[fig.]9) in ( + 3 , + 1 ) a length of 3 units is to be measured along X O X ′ in the positive direction, that is from O towards X , and a length + 1 measured along Y O Y ′ in the positive direction, that is from O towards Y . Similarly in ( − 3 , + 1 ) the length of 3 units is to be measured from O towards X ′ , and of 1 unit from towards Y . Also in ( − 3 , − 1 ) the two lengths are to be measured along O X ′ and O Y ′ respectively, and in ( + 3 , − 1 ) along O X and O Y ′ respectively. Let us for the moment call such a pair of numbers an "ordered couple." Then, from the two numbers 1 and 3 , eight ordered couples can be generated, namely

Each of these eight "ordered couples" directs a process of measurement along $X O X^{'}$ and $Y O Y^{'}$ which is different from that directed by any of the others.

The processes of measurement represented by the last four ordered couples, mentioned above, are given pictorially in the figure. The lengths $O M$ and $O N$ together correspond

to $(+ 3, + 1)$ , the lengths $O M^{'}$ and $O N$ together correspond to $(- 3, + 1)$ , $O M^{'}$ and $O N^{'}$ together to $(- 3, - 1)$ , and $O M$ and $O N^{'}$ together to $(+ 3, - 1)$ . But by completing the various rectangles, it is easy to see that the point $P$ completely determines and is determined by the ordered couple 9 $(+ 3, + 1)$ , the point $P^{'}$ by $(- 3, + 1)$ , the point $P^{″}$ by $(- 3, - 1)$ , and the point $P^{‴}$ by $(+ 3, - 1)$ . More generally in the previous figure ([fig:8]8), the point $P$ corresponds to the ordered couple $(x, y)$ , where $x$ and $y$ in the figure are both assumed to be positive, the point $P^{'}$ corresponds to $(x^{'}, y)$ , where $x^{'}$ in the figure is assumed to be negative, $P^{″}$ to $(x^{'} y^{'})$ , and $P^{‴}$ to $(x, y^{'})$ . Thus an ordered

couple $(x, y)$ , where $x$ and $y$ are any positive or negative numbers, and the corresponding point reciprocally determine each other. It is convenient to introduce some names at this juncture. In the ordered couple $(x, y)$ the first number $x$ is called the "abscissa" of the

corresponding point, and the second number $y$ is called the "ordinate" of the point, and

the two numbers together are called the "coordinates"

of the point. The idea of determining the position of a point by its "coordinates" was by no means new when the theory of "imaginaries" was being formed. It was due to Descartes, the great French

mathematician and philosopher, and appears in his Discours published at Leyden in 1637 . The idea of the ordered couple as a thing on its own account is of later growth and is the outcome of the efforts to interpret imaginaries in the most abstract way possible.

It may be noticed as a further illustration of this idea of the ordered couple, that the point $M$ in [fig.]9 is the couple $(+ 3, 0)$ , the point $N$ is the couple $(0, + 1)$ , the point $M^{'}$ the couple $(- 3, 0)$ , the point $N^{'}$ the couple $(0, - 1)$ , the point $O$ the couple $(0, 0)$ .

Another way of representing the ordered couple $(x, y)$ is to think of it as representing the dotted line $O P$ (cf. [fig.]8), rather than the point $P$ . Thus the ordered couple represents a line drawn from an "origin," $O$ , of a certain

length and in a certain direction. The line $O P$ may be called the vector line from $O$ to $P$ , or the step from $O$ to $P$ . We see, therefore, that we have in this chapter only extended the interpretation which we gave formerly of the positive and negative numbers. This method of representation by vectors is very