Further, if the proposed number be a fraction, as 7, the cube root of it must be ; and that of is 4. Lastly, the cube root of a mixed number 21 must be, or 13: because 219 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, 34, or to be the cube root required, the error would be; for the cube of is only 33, or 427.

162. This therefore shews, 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;

3

which induces 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

3

simply the root. 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.

1

164. If the proposed number be a real cube, our expressions

3

3

3

become rational; V is equal to 1 ; Võ is equal to 2; √27 is

3

equal to 3; and, generally, Va a a is equal to a.

3

3

165. If it were proposed to multiply one cube root, va, by another,

3

√, 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 divide ✔ã by Vĩ, the quo

tient will be

3

3

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

2 is equivalent to 8, that 3 is equal to 27a, and b is

3

equal to 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 root before the sign. For example, instead of 6 we may write 4; and 5 instead of 125a.

3

3

Hence 16 is equal to 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, that—12 is the same as 12, and that ✔―

may be expressed by a. 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 1 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:

But the powers of the number 10 are the most remarkable; for on these powers the system of our arithmetic is founded. A few of them arranged in order, and beginning with the first power, are as follows :

10.

III.

100, 1000,

10000, 100000, 1000000, &c.

171. In order to illustrate this subject, and to consider it in a more general manner, we may observe, that the powers of any number, a, succeed each other in the following order.

But we soon feel the inconvenience attending this manner of writing powers, which consists in the necessity of repeating the same letter very often, to express high powers; and the reader also would have no less trouble, if he were obliged to count all the letters, to know what power is intended to be represented. The hundredth power, for example, could not be conveniently written in this manner; and it would be still more difficult to read it.

172. To avoid this inconvenience, a much more commodious method of expressing such powers has been devised, which from

its extensive use deserves to be carefully explained; viz. To express, for example, the hundredth power, we simply write the number 100 above the number whose hundredth power we would express, and a little towards the right-hand; thus a 100 means a raised to 100, and represents the hundreth power of a. It must be observed, that the name exponent is given to the number written above that whose power, or degree, it represents, and which in the present instance is 100.

173. In the same manner, a2 signifies raised to 2, or the second power of a, which we represent sometimes also by aas because both these expressions are written and understood with equal facility. But to express the cube, or the third power aaa, we write as according to the rule, that we may occupy less room. So a signifies the fourth, as the fifth, and a the sixth power of a.

174. In a word, all the powers of a will be represented by a, a3, a3, a*, u3, ao, a”, a3, ao, a10, &c. Whence we see that in this manner we might very properly have written a instead of a for the first term, to shew the order of the series more clearly. In fact a1 is no more than a, as this unit shews that the letter a is to be written only once. Such a series of powers is called also a geometrical progression, because each term is greater by one than the preceding.

175. As in this scries of powers each term is found by multiplying the preceding term by a, which increases the exponent by 1; so when any term is given, we may also find the preceding one, if we divide by a, because this diminishes the exponent by 1. This shews that the term which precedes the first term a1 must necessarily be, or 1; now, if we proceed according to the exponents, we immediately conclude, that the term which precedes the first must be ao. Hence we deduce this remarkable property; that a° is constantly equal to 1, however great or small the value of the number a may be, and even when a is nothing; that is to say, a° is equal to 1.

176. We may continue our series of powers in a retrograde order, and that in two different ways; first, by dividing always by a, and secondly by diminishing the exponent by unity. And Eul. Alg.