third power, or cube of 3, is 3×3×3 = 27; that is, the cube of 3 is 27: that the fourth power of 3 is 3×3×3×3 = 81; that the fifth power is 3 × 3 × 3 × 3 × 3 = 243; and so on, to any extent. You see that it would soon become inconvenient and tedious to repeat the equal factors in this way, when high powers are to be indicated; and to avoid this, a very neat and brief form of notation for powers has been devised the number to be involved or raised to the proposed power, is simply written down; and then, over the right-hand upper corner of it, is placed, in smaller type, another number, expressing the power intended: thus, 32 indicates the square, or second power of 3; 33 indicates the cube, or third power of 3; 34 indicates the fourth power of 3; 38 the eighth power of 3; 312 the twelfth power; and so on. The number which, involved in this way, produces a power, is called the root of that power: thus, 3 is the square-root of 9; it is the cuberoot of 27; the fourth root of 81; and so on: the notation for a root is the symbol, which, when no small figure is attached to it, implies, simply, the square-root; when the cube-root is intended, a little 3 is connected with the symbol, thus; when the fourth root is meant, a little 4 is used, thus; and so on. You will now have no difficulty in making out the meaning of the following statements.

Since 329. √√9=3; since 33 = 27 :. 3/27=3; since 3481../81=3; &c.

Since 5225.25 = 5; since 4364 .. 3/64 = 4; since 742401.4/2401=7; &c.

(95.) There is one thing that must occur to you in looking over these particulars: it is this,-that although the power proposed is very easily got from knowing the root, yet that it is not so easy to discover the root when we know only the power; thus, you would find it no easy matter to get the fourth root of 2401, had you not previously seen that it was produced by successive multiplications of 7 by itself. This reverse operation, by which any proposed root of a number is found, is called Evolution. The process of Involution is uniform, whatever be the power to which a number is to be raised; but it is not so with Evolution: the rule for the square-root would help you but little towards finding the cube-root, the fifth root, &c. I am now going to show you how the square-root of a number is to be found, and afterwards how the cube-root is to be found; but I must previously tell you that comparatively few numbers are really

squares or cubes; that is, numbers actually produced by squaring or cubing other numbers. We cannot, of course, find by rules what does not strictly exist; but by aid of decimals we can obtain approximate square-roots, and cuberoots of all numbers: that is, by applying our general rules we can, by means of decimals, obtain a number which, when squared or raised to the second power, shall produce a number differing from the number whose square-root is required by a fraction or decimal as small as we please. And we can also obtain a number which, when cubed or raised to the third power, shall produce a number differing from the number whose cube-root is required by a decimal as small as we please; so that such square and cube roots may be taken as true square and cube roots, without any sensible error. Decimals are very useful in all calculations, where approximate values only are attainable. The following is the rule for finding the square-root accurately of a number, whenever that number is strictly a square, and for finding the square-root approximately to any extent of decimals when the number is not an exact square.

(96.) To extract the Square-root of a Number.

RULE 1. Prepare the number for the operation thus:Commencing from the decimal point, mark off the two final figures of the integral portion of the number; then the two figures which precede them; then the two before these; and so on, cutting up the integers in this way into as many periods, of two figures each, as you can: and, returning to the decimal point, mark off pairs of decimals, proceeding from left to right, in the same way. As we are at liberty to put a 0 at the end of the given decimals, we may always make the number of them even; so that the decimals will consist of complete periods, without any odd figure over; but if the integers be odd in number, then, besides the periods of two figures each, there will be the leading figure standing singly; this leading figure, however, is still called the first period.

2. Attend only to the first period, and find the greatest number whose square does not exceed the number in that period. This can never be matter of the slightest difficulty; for as the period can never be a number of more than two figures, it will be very easy to see which of the nine digits multiplied by itself approaches nearest to it. The greatest

number thus found is the first figure of the root: write it in the place in which you would put it if it were the first figure of a quotient; that is, to the right of the proposed number; subtract the square of it from the first period, and to the remainder annex the second period, and you will have a number which may be called the first dividend.

3. After this leading step the operation assumes a new form. To the left of the first dividend mark off a place for the corresponding first divisor, which you find thus:-Put twice the root-figure just found in the divisors place; the leading figure, or if the root-figure exceed 4, the two leading figures of the required divisor will thus be found, and you will now have a dividend, and the leading figure or figures of its divisor, to find the corresponding quotient-figure; and you know, from common division, that it is mainly the leading figures of a divisor which suggest the first figure of the quotient. Find then the quotient-figure from this incomplete divisor; the quotient-figure thus found forms the second figure of the root, and, annexed to the incomplete or trial divisor, it renders it complete; you have only then, as in division, to multiply the complete divisor by the figure just found, to subtract the product from the dividend, and to annex to the remainder the third period; you will thus have the second dividend.

