two, and having found the divisor in the usual way, proceed according to the contracted method of dividing decimals.

Thus, suppose it had been required, in the preceding example to find the answer true to nine places of decimals; these and the two places of whole numbers are eleven figures in all; and, therefore, before commencing the contraction, it is necessary to find six figures. These have been already found 19.1049. Taking therefore 382098 and 279599, the divisor and remainder already found, and also the quotient 19.1049, the continuation of

382098)2795990(19-104973174

2674691

121299

114630

6669

3821

2848

to be

2675

173

153

20

the preceding work will stand as in the margin, cipher being added to the dividend, according to the rule. The first figure that results from the division is 7, which, in working the operation at full length, must have been annexed to the divisor: we therefore carry 5, for 7 times 7, to 56, the product of 7 and 8. After this, the work proceeds exactly as in the contracted mode of dividing decimals. By this means the root is found to be 19-104973174.

RULE III. To extract the root of a vulgar fraction, reduce it to its simplest form, if it be not so already, and extract the roots of both terms, if they be complete powers: otherwise divide the root of their product by the denomina

tor.

The root may also be found by reducing the fraction to a decimal, if it be not such already, and taking the root of the decimal.

Thus, the second root of is. This result may be obtained either by taking the roots of both terms, or by reducing the given fraction to the decimal 25, the second root of which is 5, or, or, the same as before.

In like manner, the root of 21, or 2, is, or 1. This result might also be obtained by extracting the root of 2.25. This would be found

to be 1.5, or 11, as before.

Again, if it be required to find the second root of; let the square root of $5 (5X7,) which will be found to be 5 9160798, be divided by the denominator 7, and there will result 84515425, the root required. The same result would be obtained by extracting the root of 71′4285/71, the decimal equivalent to the given fraction.

The more advanced pupil may sometimes find the following contractions useful:—1. If the denominator be an exact square, and the numerator

not, divide the square root of the numerator by the square root of the denominator. 2. If the numerator be an exact square and the denominator not, divide the product of the square roots of the numerator and denominator by the denominator.

The following rule, which is only a particular application of the general rule to be given hereafter for finding the roots of powers in general, may be found useful in carrying a root out to a great number of figures, after it has been carried to a considerable number by the common rule: (1.) Find the square of the part of the root already found. (2.) Then make the sum of the given number and three times this square the first term of an analogy; the sum of this square and three times the given number, the second term; and the part of the root already found, the third term: the fourth term will be the required root true to nearly three times the number of figures in the part before found.

Thus, the square root of 2 is found by the common method to be 1414, the square of which is 1.999396. Then, as 3X1.999396+2: 3x2+1-999396: 1.414: 141421356237, which is true, except the last figure; and if we should repeat the operation, using this number and its square, the root would be found true to about thirty places.

We might employ this rule from the beginning of the operation, by estimating the root as nearly as possible. It will in general be easier, however, to find the leading figures of the root by the common method. That method also, contracted in the way already shown, will for the most part be preferable to this, when the root is not required to be carried beyond ten or twelve figures.

Exercises. Required the square roots of the following numbers:

EXTRACTION OF THE THIRD, OR CUBE ROOT.

RULE I. To extract the third, or cube root of a given number: (1.) Find by trial a number nearly equal to the required root, and call it the assumed root: (2.) Find the cube of this root (3.) Then as twice this cube added to the given number, is to twice the given number added to the same cube, so is the assumed root to the true root, nearly: (4.) By employing the approximate root thus found, and repeating the process, a number still nearer the true root will be obtained; and thus the process may be repeated as often as may be thought necessary.*

The operation is proved by involving the root, when found, to the third power: and the more nearly the result agrees with the given number, the more ncarly correct is the root.

In the use of this rule, each result will generally be true to between two and three times the number of figures to which the assumed root was true. It may be observed also, that each result is too nearly equal to the assumed root; or, in other words, the correction is too small, as if the assumed root be too great, the result is also too great; otherwise, it is too small.

Exam. 1. Required the third or cube root of 34567.

Here, by cutting off three figures towards the right hand, we find for the first period 34, the root of which is above 3, since the third power of 3 is only 27; and hence, since the root must have

* This rule, as well as the approximative rule already given for the extraction of the square root, is only a particular application of the general rule to be given hereafter, for the extraction of roots. The rule which is commonly employed in extracting the cube root is here subjoined for the use of any who may be disposed to prefer it. The rule giver. above, however, abridges the labour very greatly, as will appear by the application of both rules to any particular exercise. The labour by the common ule, indeed, of finding the cube root, of a number true even to seven or eight figures, is so very great as to ren der the operation formidable in a high degree, and to make any shorter method very desirable. The following is the

Common Rule. (1.) Commencing at the units, divide the given number into periods of three figures each. (2) The first figure of the required root will be the cube root of the first period, or of the greatest cube contained in it, if it be not a cute itself. (3.) Subtract the third power of this figure from the first period, and to the remainder annex the next period for a dividend, and for part of the divisor take 300 times the square of the part of the root already obtained. (4.) Try how often this part of the divisor is contained in the dividend, and annex the figure thus found to the part of the root already determined. (5.) Then, to find the complete divisor, add to the part already found 30 times the last figure of the root multiplied by the part of the root before it, and add also the square of the last figure. (6.) Then inultiply and subtract as in Division; to the remain. der bring down the next period, and proceed as before. (7.) Repeat the operation, till all the figures in the given number have been used; and, if any thing remain, continue the operation in the same manner to find decimals, adding, to find each figure, three ciphers: er, if there be an interminate decimal, the three figures that would next arise from its continuation.

34328-125 34567

two figures, (one for each period,) it must be greater than 30.* We may suppose it therefore to be about 32 or 33. The cube of the former we find to be 32768, and that of the latter 35937. Hence, the given number differing from each by pretty nearly the same quantity, we may suppose its root to be about 32.5. The cube of this, is found to be 34328-125. The first part of the remaining work may conveniently stand as in the margin. The work is then to proceed as in the Rule of Proportion, and the result is found to be 32.5752101, which is the required result, true except the last figure, the

2

2

69134

34328.125

68656.250

34567

As 103223-250: 103462-125::32.5: 32.5752101, Answ.

true result being 32.57521043, &c., as would be found by repeating the operation with 32-575, as the assumed root.

Exam. 2. Required the cube root of 100.

The root here is evidently between 4 and 5, but nearer the latter, as 100 differs less from 125, the third power of 5, than from

64, the third power

of 4. Let then 4.6, the cube of which is 97 336, be assumed. After this, the work may stand as

in the margin; and

the result will be

4-64158, the required root nearly. By repeating the operation with 4-64, as the assumed root, there will result 4-641588833 for the required root still more correctly.

Exam. 3. Required the cube root of 782140.

Here the root proper

to be assumed is readily found to be 92, the cube of which is 778688. The work will then stand as in the margin, and there

will result 92.13575 for As 2339516: 2342968 :: 92: 92-13575 the required root nearly.

By repeating the operation with 92.136, there will be found, for the required root, very nearly true, 92-135747933.

* It is scarcely necessary to remind the Mathematical pupil, that the root proper to` be assumed, both in this article and the next, will be found with great ease by means of logarithms. The finding of it will also be facilitated, as in the preceding example, by dividing the given number into periods as in the Square Root, but each consisting of three figures instead of two, and by considering that there must be a figure in the root for each period.

RULE II. To extract the cube root of a vulgar fraction, reduce it to a decimal, and then extract the root; or multiply the numerator by the square of the denominator, find the cube root of the product, and divide it by the denominator.

The cube root of a mixed number is generally best found by reducing the fractional part to a decimal, if it be not so already, and then extracting. It may also be found by reducing the given number to an improper fraction, and then working according to the preceding directions.

Exercises. Find the cube roots of the following numbers:

The following article is intended only for the Mathematical or the more advanced Arithmetical pupil. For others, besides it being in a considerable degree difficult, it is seldom useful.

RULE. To extract any root whatever: (1.) Call the index of the given power; and find by trial a number nearly equal to the required root, and call it the assumed root. (2.) Raise the assumed root to the power whose index is n. (3.) Then, as +1 times this power added to n—1 times the given number, is to n―l times this power added to n+1 times the given number, so is the assumed root to the true root nearly. (4.) The number thus found may be employed as a new assumed root, and the operation repeated to find a result still nearer the true root.

For the mode of investigating this rule, which is perhaps the best and most convenient that has been discovered, see the tenth of Dr. Hutton's Tracts, where it was first given.

Exam. 1. Required the 365th root of 1·06.

Here we may take 1 for the assumed root, the 365th power of which is 1; and n being 365, we have n+1 366, and n−1 = 364. The work will then proceed in the following manner, and the answer is found to be 1·0001596.

In extracting the fourth root, we may either use the preceding rule, or we may extract the second root of the given number, and the second root of the result. In extracting the sixth root also, we may either use