Elements of Algebra

8 ELEMENTS SECT. I. as a rule, that $+$ by $+$ makes $+$ or *plus*; and that, on the contrary, $+$ by $-$, or $-$ by $+$, gives $-$, or *minus*. 33. It remains 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 $+*$. * A farther illustration of this rule is generally given by algebraists as follows: First, we know that $+a$ multiplied by $+b$ gives the product $+ab$; and if $+a$ be multiplied by a quantity less than $b$, as $b-c$, the product must necessarily be less than $ab$; in short, from $ab$ we must subtract the product of $a$, multiplied by $c$; hence $a \times (b-c)$ must be expressed by $ab-ac$; therefore it follows that $a \times -c$ gives the product $-ac$. If now we consider the product arising from the multiplication of the two quantities $(a-b)$, and $(c-d)$, we know that it is less than that of $(a-b) \times c$, or of $ac-bc$; in short, from this product we must subtract that of $(a-b) \times d$: but the product $(a-b) \times (c-d)$ becomes $ac-bc-ad$, together with the product of $-b \times -d$ annexed; and the question is only what sign we must employ for this purpose, whether $+$ or $-$. Now, we have seen that from the product $ac-bc$ we must subtract the product of $(a-b) \times d$; that is, we must subtract a quantity less than $ad$. We have therefore subtracted already too much by the quantity $bd$; this product must therefore be added; that is, it must have the sign $+$ prefixed; hence we see that $-b \times -d$ gives $+bd$ for a product; or $-$ *minus* multiplied by $-$ *minus* gives $+$ *plus*. See Art. 273, 274. Multiplication has been erroneously called a compendious method of performing addition: whereas it is the taking, or repeating of one given number as many times, as the number by which it is to be multiplied, contains units. Thus, $9 \times 3$ means that 9 is to be taken 3 times; or that the measure of multiplication is 3; again $9 \times \frac{1}{2}$ means that 9 is to be taken half a time, or that the measure of multiplication is $\frac{1}{2}$. In multiplication there are two factors, which are sometimes called the multiplicand and the multiplier. These, it is evident, may reciprocally change places, and the product will be still the same: for $9 \times 3 = 3 \times 9$, and $9 \times \frac{1}{2} = \frac{1}{2} \times 9$. Hence it appears, that numbers may be diminished by multiplication, as well as increased, in any given ratio; which is wholly inconsistent with