4. Proceed step after step in this way, writing against every dividend twice the number formed by the root-figures previously found; you will thus always get the incomplete or trial divisor belonging to that dividend, and thence a new root-figure; with which, as before, the incomplete divisor is to be completed.

You know that in common division the quotient-figure suggested by the leading figures of a divisor is not always the true quotient-figure; for we cannot always foresee the full influence of the carryings. So here, the root-figure, suggested by an incomplete divisor, may prove to be erroneous when that divisor is completed, and the multiplication of it by the figure under trial executed. In such a case we do exactly as we would in common division. (See page 30.) An example worked at length will sufficiently illustrate the rule.

5,67,82, 43,20(238.2906

4

43)167

129

468) 3882

3744

Ex. 1. Extract the square-root of 56782-432. Here, the proposed number divided into periods of two figures each, as the first precept of the rule directs, is 5,67,82, 43,20, and the first root-figure is 2, this being the greatest number, whose square (4) does not exceed the first period, 5: the square of this 2, subtracted from the first period, leaves for remainder 1, which becomes 167, when the next period is brought down. Hence, 167 is the first dividend, and 4, the double of the root-figure, is the first trial, or incomplete divisor. Looking at this 4, in reference to the 16 in the dividend, 4 is suggested as

4762)13843

9524

47649)431920

428841

4765806)30790000

28594836

2195164, &c.

the quotient-figure; but, foreseeing that unit would have to be carried, we know that 4 will be too great. Putting, therefore, 3 for the second root-figure, and the same 3 against the incomplete divisor, we proceed, as in division, and obtain the second remainder, 38, which, when the next period is brought down, becomes 3882, the second dividend. Doubling the 23, the number formed by the root-figures already found, we have 46 for the incomplete divisor of 3882; so that the corresponding quotient-figure-that is, the third root-figure

is 8, which placed against the 46, gives 468 for the true divisor: hence, the third remainder is 138, and another period being brought down, the third dividend is 13843. The trial-divisor of this, that is, the double of the root, thus far found, is 476; and, therefore, the quotient-figure, that is, the fourth root-figure, is 2, and, therefore, the complete divisor is 4762. The next dividend is 431920, and the trialdivisor belonging to it is 4764, so that 9 is the fifth figure of the root completing the divisor with this 9, we get the next remainder, 3079, which, with another period brought down, namely, 00, gives the next dividend, 307900, the incomplete divisor of which, the double of 23829, is 47658. It is

obvious, from the leading figure, that this, when completed, will be greater than the dividend: hence, the next rootfigure is 0; and the next dividend, formed by annexing another period of zeros, is 30790000, the corresponding incomplete divisor being 476580, the double of the root, so far as it goes. The incomplete divisor gives 6 for the next root-figure, and 4765806 for the complete divisor; and, by bringing down zero periods in this way, step after step, we may extend the root to as many decimal places as we please. In the work in the margin, it has been carried to four places of decimals, and the number 238.2906, thus determined, is said to be the approximate square-root of 56782-432; and the decimals are all true, as far as these four places. Nevertheless, as more decimals would follow if we were to continue the work, the final figure, 6, needs a fractional correction; we know that it is too small, by some fraction, or decimal of a unit, in the fourth place. On this account, we could not expect to recover the proposed number exactly, by squaring this incomplete root.

238.2906

6092832

4765812

713872

190632

4766

2144

From the principles already taught (page 120), you know that if you were to multiply 238.2906 by itself, the result could not be depended upon beyond the first decimal; for, as there are seven figures in each factor, and but eight decimals in the complete product, the number of decimals to be depended upon is only 8-7 = 1, the other seven decimals being necessarily inaccurate. You must always keep in remembrance the influence of this error, in the last decimal of an approximate result, whenever you have occasion to use it in multiplication or division; and be careful to avoid the common mistake of supposing that, because you have got the square-root of a number true to several places of decimals, that the square of that root can be true to anything like the same extent. The square of the root just obtained, as far as one decimal place, which is all that can be safely depended on, is found by contracted multiplication, as in the margin.* If you take the trouble to multiply the root by itself, without any contraction, you will find the product to be 56782-41004836. By extending the decimals of the

* See note, page 125.

14

55782.4