another over the place of hundreds, and so on, over every second figure, both toward the left hand in whole numbers, and toward the right hand in the Decimal places.-When the number of integral places is odd, the first, or left hand period, will consist of one figure only.

Find the greatest square in the first period on the left hand, and write its root on the right hand of the given number, in the manner of a quotient figure in division.

Subtract the square, thus found, from the said period, and to the remainder annex the two figures of the next following period, for a divi

dend.

Double the root above mentioned for a divisor, and find how often it is contained in the said dividend, exclusive of its right hand figure, and set this quotient both in the place of the quotient and in the divisor.-The best way of doubling the root, to form each new.divisor, is to add the last figure always to the last divisor, as it is done in the subsequent examples.

Multiply the whole augmented divisor by this last quotient figure, and subtract the product from the said dividend, bringing down to it the next period of the given number for a new dividend.

Repeat the same operation again; that is, find another new divisor, by doubling all the figures now found in the root; from which, and the last dividend, find the next figure of the root as before; and so on through all the periods to the last.

Note 1. After the figures belonging to the given number are all exhausted, the operation may be continued in dicimals, by annexing any number of periods or ciphers to the remainder.

2. The number of integral places in the root

is always equal to the number of periods in the integral part of the resolvend.

3. When vulgar fractions occur in the given power, or number, they may be reduced to decimals, then the operation will be the same as before dictated.

Required the square root of 16007.3104.

1 | 16007.3104(126.52=Answer.

1

1

EXAMPLES FOR EXERCISE.

Required the square root of 298116. Ans 546. Required the square root of 348.17320836. Ans.

18.6594.

Required the square root of 17.3056. Ans. 4.16. Required the square root of .000729. Ans. .027. Required the square root of 17. Ans. 4.168333+

TO EXTRACT THE CUBE ROOT.

1. By the Common Rule.*

1. Separate the given number into periods of three figures each, by putting a point over the unit figure, and also over every third figure, from thence to the left hand in whole numbers, and to the right in decimals.

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

3. Subtract the cube thus found, from the said period, and to the remainder bring down the next period, and call this the resolvend..

4. 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.

5. Seek how often the divisor may be had in the resolvend, and place the result in the quotient.

6. Multiply the triple square by the last quotient figure, and note the product; Multiply the square of the last quotient figure by the triple quotient, and place this product under the last; under both, set the cube of the last quotient figure, and call their sum the subtrahend.

Among all the Rules that have been given by different Authors, I find that youth, (for whom this is chiefly designed,) can understand the Method of extracting the cube root by this with greater facility than any other Rule.

7. Subtract the subtrahend from the resolvend, and to the remainder bring down the next period for a new resolvend, with which proceed as before.

Note. The same rule must be observed for continuing the operation, as in the square root. EXAMPLES.

1. Required the cube root of 48228.544. 32×300=2700 | 48228.544.(36-41 root. 3 x30 = 90 27

Ex. 2. What is the cube root of 62570773? Ans. 397.

Ex. 3. What is the cube root of 51478848? Ans. 372.

Ex. 4. What is the cube root of 84 604519? Ans. 4.39.

Ex. 5. What is the cube root of 16974593? Ans. 257.

2. To Extract the Cube Root by another Method.*

1. By trials find the nearest rational cube to the given number, whether it be greater or less ; and call it the assumed cube.

2. Then say, by the Rule of Three, as the sum of the given number and double the assumed cube, is to the sum of the assumed and double the given number, so is the root of the assumed cube, to the root required, nearly. Or, as the first sum is to the difference of the given and assumed cube, so is the assumed root to the difference of the roots, nearly.

3. By using, in like manner, the cube of the root last found as a new assumed cube, another root will be obtained still nearer. And so on as far as we please; using always the cube of the last found root, for the assumed cube.

EXAMPLES.

1. To find the cube root of 21035.8.

Here the root is soon found between 27 and 28. Taking therefore 27, its cube is 19683, which is the assumed cube.

Then,

As. 60401.8 61754.6: 27: 27.6047,
Therefore 27.6047. is the root nearly.

Again by repeating the operation, and taking

* This Rule is found in Hutton's Mathematics. There has been different Rules given for Extracting the cube root, among which, this and another Rule given in Pike's Arithmetic (by approximation), are very expeditious.