culation of imaginary quantities is of the greatest importance: questions frequently arise, of which we cannot immediately say, whether they include any thing real and possible, or not. Now, when the solution of such a question leads to imaginary numbers, we are certain that what is required is impossible.*

CHAPTER XIV.

Of Cubic Numbers.

152. WHEN a number has been multiplied twice by itself, or, which is the same thing, when the square of a number has been multiplied once more by that number, we obtain a product which is called a cube, or a cubic number. Thus, the cube of a is a a a, since it is the product obtained by multiplying a by itself, or by a, and that square a a again by a.

The cubes of the natural numbers therefore succeed each other in the following order.

Numbers 1 2 3 4 5

6 7 8 9 10

Cubes

1827641252163435127291000

153. If we consider the differences of these cubes, as we did those of the squares, by subtracting each cube from that which comes after it, we shall obtain the following series of numbers:

7, 19, 37, 61, 91, 127, 169, 217, 271.

At first we do not observe any regularity in them; but if we take the respective differences of these numbers, we find the following

series :

* This is followed in the original by an example intended to illustrate what is here said. It is omitted by the editor, as it implies a degree of acquaintance with the subject, which the learner cannot be supposed to possess at this stage of his progress.

12, 18, 24, 30, 36, 42, 48, 54, 60;

in which the terms, it is evident, increase always by 6.

154. After the definition we have given of a cube, it will not be difficult to find the cube of fractional numbers; is the cube of;

is the cube of; and is the cube of. In the same manner, we have only to take the cube of the numerator and that of the denominator separately, and we shall have as the cube of, for instance, .

27

155. If it be required to find the cube of a mixed number, we must first reduce it to a single fraction, and then proceed in the manner that has been described. To find, for example, the cube of 1, we must take that of, which is 27, or 3 and 3. So the cube of 1, or of the single fraction, is 25, or 181; and the cube of 31, or of 13 is 227, or 341.

2197
64

64

156. Since a a a is the cube of a, that of ab will be a a abb b whence we see, that if a number has two or more factors, we may find its cube by multiplying together the cubes of those factors. For example, as 12 is equal to 3 x 4, we multiply the cube of 3, which is 27, by the cube of 4, which is 64, and we obtain 1728, for the cube of 12. Further, the cube of 2 a is 8 a a a, and consequently 8 times greater than the cube of a: and likewise, the cube of 3 a is 27 a a a, that is to say, 27 times greater than the cube of a.

157. Let us attend here also to the signs + and -. It is evident that the cube of a positive number + a must also be positive, that is aaa. But if it be required to cube a negative number a, it is found by first taking the square, which is + a a, and then multiplying, according to the rule, this square by a, which gives for the cube required ααα. In this respect, therefore, it is not the same with cubic numbers as with squares, since the latter are always positive: whereas the cube of 1 is 1, that of -2 is

CHAPTER XV.

Of Cube Roots, and of Irrational Numbers resulting from them.

158. As we can, in the manner already explained, find the cube of a given number, so, when a number is proposed, we may also reciprocally find a number, which, multiplied twice by itself, will produce that number. The number here sought is called, with relation to the other, the cube root. So that the cube root of a given number is the number whose cube is equal to that given number.

159. It is easy therefore to determine the cube root, when the number proposed is a real cube, such as the examples in the last chapter. For we easily perceive that the cube root of 1 is 1; that of 8 is 2; that of 27 is 3; that of 64 is 4, and so on. And in the same manner, the cube root of - 27 is 3; and that of

343

- 125

Further, if the proposed number be a fraction, as, the cube root of it must be ; and that of is. Lastly, the cube root of a mixed number 21 must be, or 1: because 21 is equal to 4.

160. But if the proposed number be not a cube, its cube root cannot be expressed either in integers, or in fractional numbers. For example, 43 is not a cubic number; I say therefore that it is impossible to assign any number, either integer or fractional, whose cube shall be exactly 43. We may however affirm, that the cube root of that number is greater than 3, since the cube of 3 is only 27; and less than 4, because the cube of 4 is 64. We know, therefore, that the cube root required is necessarily contained between the numbers 3 and 4.

