Part I. Section I. Chapter 3. “Of the Multiplication of Simple Quantities.”

23 When there are two or more equal numbers to be added together, the expression of their sum may be abridged: for example,

\(a + a\) is the same with \(2 \cdot a\),

\(a+a+a\) is the same with \(3 \cdot a\),

\(a+a+a+a\) is the same with \(4 \cdot a\), and so on.

In this manner we may form an idea of multiplication; and it is to be observed that,

\(2 \cdot a\) signifies 2 times \(a\)

\(3 \cdot a\) signifies 3 times \(a\)

\(4 \cdot a\) signifies 4 times \(a\), etc.

24 If therefore a number expressed by a letter is to be multiplied by any other number, we simply put that number before the letter, thus;

\(a\) multiplied by 20 is expressed by \(20a\), and

\(b\) multiplied by 30 is expressed by \(30b\), etc.

It is evident, also, that \(c\) taken once, or \(1c\), is the same as \(c\).

25 Farther, it is extremely easy to multiply such products again by other numbers; for example:

2 times \(3a\), makes \(6a\)

3 times \(4b\), makes \(12b\)

5 times \(7x\) makes \(35x\),

and these products may be still multiplied by other numbers at pleasure.

26 When the number by which we are to multiply is also represented by a letter, we place it immediately before the other letter; thus, in multiplying \(b\) by \(a\), the product is written \(ab\); and \(pq\) will be the product of the multiplication of the number \(q\) by \(p\). Also, if we multiply this \(pq\) again by \(a\), we shall obtain \(apq\).

27 It may be farther remarked here, that the order in which the letters are joined together is indifferent; thus \(ab\) is the same thing as \(ba\); for \(b\) multiplied by \(a\) is the same as \(a\) multiplied by \(b\). To understand this, we have only to substitute, for \(a\) and \(b\), known numbers, as 3 and 4; and the truth will be self-evident; for 3 times 4 is the same as 4 times 3.

28 It will not be difficult to perceive, that when we substitute numbers for letters joined together, in the manner we have described, they cannot be written in the same way by putting them one after the other. For if we were to write 34 for 3 times 4, we should have 34 and not 12. When therefore it is required to multiply common numbers, we must separate them by a point: thus, 3 ⋅ 4, signifies 3 times 4; that is, 12. So, 1 ⋅ 2 is equal to 2; and 1 ⋅ 2 ⋅ 3 makes 6. In like manner, 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5 ⋅ 6 makes 1344; and 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5 ⋅ 6 ⋅ 7 ⋅ 8 ⋅ 9 ⋅ 10 is equal to 3628800, etc.

29 In the same manner we may discover the value of an expression of this form, \(5 \cdot 7 \cdot 8 \cdot abcd\). It shows that 5 must be multiplied by 7, and that this product is to be again multiplied by 8; that we are then to multiply this product of the three numbers by \(a\), next by \(b\), then by \(c\), and lastly by \(d\). It may be observed, also, that instead of 5 ⋅ 7 ⋅8, we may write its value, 280; for we obtain this number when we multiply the product of 5 by 7, or 35, by 8.

30 The results which arise from the multiplication of two or more numbers are called products; and the numbers, or individual letters, are call factors.

31 Hitherto we have considered only positive numbers, and there can be no doubt, but that the products which we have seen arise are positive also: namely \(+a\) by \(+b\) must necessarily give \(+ab\). But we must separately examine what the multiplication of \(+a\) by \(-b\), and of \(-a\) by \(-b\), will produce.

32 Let us begin by multiplying \(-a\) by 3 or +3. Now, since \(-a\) may be considered as a debt, it is evident that if we take that debt three times, it must thus become three times greater, and consequently the required product is \(-3a\). So if we multiply \(-a\) by \(+b\), we shall obtain \(-ba\), or, which is the same thing, \(-ab\). Hence we conclude, that if a positive quantity be multiplied by a negative quantity, the product will be negative; and it may be laid down as a rule, that + by + makes + or plus; and that, on the contrary, + by - , or - by +, gives -, or minus.

33 It remams to resolve the case in which - is multiplied by -; or, for example, \(-a\) by \(-b\). It is evident, at first sight, with regard to the letters, that the product will be \(ab\); but it is doubtful whether the sign +, or the sign -, is to be placed before it; all we know is, that it must be one or the other of these signs. Now, I say that it cannot be the sign -: for \(-a\) by \(+b\) gives \(-ab\), and \(-a\) by \(-b\) cannot produce the same result as \(-a\) by \(+b\) but must produce a contrary result, that is to say, \(+ab\); consequently, we have the following rule: - multiplied by - produces +, that is, the same as + multiplied by +.

34 The rules which we have explained are expressed more briefly as follows:

Like signs, multiplied together, give +; unlike or contrary signs give -. Thus, when it is required to multiply the following numbers; \(+a\), \(-b\), \(-c\), \(+d\); we have first \(+a\) multiplied by \(-b\), which makes \(-ab\); this by \(-c\), gives \(+abc\); and this by \(+d\), gives \(+abcd\).

35 The difficulties with respect to the signs being removed, we have only to show how to multiply numbers that are themselves products. If we were, for instance, to multiply the number \(ab\) by the number \(cd\), the product would be \(abcd\), and it is obtained by multiplying first \(ab\) by \(c\), and then the result of that multiplication by \(d\). Or, if we had to multiply 36 by 12; since 12 is equal to 3 times 4, we should only multiply 36 first by 3, and then the product 108 by 4, in order to have the whole product of the multiplication of 12 by 36, which is consequently 432.

36 But if we wished to multiply \(5ab\) by \(3cd\), we might write \(3cd 5ab\). However, as in the present instance the order of the numbers to be multiplied is indifferent, it will be better, as is also the custom, to place the common numbers before the letters, and to express the product thus: \(5 \cdot 3 abcd\), or \(15abcd\); since 5 times 3 is 15.

So if we had to multiply \(12pqr\) by \(7xy\), we should obtain \(12\cdot 7pqrxy\), or \(84pqrxy\).

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.