Part I. Section II. Chapter 1. “Of the Addition of Compound Quantities.”

256 When two or more expressions, consisting of several terms, are to be added together, the operation is frequently represented merely by signs, placing each expression between two parentheses, and connecting it with the rest by means of the sign +. Thus, for example, if it be required, to add the expressions \(a+b+c\) and \(d+e+f\), we represent the sum by

\[(a+b+c)+(d+e+f).\]

257 It is evident that this is not to perform addition, but only to represent it. We see, however, at the same time, that in order to perform it actually, we have onlv to leave out the parentheses; for as the number \(d+e+f\) is to be added to \(a+b+c\), we know that this is done by joining to it first \(+d\), then \(+e\), and then \(+f\); which therefore gives the sum

\[a+b+c+d+e+f;\]

and the same method is to be observed, if any of the terms are affected by the sign -; as they must be connected in the same way, by means of their proper sign.

258 To make this more evident, we shall consider an example in pure numbers, proposing to add the expression 15 - 6 to 12 - 8. Here, if we begin by adding 15, we shall have 12 - 8 + 15; but this is adding too much, since we had only to add 15 - 6, and it is evident that 6 is the number which we have added too much; let us therefore take this 6 away by writing it with the negative sign, and we shall have the true sum,

12 - 8 + 15 - 6;

which shows that the sums are found by writing all the terms, each with its proper sign.

259 If it were required therefore to add the expression \(d-e+f\) to \(a-b+c\), we should express the sum thus,

\[a-b+c+d-e-f;\]

remarking, however, that it is of no consequence in what order we write these terms; for their places may be changed at pleasure, provided their signs be preserved; so that this sum might have been written thus,

\[c-e+a-f+d-b.\]

260 It is evident, therefore, that addition is attended with no difficulty, whatever be the form of the terms to be added: thus, if it were necessary to add together the expressions \(2a^3+6\surd b-4\log c\) and \(5 \sqrt[5]{\vphantom{a}} a-7c\), we should write them

\[2a^3+6\surd b-4\log c+5 \sqrt[5]{\vphantom{a}} a-7c,\]

either in this or in any other order of the terms; for if the signs are not changed, the sum will always be the same.

261 But it frequently happens that the sums represented in this manner may be considerably abridged, as is the case when two or more terms destroy each other; for example, if we find in the same sum the terms \(+a-a\), or \(3a-4a+a\): or when two or more terms may be reduced to one, etc. Thus, in the following examples

\[\begin{array}{lll} 3a+2a=5a,&7b-3b=+4b,&-6c+10c=+4c,\\ 5a-8a=-3a,&-7b+b=-6b,&-3c-4c=-7c,\\ 2a-5a+a=-2a,&-3b-5b+2b=-6b.& \end{array}\]

Whenever two or more terms, therefore, are entirely the same with regard to letters, their sum may be abridged; but those cases must not be confounded with such as these, \(2a^2 + 3a\), or \(2b^3 - b^4\), which admit of no abridgment.

262 Let us consider now some other examples of reduction, as the following, which will lead us immediately to an important truth. Suppose it were required to add together the expressions \(a+b\) and \(a-b\); our rule gives \(a+b+a-b\); now \(a + a = 2a\), and \(b-b=0\); the sum therefore is \(2a\): consequently, if we add together the sum of two numbers \((a + b)\) and their difference \((a - b)\), we obtain the double of the greater of those two numbers.

This will be better understood perhaps from the following examples:

\[\begin{array}{rrr} 3a&-2b&-c\\ 5b&-6c&+a\\ \hline 4a&+3b&-7c \end{array}\] \[\begin{array}{rrrr} a^3&-2a^2b&+2ab^2&\\ &-a^2b&+2ab^2&-b^3\\ \hline a^3&-3a^2b&+4ab^2&-b^3. \end{array}\]

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.