this principle, that the square of a number ought necessarily to contain twice as many decimals as there are in the root.

THE SQUARE OF FRACTIONS, AND

THE EXTRACTION OF THEIR ROOTS.

To obtain the square of a fraction, we must raise separately its numerator and denominator to their squares, since to square a fraction is to multiply it by itself. We see, then, that to obtain the square root of a fraction, we must take separately the square root of its two terms, and that might always be thus executed without embarrassment, if the fraction proposed were a perfect square.

But, when that is not the case, or rather when the denominator is not a perfect square, we must always render it such, by multiplying the two terms of the fraction by its denominator: then the root is always just within a fractional unit.

But the most simple method, when we are at liberty to employ it, is to express the value of the fraction in decimals, and to extend the division in such a manner, as to have twice as many decimals in the quotient, as we wish to have in the root.

If we have to extract the root of a square, which has an odd number of decimals, such as 45,7, we must supply the deficiency by putting a cypher to the right of the number, which does not change its value, and renders the number of decimals even, which is absolutely necessary, the number proposed being considered as a square.

EXTRACTION OF THE CUBE ROOT.

It is also on the knowledge we have of the composition of a cube, that is founded the method of the extraction of the cube root of any number whatever. But we must first observe, that every cube, which is not expressed by more than three figures, can have but one figure for its root; for 1000, which is the smallest number of 4 figures, has for its cube root 10, which is the smallest number of two figures; thus every number below 1000, will have but one figure for its root; then to find the cube root of a number of three figures and below, there is no necessity of method, we must have recourse to a table which contains the cube of all the simple numbers; the question then is, to find the cube root of a number composed of more than three figures.

This number ought necessarily to contain two figures in its root; let us see, then, how a cube is formed, which contains tens and units in its root; let us suppose this root to be 24; I form at first its square, which gives for the value of Double product

the square,

Square
of the
16

units.

of the tens
160

by the units.

Square
of the

400

tens.

We must, to have the cube, multiply each of the parts of the square, by the tens and the units of the root; beginning with multiplying all the parts of the square, by the units, and then by the tens; let us also set down separately all the partial products; we shall have six products.

I now resume all these products,

by the units.

tens.

beginning by

the right, and put together the 5th and the 3d, as

also the 4th and the 2nd ; thus I see, that a cube is composed of 4 parts; viz. of the cube of the tens; of 3 times the square of the tens by the units; of 3 times the tens by the square of the units; and, lastly, of the cube of the units.]

With this knowledge, we can find the cube root of any number whatever.

thousands, which has three places to its right; therefore if we separate three figures to the right of the number proposed, the part on the left will contain the cube of the tens,

But this number, being itself composed of more than three figures, contains also tens and units in its root, and is, consequently, composed of the cube of the tens, of three times the square of the tens by the units, of &c. therefore, if we cut off three figures towards the right of this number, the part on the left will contain the cube of the tens; and, as this part contains less than 4 figures, I can know the root by means of the table of the cubes of the single figures.

I see that the root of the greatest cube, comprised in this number, is 4; I write this figure as we see, then I subtract its cube from 94, and to the remainder 30 I bring down the next period.

Now I consider, that the number 30897, contains only three parts of the cube, since from the number 94897, we have subtracted the cube of the tens; these three parts, are therefore three times the square of the tens by the units, more three times the tens by the square of the units, and the cube of the units.

the

But three times the square of the tens by the units, is a number of hundreds, which has two places to its right; if, therefore, we separate two figures to the right of the number of which we are speaking, part on the left will contain three times the product of the square of the tens by the units; and, if we divide this number by three times the square of the tens, we shall have the units in the quotient, 48 is three times the square of the tens; we must then make use of it to divide 308.

The most expeditious method of trying the figure of the quotient, is to cube the root, supposed to be known, and to subtract this cube from the number, the root of which it represents; we shall find here, that 5 is the proper quotient; having then subtracted the cube of 45, from the number 94397, we must bring down to the remainder the next period, and continuing to operate, as we have just done, we shall find, that the figure of the units is 6, so that 456 is the cube root of the number proposed, with a remainder of 78768.

If we were desirous of having the root of this number more exactly, within a tenth, or a hundredth &c. we must continue the operation, by placing to the

right of this number, three times as many cyphers as we wish to have decimals in the root, since the cube of a number, ought to contain three times as many decimals as its root.

As to the extraction of the cube roots of fractions, we are to be conducted by the same method of reasoning, and according to the same principles, which we have laid down for obtaining their square root.

If we have to extract the cube root of a number which contains decimals, but which cannot be separated into periods of three decimals, we must supply the deficiency, by putting cyphers to the right hand of the number.