4. Consider what figure must be annexed to the divisor, so that if the result be multiplied by it, the product may be equal to, or next less than the dividend, and it will be the second figure of the root.

5. Subtract the said product from the dividend, and to the remainder bring down the next period for a new dividend.

6. Find a divisor as before, by doubling the figures already in the root; and from these find the next figure of the root, as in the last article; and so on through all the periods to the last.

NOTE 1. When the root is to be extracted to a great number of places, the work may be much abbreviated thus: having proceeded in the extraction by the common method till you have found one more than half the required number of figures in the root, the rest may be found by dividing the last remainder by its corresponding divisor, annexing a cypher to every dividual, as in division of decimals; or rather,

.2

Then a+b+ca2+2ab+b2+2ac+2bc+c2, and the manner of finding a, b, and c will be, as before: thus,

1st divisor a)a2+2ab+b2+2ac+2bc+c2(a+b+c=root,

a2

2d divisor 2a+b)2ab+b2

2ab+b2

3d divisor 2a+2b+c)2ac+2bc+c2

2ac+2bc+c2

Now the operation in each of these cases exactly agrees with

the rule, and the same will be found to be true, when of any number of periods whatever.

consists

without annexing cyphers, by omitting continually the first figure of the divisor on the right, after the manner of contraction in division of decimals.

NOTE 2. By means of the square root we readily find the fourth root, or the eighth root, or the sixteenth root, &c. that is, the root of any power, whose index is some power of the number 2; namely, by extracting so often the square root, as is denoted by that power of 2; that is, twice for the fourth root, thrice for the eighth root, and so on.

TO EXTRACT THE SQUARE ROOT OF A VULGAR FRACTION.

RULE.

First prepare all vulgar fractions, by reducing them to their least terms, both for this and all other roots. Then

1. Take the root of the numerator and that of the denominator for the respective terms of the root required. And this is the best way, if the denominator be a complete power. But if not, then

2. Multiply the numerator and denominator together; take the root of the product: this root, being made the numerator to the denominator of the given fraction, or the denominator to the numerator of it, will form the fractional root required.

And this rule will serve, whether the root be finite or infinite.

Or 3. Reduce the vulgar fraction to a decimal, and extract its root.

EXAMPLES.

1. Required the square root of 5499025.

5499025(2345 the root.

4

43/149
3129

464/2090
4/1856

4685 23425
23425

2. Required the square root of 184:2,

184.2000(13.57 the root.

1

23/84
3/69

265 1520

51325

2707|19500
18949

551 remainder,

21

3. Required the square root of 2 to 12 places.

2(1.41421356237 +root.

5. What is the square root of ⚫oco 32754?

1. Having divided the given number into periods of 3 figures, find the nearest less cube to the first period by the table of powers or trial; set its root in the quotient, and subtract the said cube from the first period; to the remainder bring down the second period, and call this the resolvend.

2. To three times the square of the root, just found, add three times the root itself, setting this one place more to the right than the former, and call this sum the divisor. Then divide the resolvend, wanting the last figure, by the divisor,

* The reason of pointing the given number, as directed in the rule, is obvious from Cor. 2, to the Lemma, used in demonstrating the Square Root; and the rest of the operation will be best understood from the following analytical process.

Suppose N, the given number, to consist of two periods, and let the figures in the root be denoted by a and b.

Then a+b=a+3a2b+3ab3+b3=N= given number, and to find the cube root of Nis the same as to find the cube root of a3+3a2b+3ab2+b2; the method of doing which is as fol lows: