Elements of Algebra

CHAP. XIV. OF ALGEBRA. 45 CHAP. 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 $aaa$, since it is the product obtained by multiplying $a$ by itself, or by $a$, and that square $aa$ 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. | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1000 153. If we consider the differences of those cubes, as we did of the squares, by subtracting each cube from that which comes after it, we obtain the following series of numbers: 7, 19, 37, 61, 91, 127, 169, 217, 271. Where we do not at first observe any regularity in them; but if we take the respective differences of these numbers, we find the following series: 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 cubes of fractional numbers; thus, $\frac{1}{8}$ is the cube of $\frac{1}{2}$; $\frac{1}{27}$ is the cube of $\frac{1}{3}$; and $\frac{8}{27}$ is the cube of $\frac{2}{3}$. In the same manner, we have only to take the cube of the numerator and that of the denominator separately, and we shall have $\frac{64}{125}$ for the cube of $\frac{4}{5}$. 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 $1\frac{1}{2}$, we must take that of $\frac{3}{2}$, which * We are indebted to a mathematician of the name of J. Paul Buchner, for Tables published at Nuremberg in 1701, in which are to be found the cubes, as well as the squares, of all numbers from $1$ to $12000$. F. T.