in turn leads to 'the notion which is really the fundamental one... underlying and pervading the whole of modern analysis that of imaginary magnitude. . . .'

It is this question, and the further analysis of elementary algebraic forms to which it gives rise, which constitute the subject-matter of the next chapter.

CHAPTER VIII

THE CONCEPTIONS AND SYMBOLISM OF ELEMENTARY ALGEBRA (continued)

Analysis of the relations implied by the use of the correlative terms Power and Root in Arithmetic and in Algebra.-The actual use of the power-index is inconsistent with the implied definition of Power and Root in Algebra.-But this inconsistency is convenient because it confers brevity and symmetry upon the symbolic system; and its real sanction lies in its arithmetical interpretability.-The convention, once admitted, leads by strict analogy to a similar use of the root-index, and suggests the pseudo-concept of Imaginary Quantity.—The textbook explanation of Imaginary Quantity.-The sophisms which this explanation involves.— Recapitulation of the two ways of interpreting the development of symbolism in Elementary Algebra.-Argand's geometrical representation of Imaginary Quantities.

'In addition to the four species (of arithmetical operation) it is usual, even in arithmetic, to introduce another pair of mutually inverse operations, viz. Involution (Raising to a Power), and Evolution (Radication or Root Extraction). In the first instance, at least, these new operations are not independent of those already enumerated. Involution is, in fact, repeated multiplication: thus 3, 3 × 3, 3 × 3 × 3, 3 × 3 × 3 × 3, . . . are represented by 31, 32, 33, 34, . and are described as three to the first power, three to the second power or three square, three to the third power or three cube, three to the fourth power, . . . and in general a×a×a... (n factors), n being of course an arithmetical integer, is contracted into an. The quantity whose nth power is a is called the nth root of a, and is denoted by a, . . .' î

So far as arithmetical quantities are concerned we have here a clear explanation of the numerical relations which are intended to be conceived under the correlative terms 'root' and 'power'. But I am unable to find, anywhere in the textbook under consideration, an explicit definition of the relations intended to be conceived when the terms 'root' and 'power' refer to algebraic quantity. I suppose, however, that the definition would run somewhat as follows: An algebraic quantity is termed a power in relation to another algebraic quantity, which is termed 1 Introduction to Algebra, p. 5.

a root, when the former arises from the involution of the latter; or, conversely, an algebraic quantity is termed a root in relation to another algebraic quantity, which is termed a power, when the former arises from the evolution of the latter.

This definition, however, is of no use to us unless we know the exact sense in which the words 'involution' and 'evolution' are used in relation to algebraic quantity. But if, in order to find this out, we turn to the Index of Technical Words and Phrases given in the beginning of the book, we find ourselves referred to p. 5, that is, to the explanation already quoted, in which the meanings of the terms 'involution' and' evolution' are explained only in relation to arithmetical quantity.

In these circumstances we must suppose that what algebraists precisely conceive under the correlatives 'root' and 'power', when these terms refer to algebraic quantity, is intended to be revealed by the study of the contexts in which these terms appear, by the use to which they put indices or exponents of power and the radical sign, and by the manner in which they combine these new conventions with the others, especially with the rule of signs. Let us take the simplest examples possible, and, using the current phraseology of algebraic exposition, consider how far they reveal a clear process of conceiving and of symbolizing.

First, according to the convention of indices, so far as actually explained, we have

ax a = a2

(I)

The same convention, if we extend it to algebraic quantities gives

Now it is the actual practice of algebraists to equate (4), (5), and (6) respectively to +a2, +a2, -a2. I suppose the explanation of this to be that by equation (1) we may substitute a2 for (axa) in (4), (5), and (6). But it will be seen, by comparing (2) and (3) with (4), (5), and (6), that in effecting this substitution

we introduce an ambiguity in the use of the index. That is to say, an index being an exponent of power, and the terms power and root being used as correlatives, we fail so far to form a definite and stable conception of the relation implied by these correlative terms as applied to algebraic quantity. For in (2) and (3), constructed analogically from (1), the index 2 clearly refers to the number of algebraic quantities' (+a) or (-a); but in (4), (5), and (6), when equated to +a2, +a2, -a2, the index appears to refer to the number of arithmetical quantities a, and in (6) certainly cannot refer to two ' algebraic quantities', +a, or -a.

Now if, on the one hand, it is agreed that the numbers 1, 2, 3,... n, employed in this particular manner, shall indicate 1, 2, 3, . . . n identical algebraic quantities multiplied together', or, in other words, shall be indices of the successive orders of power of the identical quantity or root-an agreement which definitely fixes the relation intended to be conceived in the correlation of the terms 'root' and 'power', as predicated of algebraic quantity-then it is inconsistent to transform the expression (+a) (-a) into a2, since the number 2, thus used, is by definition an index or exponent of power, that is, designates an algebraic quantity which stands to some other algebraic quantity in the relation of power to root, which is not here the case, since +a and -a are not identical. If, on the other hand, it be agreed that integers, thus used, are to indicate the number of arithmetical quantities multiplied together, then the relation conceived under the correlated terms 'power' and 'root' is an arithmetical relation, and it is meaningless to speak of an algebraic quantity as a power or as a root.

But algebraists do speak and write, throughout algebra, of roots and powers as algebraic quantities; thus, notwithstanding the inconsistency which attends their employment of indices, we cannot doubt that the relation which they conceive between two algebraic quantities which they respectively term root and power is that indicated in the first of these alternatives. But let me add here that I am far from asserting that a use of indices which is inconsistent with the nature of the relation conceived is necessarily a vicious use. It need not be so, provided we are clearly aware of the inconsistency, and deliberately accept it for the sake of the brevity and symmetry it confers upon the system. The danger to clear thinking, and the tendency to

pseudo-conceiving, lie in not being aware of the inconsistency, or in shutting our eyes to it, for this is to remain defenceless against a prejudicial reaction of symbolic forms on the process of conception.

We may at this point pause to recall for a moment Mr. Whitehead's explanation of the enigma involved in the doctrine of imaginaries in algebra-especially the view that the laws of algebra, though suggested by arithmetic, are independent of it, and that once algebra is conceived as an independent science dealing with the relations of certain marks conditioned by the observance of certain conventional laws, the difficulty vanishes. In my criticism of this view I remarked that so to conceive algebra is just as great a difficulty as the difficulty it is supposed to remove, because no intelligible account can then be given of the invention, the raison d'être, of the relations, conditions, and laws or conventions affecting the symbols or marks. But it seems to be at least involved in this conception of algebra that its laws develop in self-consistency, however obscure the motive or cause of this development may be. The inconsistency in development which has just been pointed out seems, then, to be absolutely fatal to this view. The symbolism of algebra develops here in defiance of logical consistency; and the real sanction of this inconsistency in symbolism plainly is that it is arithmetically intelligible or interpretable.

The dilemma in which the orthodox exposition of root and power leaves us is quite clear. We cannot escape from the choice between two equally embarrassing conclusions. The correlative terms root and power either are or are not intended in algebra to indicate a relation which is different from that indicated in arithmetic by this correlation. While the orthodox exposition makes it clear that the correlation intended is not simply arithmetical, but algebraic, we cannot admit it without at the same time condemning as illogical the actual use of indices or exponents of power. On the other hand we can regard the actual use of exponents of power as logical, but we can then admit no difference between arithmetical and algebraic root and power, and it becomes meaningless to say that +a2 has two roots and -a2 none, and absurd to go on to invent two roots for a2. Here again we have no option but to abandon the orthodox, which is also the mystical, standpoint, and return to common sense. But we may