We must premise that, as in all other cases where the first principles of a science have been matter of dispute, it by no means follows that one view of the subject is the most easy to every mind. Something must depend on the intellectual constitution of the individual ; and if this be most probably true in geometry, the remark applies with still greater force to algebra.
The first abstraction which meets us in arithmetic follows the transition from actual magnitudes (concrete numbers, so called) to their numerical representations. We then find general properties of numbers, in which we learn to consider number independently of a specific concrete unit.. Thus we see in 7 + 5-3 = 7-3 + 5 a relation equally true. whatever may be the nature or magnitude of the unit. When we drop the concrete number and rise to the abstract, we gain something more by the transition than immediately appears; and this the student should particularly note, because some of the succeeding difficulties which attend the passage into algebra are very similar in character, though preceded by a stranger and harder process. The operation of multiplication takes a power and a property which it had not before : thus if we denote concrete number by Roman numerals, and if we speak of yards, it is clear that 5 x VII = XXXV, or 7 yards taken 5 times is 35 yards. But we may not, therefore, say that VII x 5 = XXXV, for VII x 5, the number 5 multiplied by 7 yards [MULTI riseartos], Is an incongruous and unmeaning set of words, and it would be equally improper to say that it is and it is not 35 yards. In abstract numbers no such caution is necessary : 7 x 5 and 5 x 7 are both the same. If men had never considered number independently of magnitude measured or repeated by it, the arithmetician would have confounded VII x 5 and 5 x VII, because he would soon have found that no false results would have sprung therefrom; while VII x 5 would have been a sort of impossible quantity, useful in practice and difficult in theory.
We are now on the ground of abstract arithmetic, and on examining the four fundamental opemtions,we see no difficulty in either addition, multiplication, or division. So soon as we have mastered the subject of fractions, and have clearly admitted the introduction of a part of a repetition PIcsvirsicavtos], we say as follows :—Let a and b be any two numbers or fractions, and a + b,. ab, and a : b must be real
numbers or fractions, assignable by demonstrated operations so soon as a and b are assigned. But there is still a restriction upon the possibility of subtraction a—b has no imaginable existence, unless a be greater titan b ; when a = b, the magnitude of a—b vanishes entirely, and when a is lugs than 5, the direction to perform a—b just the same as asking for a part which shall be greater than the whole of which it is a part.. If we confined ourselves to particular arithmetic, in which all numbers used have specific values, it would most likely be thought of no use to carry the subject further, and in one point of view correctly; that is, it would be of little moment to deduce methods by which an individual so careless as to write down and operate upon such a symbol as 3 — 4 might be enabled to arrive at a subsequent correction of the mistake which a glance at the symbol should show him lie luta made. But when wo use general symbols of number, we are liable to mistakes of two kinds, both dependent upon our liability to Invert the order of terms of which the less should be subtaicted from the greater.
First, we may mistake the nature of the quantity which results : thus if it be part of the conditions of a problem that I pay La and receive -Us and if tho application of the conditions requires that should state how much I gain or lose, the answer should be either a lows of £(14—b) or a gain of (.0—a), according as a or b is the greater. We have then the choice between adopting one of these with the chance of being entirely wrong, or of working the problem in two distinct ways. And if it should happen that the conditions of the roblem !regent this alternative in six distinct instances (and (401218 CMG!! it happens oftener), there would be no less than 04 eases of solution, all, arithmetically speaking, essentially different in the mode fr.l obtaining the answer, whether the answers obtained be the same or different.