SUM AND DIFFERENCE. There is no need to define the arithmetical meaning of these terms : a few words only are necessary to put them in their proper position in algebra. When quantities receive their proper algebraical signs, and those signs their iuterpreta tions [ALGEBRA; SION; ke.], they are said to,be added to a quantity when they are allowed to produce their effect, and Subtracted when they are allowed to produce a contrary effect. And when quantities are put together so that each produces its simple effect, they are said to be added together ; while any parcel which is either withdrawn, or compensated by others of equal and opposite effects, is said to be sub tracted. We are not here discussing principles, but settling terms ; and it is enough if the notions appended to them be proper founda tions for clear and good deduction ; and an additional advantage if common ideas and received phraseology be also suited, provided that nothing be assumed from such ideas and phraseology to the prejudice of the dependence of the deduction upon the prescribed definitions.
To form a just idea of the property of any person, we take the sums which he owes away from his assets; that is, we take away, not his debts, but sums out of his assets equal to his debts. To say that this
is taking away his debts would not be correct ; for taking away his debts would be merely destroying his liabilities, without making his assets answerable : a person who himself pays another's debts takes them away. A court of justice which decides a claim against the assets of any one, annexes or puts on a liability; and this is in algebra adding : if the decision should be reversed on appeal, this liability to pay is removed, and this is in algebra subtracting. In the phrase .‘" to gain a loss," the word " gain " is used in the preceding sense of simple adjunction : if it were as common to talk of losing a loss, the verb to lose would be used in the sense of to remove or to get rid of : the other form of the word would be less of a bull, for to loose a loss would be to detach it. In a third form, the idiom is still plainer ; to release [from] a loss would be precisely the idea of algebra, answering to sub tracting a loss.