Other examples might have been added, but I believe the above are sufficient to convey a knowledge of the manner of performing such operations, preparatory to the study of Mensuration.

SQUARE ROOT.

THE square or second root of any number or power, is such a number as being multiplied by itself as a factor will produce the power or number proposed. Thus, 2 is the square or second root of 4, because 22 or 2 x 24.

Any power of a given number may be found exactly, but there are many numbers of which a given root can never be precisely determined; although by the help of decimals we can approximate towards it to any assigned degree of exactness.

The roots which approximate are called surd roots, and those which are perfectly accurate rational roots; thus the square root of 2 is a surd root.

The squares or second powers of the nine digets are expressed in the following

TABLE

To extract the square root.

RULE. Divide the given power or number into periods of two figures each, by placing a dot over units, another over hundreds and so on.

Find the greatest square in the first period and set its root in the right hand, as a quotient figure in division. Subtract the square thus found, and to the remainder annex the succeeding period for a new dividend. Double the root for a divisor, and examine how often it is contained in the dividend, exclusive of the place of units, and put the result in the quotient and also in the units place of the divisor. Multiply the divisor thus increased by the new quotient figure, and subtract the product from the dividend. Bring down the next period, find a divisor as before by doubling the figures already in the root, and proceed as before and so on through all the periods to the last..

The best way of doubling the root, to form the new divisors, is by adding always the last figure of the last divisor to that divisor. And after the figures belonging to the given num. ber are all exhausted, the operation may be continued into decimals at pleasure, by adding any number of periods of ciphers, two in each period. *

* For a demonstration of the above rule, and other observations on this subject see John Walker's Philosophy of Arithmetic, pages 99

to 105.

Find the square or second roots of the following numbers.

Find the square roots of the following numbers respectively.

25. 1073741824

26. 481890304

Find the square root of 674193419 (25965,234, &e.

When the root is to be extracted to many places of figures the work may be considerably shortened, thus: having proceeded after the common method 'till there are found half the required number of figures in the root, or one figure more ; then for the rest, divide the last remainder by its corresponding divisor after the manner of the second contraction in Division of Decimals.

Examples.

Find the square root of 5 to 6 places of decimals, 5 (2,26068, &c,

37. Find the square root of 2 to 9 places. 38. Find the square root of 21 to 7 places. 39. Find the square root of 11 to 9 places. 40. Find the square root of 31 to 7 places. 41. Find the square root of 399 to 9 places.' 42. Find the square root of 501 to 6 places.

To find the square root of a vulgar fraction or mixed number.

Prepare all vulgar fractions by reducing them to their least terms. Then take the root of the numerator and of the denominator for the respective terms of the root required, whichis the best way if both terms of the fraction are complete squares; but if not, then multiply the numerator and denominator together and find the root of the product: this root being made the numerator to the denominator of the given fraction, or made the denominator of the numerator of it, will form the fractional root required.

Or Reduce the vulgar fraction to a decimal and extract its

root.

Mixed numbers may be either reduced to improper fractions and the root extracted by the first or second rules or the vul

gar fraction may be reduced to a decimal, then joined to the integers, and the root of the whole extracted.

64 = ¥ & √ ¥=+=288,25 & √ 8,25-2,87228132. Find the square roots of the following fractions and mixed numbers respectively.

NOTE.-This mark V is the sign of the square root and signifies that the root of the number before which it is placed is to be extracted. Or it expresses the square root of any number or numbers which immediately follow it,

'CUBE ROOT.

THE cube or third root of any number or power is such a number as being multiplied by itself, and then again into the product; or being multiplied by its square or second power will produce the number or power proposed. Thus 43 = 64≈ 42 x 4 = 16 x 4 = 4 × 4 × 4.

The cubes or third powers of the nine digits are expressed in the following

TABLE

RULE.-Separate the given number or power into periods of three figures each, beginning from the units place, then from the first period, subtract the greatest cube it contains, put the root in the quotient, and to the remainder bring down the next