interpretation of the symbols so that a always has meaning whether a be positive or negative. Of course, the interpretation must be such that all the ordinary formal laws for addition, subtraction, multiplication, and division hold good; and also it must not interfere with the

generality which we have attained by the use of the positive and negative numbers. In fact, it must in a sense include them as special cases. When $a$ is negative we may write $- c^{2}$ for it, so that $c^{2}$ is positive. Then

$\sqrt{a} & = \sqrt{(- c^{2})} = \sqrt{{(- 1) \times c^{2}}} & = \sqrt{(- 1)} \sqrt{c^{2}} = c \sqrt{(- 1)} .$

Hence, if we can so interpret our symbols that $\sqrt{(- 1)}$ has a meaning, we have attained our object. Thus $\sqrt{(- 1)}$ has come to be looked on as the head and forefront of all the imaginary quantities.

This business of finding an interpretation for $\sqrt{(- 1)}$ is a much tougher job than the analogous one of interpreting $- 1$ . In fact, while the easier problem was solved almost instinctively as soon as it arose, it at first hardly occurred, even to the greatest mathematicians, that here a problem existed which was perhaps capable of solution. Equations like $x^{2} = - 3$ , when they arose, were simply ruled aside as nonsense.

However, it came to be gradually perceived during the eighteenth century, and even earlier, how very convenient it would be if an interpretation could be assigned to these nonsensical symbols. Formal reasoning with these symbols was gone through, merely assuming that they obeyed the ordinary

algebraic laws of transformation; and it was seen that a whole world of interesting results could be attained, if only these symbols might legitimately be used. Many mathematicians were not then very clear as to the logic of their procedure, and an idea gained ground that, in some mysterious way, symbols which mean nothing can by appropriate manipulation yield valid proofs of propositions. Nothing can be more mistaken. A symbol which has not been properly defined is not a symbol at all. It is merely a blot of ink on paper which has an easily recognized shape. Nothing can be proved by a succession of blots, except the existence of a bad pen or a careless writer. It was during this epoch that the epithet "imaginary" came to be applied to $\sqrt{(- 1)}$ . What these mathematicians had really succeeded in proving were a series of hypothetical propositions, of which this is the blank form: If interpretations exist for $\sqrt{(- 1)}$ and for the addition, subtraction, multiplication, and division of $\sqrt{(- 1)}$ which make the ordinary algebraic rules (e.g. $x + y = y + x$ , etc.) to be satisfied, then such and such results follows. It was natural that the mathematicians should not always appreciate the big "If," which ought to have preceded the statements of their results.

As may be expected the interpretation,

when found, was a much more elaborate affair than that of the negative numbers and the reader's attention must be asked for some careful preliminary explanation. We have already come across the representation of a point by two numbers. By the aid of the 8 positive and negative numbers we can now represent the position of any point in a plane by a pair of such numbers. Thus we take the pair of straight lines $X O X^{'}$ and $Y O Y^{'}$ , at right angles, as the "axes" from which we start all our measurements. Lengths measured along $O X$ and $O Y$ are positive, and measured backwards along $O X^{'}$ and $O Y^{'}$ are negative. Suppose that a pair of numbers, written in order, e.g. $(+ 3, + 1)$ , so that there

is a first number ( + 3 in the above example), and a second number ( + 1 in the above example), represents measurements from O along X O X ′ for the first number, and along Y O Y ′ for the second number. Thus (cf.