Take a square, one of those shown in Fig. 77 and give it a neutral colour, let this colour be called “null,” and be such that it makes no appreciable difference to any colour with which it mixed. If there is no such real colour let us imagine such a colour, and assign to it the properties of the number zero, which makes no difference in any number to which it is added. Above this square place a red square. Thus we symbolise the going up by adding red to null. Away from this null square place a yellow square, and represent going away by adding yellow to null. To complete the figure we need a fourth square. Colour this orange, which is a mixture of red and yellow, and so appropriately represents a going in a direction compounded of up and away. We have thus a colour scheme which will serve to name the set of squares drawn. We have two axes of colours—red and yellow—and they may occupy as in the figure the direction up and away, or they may be turned about; in any case they enable us to name the four squares drawn in their relation to one another. Now take, in Fig. 78, nine squares, and suppose that at the end of the going in any direction the colour started with repeats itself. [Illustration: Fig. 78.] We obtain a square named as shown. Let us now, in fig. 79, suppose the number of squares to be increased, keeping still to the principle of colouring already used. Here the nulls remain four in number. There are three reds between the first null and the null above it, three yellows between the first null and the null beyond it, while the oranges increase in a double way. [Illustration: Fig. 79.] Suppose this process of enlarging the number of the squares to be indefinitely pursued and the total figure obtained to be reduced in size, we should obtain a square of which the interior was all orange, while the lines round it were red and yellow, and merely the points null colour, as in fig. 80. Thus all the points, lines, and the area would have a colour. [Illustration: Fig. 80.] We can consider this scheme to originate thus:—Let a null point move in a yellow direction and trace out a yellow line and end in a null point. Then let the whole line thus traced move in a red direction. The null points at the ends of the line will produce red lines, and end in null points. The yellow line will trace out a yellow and red, or orange square. Now, turning back to fig. 78, we see that these two ways of naming, the one we started with and the one we arrived at, can be combined. By its position in the group of four squares, in fig. 77, the null square has a relation to the yellow and to the red directions. We can speak therefore of the red line of the null square without confusion, meaning thereby the line AB, fig. 81, which runs up from the initial null point A in the figure as drawn. The yellow line of the null square is its lower horizontal line AC as it is situated in the figure. [Illustration: Fig. 81.]