

"Subtract the cube thus found, from the said period, and to the remainder bring down the next period, and call this the dividend. "Multiply the square of the quotient by 300, calling it the triple square, and the quotient by 30, calling it the triple quotient, and the sum of these call the divisor.

"Seek how often the divisor may be had in the dividend, and place the result in the quotient.

66 Multiply the triple square by the last quotient figure, and write the product under the dividend; multiply the square of the last quotient figure by the triple quotient, and place this product under the last; under all, set the cube of the last quotient figure, and call their sum the subtrahend.

"Subtract the subtrahend from the dividend, and to the remainder bring down the next period for a new dividend, with which proceed as before, and so on till the whole be finished."

The same rule will be observed for continuing the operation, and for extracting the roots of fractions, as in the square root.

What is the cube root of 10648 ?