161. Since the cube root of 43 is greater than 3, if we add a fraction to 3, it is certain that we may approximate still nearer and nearer to the true value of this root; but we can never assign the number which expresses that value exactly; because the cube of a mixed number can never be perfectly equal to an integer, such as 43. If we were to suppose, for example, 31, or to be the cube root required, the error would be ; for the cube ofis only 343, or 427.

162. This therefore shows, that the cube root of 43 cannot be expressed in any way, either by integers or by fractions. However,

we have a distinct idea of the magnitude of this root; which induces

3

us to use, in order to represent it, the sign, which we place before the proposed number, and which is read cube root, to distinguish it from the square root, which is often called simply the root.

3

Thus 43 means the cube root of 43, that is to say, the number whose cube is 43, or which, multiplied twice by itself, produces 43.

163. It is evident also, that such expressions cannot belong to rational quantities, and that they rather form a particular species of irrational quantities. They have nothing in common with square roots, and it is not possible to express such a cube root by a square root; as, for example, by 12; for the square of 12 being 12, its cube will be 12 12, consequently still irrational, and such cannot be equal to 43.

3

3

3

164. If the proposed number be a real cube, our expressions become rational; I is equal to 1: 8 is equal to 2; 27 is equal to 3; and, generally, Vaaa is equal to a.

3

3

165. If it were proposed to multiply one cube root, ✔a by another, V5, the product must be ab; for we know that the cube root of a product ab is found by multiplying together the cube roots of the factors (156). Hence, also, if we divideva by Vĩ, the quotient

will be

3

3

3

3

3

166. We further perceive, that 2a is equal to 8a, because 2

3

is equivalent to 8; that 3a is equal to 27a, and ba is equal

3

Vabbb. So, reciprocally, if the number under the radical sign has a factor which is a cube, we may make it disappear by placing its cube

3

root before the sign. For example, instead of 64 a we may write

3

3

3

4a; and 5 va instead of 125 a. Hence 16 is equal to 2√2, because 16 is equal to 8 × 2.

167. When a number proposed is negative, its cube root is not subject to the same difficulties that occurred in treating of square roots. For, since the cubes of negative numbers are negative, it follows that the cube roots of negative numbers are only negative. Thus 8 is equal to - 2, and 27 to - 3. It follows also,

3

3

3

that 12 is the same as-12, and that may be ex

3

pressed by Va. Whence we see that the sign, when it is found after the sign of the cube root, might also have been placed before it. We are not, therefore, here led to impossible, or imaginary numbers, as we were in considering the square roots of negative numbers.

CHAPTER XVI.

Of Powers in general.

168. THE product which we obtain by multiplying a number several times by itself, is called a power. Thus, a square which arises from the multiplication of a number by itself, and a cube which we obtain by multiplying a number twice by itself, are powers. We say also in the former case, that the number is raised to the second degree, or to the second power; and in the latter, that the number is raised to the third degree, or to the third power.

169. We distinguish these powers from one another by the number of times that the given number has been used as a factor. For example, a square is called the second power, because a certain given number has been used twice as a factor; and if a number has been used thrice as a factor, we call the product the third power, which therefore means the same as the cube. Multiply a number by itself till you have used it four times as a factor, and you will have its fourth power, or what is commonly called the bi-quadrate. From what has been said it will be easy to understand what is meant by the fifth, sixth, seventh, &c. power of a number. I only add, that the names of these powers, after the fourth degree, cease to have any other but these numeral distinctions.

170. To illustrate this still further, we may observe, in the first place, that the powers of 1 remain always the same; because whatever number of times we multiply I by itself, the product is found to be always 1. We shall, therefore, begin by representing the powers of 2 and of 3. They succeed in the following order: