Part I. Section I. Chapter 11. “Of Square Numbers.”

115 The product of a number, when multiplied by itself, is called a square; and, for this reason, the number, considered in relation to such a product, is called a square root. For example, when we multiply 12 by 12, the product 144 is a square, of which the root is 12.

The origin of this term is borrowed from geometry, which teaches us that the contents of a square are found by multiplying its side by itself.

116 Square numbers are found therefore by multiplication; that is to say, by multiplying the root by itself: thus, 1 is the square of 1, since 1 multiplied by 1 makes 1; likewise, 4 is the square of 2; and 9 the square of 3; 2 also is the root of 4, and 3 is the root of 9.

We shall begin by considering the squares of natural numbers; and for this purpose shall give the following small Table, on the first line of which several numbers, or roots, are ranged, and on the second their squares.

Numbers 1 2 3 4 5 6 7 8 9 10 11 12 13
Squares 1 4 9 16 25 36 49 64 81 100 121 144 169

117 Here it will be readily perceived that the series of square numbers thus arranged has a singular property; namely, that if each of them be subtracted from that which immediately follows, the remainders always increase by 2, and form this series;

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, etc.

which is that of the odd numbers.

118 The squares of fractions are found in the same manner, by multiplying any given fraction by itself. For example, the square of ½ is ¼,

the square of ⅓ is ⅑
the square of ⅔ is ⁴⁄₉
the square of ¼ is ¹⁄₁₆
the square of ¾ is ⁹⁄₁₆

We have only therefore to divide the square of the numerator by the square of the denominator, and the fraction which expresses that division will be the square of the given fraction; thus, ²⁵⁄₆₄ is the square of ⅝; and reciprocally, ⅝ is the root of ²⁵⁄₆₄.

119 When the square of a mixed number, or a number composed of an integer and a fraction, is required, we have only to reduce it to a single fraction, and then take the square of that fraction. Let it be required, for example, to find the square of 2½; we first express this number by ⁵⁄₂, and taking the square of that fraction, we have ²⁵⁄₄, or 6¼, for the value of the square of 2½. Also to obtain the square of 3¼, we say 3¼ is equal to ¹³⁄₄; therefore its square is equal to ¹⁶⁹⁄₁₆, or to 10⁹⁄₁₆. The squares of the numbers between 3 and 4, supposing them to increase by one fourth, are as follow:

Numbers. 3 3¼ 3½ 3¾ 4
Squares. 9 10⁹⁄₁₆ 12¼ 14¹⁄₁₆ 16

From this small Table we may infer, that if a root contain a fraction, its square also contains one. Let the root, for example, be 1⁵⁄₁₂; its square is ²⁸⁹⁄₁₄₄ or 2¹⁄₁₄₄; that is to say, a little greater than the integer 2.

130 Let us now proceed to general expressions. First, when the root is \(a\), the square must be \(aa\); if the root be \(2a\), the square is \(4aa\); which shows that by doubling the root, the square becomes 4 times greater; also, if the root be \(3a\), the square is \(9aa\); and if the root be \(4a\), the square is \(16aa\), Farther, if the root be \(ab\), the square is \(aabb\); and if the root be \(abc\), the square is \(aabbcc\); or \(a^2b^2c^2\).

131 Thus, when the root is composed of two, or more factors, we multiply their squares together; and reciprocally, if a square be composed of two, or more factors, of which each is a square, we have only to multiply together the roots of those squares, to obtain the complete root of the square proposed. Thus, 2304 is equal to 4 · 16 · 36, the square root of which is 2 · 4 · 6, or 48; and 48 is found to be the true square root of 2304, because 48 · 48 gives 2304.

132 Let us now consider what must be observed on this subject with regard to the signs + and -. First, it is evident that if the root have the sign +, that is to say, if it be a positive number, its square must necessarily be a positive number also, because + multiplied by + makes +: hence the square of \(+a\) will be \(+aa\): but if the root be a negative number, as \(-a\), the square is still positive, for it is \(+aa\). We may therefore conclude, that \(+aa\) is the square both of \(+a\) and of \(-a\), and that consequently every square has two roots, one positive, and the other negative. The square root of 25, for example, is both +5 and -5, because -5 multiplied by -5 gives 25, as well as +5 by +5.

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.