should thereby be led into more technical processes of mathematics than falls within the design of this book. But now we can immediately see that the associative law [numbered () above] for multiplication is satisfied. Consider first the length of the resultant vector; this is got by the ordinary process of multiplication for real numbers; and thus the associative law holds for it.

Again, the direction of the resultant vector is got by the mere addition of angles, and the associative law holds for this process also.

So much for multiplication. We have now rapidly indicated, by considering addition and multiplication, how an algebra or "calculus" of vectors in one plane can be constructed, which is such that any two vectors in the plane can be added, or subtracted, and can be multiplied, or divided one by the other.

We have not considered the technical details of all these processes because it would lead us too far into mathematical details; but we have shown the general mode of procedure. When we are interpreting our algebraic symbols in this way, we are said to be employing "imaginary quantities" or "complex

quantities." These terms are mere details, and we have far too much to think about to stop to enquire whether they are or are not very happily chosen.

The nett result of our investigations is that

any equations like $x + 3 = 2$ or $(x + 3)^{2} = - 2$ can now always be interpreted into terms of vectors, and solutions found for them. In seeking for such interpretations it is well to note that $3$ becomes $(3, 0)$ , and $- 2$ becomes $(- 2, 0)$ , and $x$ becomes the "unknown" couple $(u, v)$ : so the two equations become respectively $(u, v) + (3, 0) = (2, 0)$ , and ${(u, v) + (3, 0)}^{2} = (- 2, 0)$ .

We have now completely solved the initial difficulties which caught our eye as soon as we considered even the elements of algebra. The science as it emerges from the solution is much more complex in ideas than that with which we started. We have, in fact, created a new and entirely different science, which will serve all the purposes for which the old science was invented and many more in addition. But, before we can congratulate ourselves on this result to our labours, we must allay a suspicion which ought by this time to have arisen in the mind of the student. The question which the reader ought to be asking himself is: Where is all this invention of new interpretations going to end? It is true that we have succeeded in interpreting algebra so as always to be able to solve a quadratic equation like $x^{2} - 2 x + 4 = 0$ ; but there are an endless number of other equations, for example, $x^{3} - 2 x + 4 = 0$ , $x^{4} + x^{3} + 2 = 0$ , and so on without limit. Have we got to make a

new science whenever a new equation appears?

Now, if this were the case, the whole of our preceding investigations, though to some minds they might be amusing, would in truth be of very trifling importance. But the great fact, which has made modern analysis possible, is that, by the aid of this calculus of vectors, every formula which arises can receive its proper interpretation; and the "unknown" quantity in every equation can be shown to indicate some vector. Thus the science is now complete in itself as far as its fundamental ideas are concerned. It was receiving its final form about the same time as when the steam engine was being perfected, and will remain a great and powerful weapon for the achievement of the victory of thought over things when curious specimens of that machine repose in museums in company with the helmets and breastplates of a slightly earlier epoch.