Elements of Algebra

78 ELEMENTS SECT. II. $4a^2-3b+2c$ $a^4+2ab+b^2$ $3a^2+2b-12c$ $-a^4-2a^3b+3b^3$ --- --- $7a^2-b+10c$ $-2a^3b+2ab+4b^3$ --- --- CHAP. II. *Of the Subtraction of Compound Quantities.* 263. If we wish merely to represent subtraction, we enclose each expression within two parentheses, joining, by the sign $-$, the expression which is to be subtracted, to that from which we have to subtract it. When we subtract, for example, the expression $d-e+f$ from the expression $a-b+c$, we write the remainder thus: $(a-b+c)-(d-e+f)$; and this method of representing it sufficiently shews which of the two expressions is to be subtracted from the other. 264. But if we wish to perform the actual subtraction, we must observe, first, that when we subtract a positive quantity $+b$ from another quantity $a$, we obtain $a-b$: and secondly, when we subtract a negative quantity $-b$ from $a$, we obtain $a+b$; because to free a person from a debt is the same as to give him something. 265. Suppose now it were required to subtract the expression $b-d$ from $a-c$. We first take away $b$, which gives $a-c-b$: but this is taking away too much by the quantity $d$, since we had to subtract only $b-d$; we must therefore restore the value of $d$, and then shall have $a-c-b+d$; whence it is evident that the terms of the expression to be subtracted must change their signs, and then be joined, with those contrary signs, to the terms of the other expression. 266. Subtraction is therefore easily performed by this rule, since we have only to write the expression from which we are to subtract, joining the other to it without any change beside that of the signs. Thus, in the first example, where it was required to subtract the expression $d-e+f$ from $a-b+c$, we obtain $a-b+c-d+e-f$. An example in numbers will render this still more clear;