If we call the direction to the right the unknown direction, then the line he saw before, the yellow line, goes out into this unknown direction, and the line which before went into the unknown direction, comes in. It comes in in the opposite direction to that in which the yellow line ran before; the interior of the face now against the plane is pink. It is a property of two lines at right angles that, if one turns out of a given direction and stands at right angles to it, then the other of the two lines comes in, but runs the opposite way in that given direction, as in fig. 88. [Illustration: Fig. 88.] Now these two presentations of the cube would seem, to the plane creature like perfectly different material bodies, with only that line in common in which they both meet. Again our cube can be turned about the yellow line. In this case the yellow square would disappear as before, but a new square would come into the plane after the cube had rotated by an angle of 90° about this line. The bottom square of the cube would come in thus in figure 89. The cube supposed in contact with the plane is rotated about the lower yellow line and then the bottom face is in contact with the plane. Here, as before, the red line going out into the unknown dimension, the white line which before ran in the unknown dimension would come in downwards in the opposite sense to that in which the red line ran before. [Illustration: Fig. 89.] Now if we use _i_, _j_, _k_, for the three space directions, _i_ left to right, _j_ from near away, _k_ from below up; then, using the colour names for the axes, we have that first of all white runs _i_, yellow runs _j_, red runs _k_; then after the first turning round the _k_ axis, white runs negative _j_, yellow runs _i_, red runs _k_; thus we have the table:— _i_ _j_ _k_ 1st position white yellow red 2nd position yellow white— red 3rd position red yellow white— Here white with a negative sign after it in the column under _j_ means that white runs in the negative sense of the _j_ direction. We may express the fact in the following way:— In the plane there is room for two axes while the body has three. Therefore in the plane we can represent any two. If we want to keep the axis that goes in the unknown dimension always running in the positive sense, then the axis which originally ran in the unknown dimension (the white axis) must come in in the negative sense of that axis which goes out of the plane into the unknown dimension.