should only have six figures for the whole number; this root therefore is 873901,342, to within a thousandth.

We could, in general, carry the division one figure farther, that is to say, as far as the number of figures in the root al ready found; but there are some cases, though rare, where the error in the last figure would be five units; instead of which, in limiting ourselves to one figure less, as we have done, the last figure will never vary a unit.

If, after we have found the first figures of the root, by the ordinary method, the remainder be found equal to the double of the first figures, we must, to avoid all embarrassment, determine one more figure by the ordinary method, after which, we find the others by the above method, which, as we see plainly, applies also to decimals.

If any of the intermediate figures of the root should be ciphers, in the case where these ciphers should be among the number of the figures which we find by division, it might happen, if they ought to be the first figures of the quotient, that we should not perceive it, because in the division we do not mark the ciphers which ought to precede on the left of the quotient; we obtain these ciphers by observing that we ought always to have as many figures in the quotient as we have placed on the right of the remainder; and, consequently, when there are not enough, we must complete the number by ciphers placed on the left of this quotient.

The abridgment, we have here given, is a consequence of this general principle, which is easy to deduce from what we have seen,(134) namely, that the square of any quantity whatever, composed of two parts, contains the square of the first part, twice the first part multiplied by the second, and the square of the second.

EXAMPLES FOR PRACTICE,

1. What is the square root of 16641?

Ans. 129.

2. What is the square root of 549,9025? Ans. 23,45. 3. What is the square root of 8532769? Ans. 2921,09+. 4. What is the square root of 12674062,804225?

Ans. 3560,065.

OF THE FORMATION OF CUBE NUMBERS, AND THE
EXTRACTION OF THEIR ROOTS.

149. To form what is called the cube of a number, we must first multiply that number by itself, and afterward multiply by that same number the product resulting from this first multiplication.

Thus, the cube of a number is, properly speaking, the product of the square of the number multiplied by that same number: 27 is the cube of 3, because it results from the multiplication of 9, the square of 3, by the same number 3.

The number which we cube is then three times factor in the cube; it is for this reason that the cube is also called the third power, or third degree of this number.

150. Universally, we say that a number is raised to its second, third, fourth, fifth, etc. power, when we have multiplied it into itself 1, 2, 3, 4, etc. times successively, or when it is twice, 3 times, 4 times, 5 times, etc, factor in the product.

Note. The numbers 2, 3, 4, 5, etc. are placed over a number to signify the second, third, fourth, fifth, etc. power of that number. Thus, 339; 33-27; 3481, etc.

151. The cube root of a proposed cube is the number, which, multiplied by its square, produces this cube; thus, 3 is the cube root of 27.

152. We have no need then of rules to form the cube of a number; but to return from the cube to its root we must have a method. We shall deduce this method from the examen of what takes place in the formation of a cube.

Let us observe, however, that we have no need of a method to extract the cube root in whole numbers, but when the proposed number has more than three figures; for 1000 being the cube of 10, every number under 1000, and, consequently, of less than four figures, will have for its root a num ber less than 10, that is to say, less than two figures.

Therefore, any number which shall fall between any two of these

1, 8, 27, 64, 125, 216, 343, 512, 729,

will have its cube root, in integers, between the two corresponding numbers of this suite :

1, 2, 3, 4, 5,
5, 6,

7, 8, 9, of which the first contains the cubes.

153. Every number has not a cube root; but we can approach continually to a number which, being cubed, approaches also nearer and nearer to the proposed number; this we shall see after having learned to find the root of a perfect cube.

154. Let us see then of what parts the cube of a number is composed which contains tens and units.

Since the cube results from the square of a number multiplied by this same number, it is essential to recollect here (134) that the square of a number composed of tens and units, contains 1. the square of the tens ; 2. twice the product of the tens by the units; 3. the square of the units.

To form the cube, we must then multiply these three parts by the tens and by the units of this same number.

To the end that we may perceive more distinctly the products which will thence result, let us give to this feigned operation the following form:

Therefore, in collecting these six results and uniting those which are alike, we see that the cube of a number composed of tens and units, contains four parts, namely—the cube of the tens, three times the square of the tens multiplied by the units, three times the tens multiplied by the square of the units, and lastly, the cube of the units.

Let us form, from this, the cube of a number composed of tens and units, of 43, for example:

64000

14400

1080

27

79507

We will then take the cube of 4, which is 64; but as this 4 is tens, its cube will be thousands; because 103=1000 % therefore the cube of the four tens will be 64000.

3X16, or three times the square of the 4 tens, being multiplied by the three units, will give 144 hundreds, because the square of 10 is 100; therefore the product will be

14400.

3 times 4, or three times the tens, being multiplied by the square 9 of the units, will give tens; hence this product will be 1080.

Lastly, the cube of the units will terminate in the place of units, and will be 27. Uniting these four parts, we shall have 79507 for the cube of 43, which cube we should without doubt have found more easily in multiplying 43 by 43, and the product 1849 again by 43; but it is not so much the question here to find the value of the cube, as to understand

by the examen of the parts which compose it, the method of returning to its root.

155. This established, see the process of the extraction of the cube root.

EXAMPLE I.

Let it be proposed then to extract the cube root of 79507.

To have the part of this number which contains the cube of the tens of the root, I separate the three last figures of it, in which we have seen that this cube cannot be contained, since it is thousands.

I seek the cube root of 79; it is 4, which I write in the root. I cube 4, and subtract the product 64 from 79; there remains 15, which I write under 79.

To the side of 15 I bring down 507, which gives me 15507, in which there should be 3 times the square of the 4 tens found, multiplied by the units which we seek, plus 3 times these same tens multiplied by the square of the units, plus finally, the cube of the units.

I separate the two last figures 07; the part 155 which remains on the left contains 3 times the square of the tens multiplied by the units; for this reason, in order to have the units, (74) I shall divide this 155, by the triple square of the 4 tens, that is to say by 48. I find that 48 is contained three times in 155; I write therefore 3 in the root.

To prove this root, and find the remainder, if there be any, we could compose the three parts which should be found in 15507, and see if they form 15507, or by how much they differ from it; but it is as commodious to make this verification in directly cubing 43, that is to say, in multiplying 43 by 43, which produces 1849, and multiplying this product by 43, which gives finally 79507.

Therefore, 43 is exactly the cube root.

If the proposed number have more than six figures, we shall reason as in the following example: