Part I. Section I. Chapter 14. “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, ⅛ the cube of ½; ¹⁄₂₇ is the cube of ⅓; and ⁸⁄₂₇ is the cube of ⅔. In the same manner, we have only to take the cube of the numerator and that of the denominator separately, and we shall have ²⁷⁄₆₄ for the cube of ¾.

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 1½, we must take that of ³⁄₂, which is ²⁷⁄₈ or 3⅜; also the cube of 1¼, or of the single fraction ⁵⁄₄, is ¹²⁵⁄₆₄, or 1⁶¹⁄₆₄; and the cube of 3¼, or of ¹³⁄₄, is ²¹⁹⁷⁄₆₄, or 34²¹⁄₆₄.

156 Since \(aaa\) is the cube of \(a\), that of \(ab\) will be \(aaabbb\); 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 · 4, we multiply the cube of 3, which is 27, by the cube of 4, which is 64, and we obtain 1728, the cube of 12 ; and farther, the cube of \(2a\) is \(8aaa\), and consequently 8 times greater than the cube of \(a\): likewise, the cube of \(3a\) is \(27aaa\); 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 \(+aa\), and then multiplying, according to the rule, this square by \(-a\), which gives for the cube required \(-aaa\). 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 -8, that of -3 is -27, and so on.

Editions

  1. Leonhard Euler. Elements of Algebra. Translated by Rev. John Hewlett. Third Edition. Longmans, Hurst, Rees, Orme, and Co. London. 1822.
  2. Leonhard Euler. Vollständige Anleitung zur Algebra. Mit den Zusätzen von Joseph Louis Lagrange. Herausgegeben von Heinrich Weber. B. G. Teubner. Leipzig and Berlin. 1911. Leonhardi Euleri Opera omnia. Series prima. Opera mathematica. Volumen primum.