But

Argand draws attention, is a 'proportion', and that a direction perpendicular to two opposed directions is a 'mean proportional' between them, is a proposition which is neither true nor false, and thus cannot be made the subject of proof or disproof. But, given the purpose and the object of Argand's Essai, it is legitimate to propose, as a new premise or convention in symbolism, that identity of relations of direction shall be given the same name, and shall be algebraically symbolized in the same way, as identity of relations of magnitude. Any one who, reading Argand's argument, looks upon this proposition as one which ought to be demonstrated, will naturally find in the conclusion that I denotes the relation of perpendicularity nothing but the reassertion of an unproved proposition. from the moment this proposition is understood and admitted as a new convention in symbolism, it is at once seen that the employment of ✓ to denote this geometrical relation follows logically from this new premise in symbolism. For the equation ✓IX √-I = -I can, in accord with algebraic convention, be transformed into +1 : √FI ::

B

li

A

√—I: -1, and since the latter expression does not symbolize a quantitative proportion (I not being a symbol of quantity),

I can without contradiction or ambiguity be substituted for id in equation (B). Again, I or i not symbolizing a quantity, the combination a+bi does not symbolize the conception of summation, and is, thus isolated, meaningless, being nothing but a factor in an organized expression of algebraic quantity; it can thus be employed without ambiguity or contradiction to symbolize the geometrical combination of OA = a and AB bi, or the resultant, OB, of this combination: in the ordinary language of Algebra, OB represents the complex number a + bi.

I said a little while ago that in so far as these geometrical conceptions are assigned as meanings of the so-called symbols

of imaginary and complex quantity or number, Algebra may be said to be no longer simply Arithmetica Universalis. The same may be said with reference to the totally different interpretation of these forms in analytical geometry. But I should express my meaning more exactly in saying that in these cases Algebra becomes something more than, without ceasing to be, Arithmetica Universalis. For algebraic symbolism is not an expression of the development of geometrical conception; it is and it remains-notwithstanding these particular attributions of geometrical meaning-essentially and directly a language specially devised for the purpose of reasoning about number or quantity in the abstract; i.e. it is and remains what Newton called it, Arithmetica Universalis. And in order to make my meaning still more plain, if it is not already plain enough, I will add that in saying that Algebra is essentially Arithmetica Universalis, I intend to traverse the assertion that Algebra is independent of Arithmetic, unless nothing more is meant by this than to reiterate in other words the perfectly obvious and indisputable fact that the algebraic forms in question are arithmetically uninterpretable. But it is clear there is more than this in the assertion; the independence is looked upon as an inference or conclusion arising from other considerations, one of which is this particular case of uninterpretability in arithmetic: 'If there were such a dependence, it is obvious that as soon as algebraic expressions are arithmetically unintelligible all laws respecting them must lose their validity.'

But the algebraic expression ✔-I× √-I is arithmetically intelligible in the sense demanded by this passage. The point is whether it is legitimate to isolate √-I and regard it as an algebraic expression' or entertain such a question as, What does it mean? No doubt it will appear to be so if we submissively follow the orthodox exposition of the development of elementary algebra, and are either blind, or content to shut our eyes, to the sophisms and incongruities which abound in it. The argument of this and the preceding chapter is a sustained protest against the mystical bias which prompts these sophisms and incongruities and renders them not intolerable even to critical minds, against the cumulative and finally disastrous effect of an unsuspected reaction of the symbolic forms of algebra upon the conceptual process which gives them birth. Set free

the judgement from this mystical bias, disengage the process of conception from the false implications of the process of symbolism, and not only do these sophisms and incongruities disappear, but with their disappearance also vanishes the suggestion which they facilitate, viz. that such combinations as I, a+bi are, in and by themselves, algebraic expressions', that is, symbolic of mathematical notions or abstractions. They are then at once recognized as mere factors or constituent parts of actual expressions, algebraically symbolic, interpretable in terms of generalized number and abstract quantity.

CHAPTER IX

THE DOCTRINE OF IMAGINARY LOCI IN GEOMETRY

Explanation of Imaginary Points (1) by means of the Principle of Contingent Relations (Chasles), (2) by means of the Theory of Involution (von Staudt).—The Doctrine of Geometrical Imaginaries, rationally considered, is an artifice in expression which involves paradox for the sake of brevity in the statement of certain geometrical relations.-The conceptions of the point and line at infinity as leading to this paradoxical mode of expression.-Algebraic Imaginaries and Analytical Geometry.

THE common sense of this doctrine-using the term ' common sense' in place of the more pretentious word ' philosophy 'as Clifford did in the The Common Sense of the Exact Scienceslies, so it appears to me, in a rational interpretation of, e.g. the simple statement which Cayley makes on the subject in his presidential address: 'But the geometrical construction being in each case the same, we say that in the second case also the line passes through the two intersections of the circle.' This, as will be recollected, refers to the configuration composed of two circles and a straight line, and the two cases, (1) where the line is the common chord of the two circles, (2) where the line is not a chord at all.

This explanation of Cayley's, apparently not intended by him as anything more than a bare statement of the practice of geometers, really contains, in epitome and by reference to a particular case, all that Chasles has to say on the subject in the fifth chapter and note xxvi of the Aperçu historique sur l'Origine des Méthodes en Géométrie; that is, his statement and discussion of the Methode ou principe des relations contingentes, of the distinction between the accidental and permanent. properties of a system of figures. In immediate connexion with the latter phrase, he discusses (p. 205) the case of the system of two circles and line in a plane which Cayley used in illustration of the doctrine. He says:

'When the two circles intersect, this straight line is their common chord, and this fact suffices to define and to construct it this is what we call one of its contingent or accidental

properties. But when the two circles do not intersect, this property disappears although the line still exists and is of great utility in the theory of circles. We must therefore define and construct this straight line by means of some other one of its properties common to all cases of the general construction of the figure, or of the system of the two circles. This will be one of its permanent properties. It is through such considerations. that M. Gaultier, instead of calling this line the common chord of the two circles, is led to call it the radical axis; an expression prompted by a permanent property of this line, viz. that the tangents drawn from any one of its points to the two circles are equal, so that every point of this line is the centre of a circle which cuts the other two orthogonally.

'The doctrine of contingent relations seems to us to present a further advantage, that of affording a satisfactory explanation of the word "imaginary ", now used in pure geometry, where it expresses un être de raison sans existence, but which we can nevertheless endow with certain properties of which we make use temporarily as auxiliaries, and about which we reason as we do about a real and palpable object. This idea of the imaginary, which seems at first sight obscure and paradoxical, thus acquires, in the theory of contingent relations, a clear, precise, and legitimate sense.'

Chasles here refers us to note xxvi, in which he enters more fully into his view of the doctrine of contingent relations as explanatory of the use of imaginaries in pure geometry. We employ this word 'imaginary', he says, ' as a means of attaining brevity of expression; it implies that the process of reasoning applies to another general state of the figure, in which the parts which are the subject of reasoning would really exist, instead of being, as in the figure actually contemplated, imaginary. And since according to the principle of contingent relations, or, if you prefer it, the principle of continuity, the truths demonstrated for one of the two general states of the figure apply to the other state, it is seen that the use of imaginaries is completely justified.' In other words, and with reference to the system of two circles, when we reason about the general case in which the two circles do not intersect, the reasoning applies also to the general case in which the two circles do intersect, because it is directed to some relation between the two circles and the line, which is independent of the accidental circumstance of intersection, or of non-intersection. Thus when we say that the radical axis of two circles is a line which passes through the points, real or