Now these two turnings are not inconsistent. In his plane, if AB turned about A, and A´B´ about A´, the consistency of the square would be destroyed, it would be an impossible motion for a rigid body to perform. But in the turning which he studies portion by portion there is nothing inconsistent. Each line in the square can turn in this way, hence he would realise the turning of the whole square as the sum of a number of turnings of isolated parts. Such turnings, if they took place in his plane, would be inconsistent, but by virtue of a third dimension they are consistent, and the result of them all is that the square turns about the line AC and lies in a position in which it is the mirror image of what it was in its first position. Thus he can realise a turning about a line by relinquishing one of his axes, and representing his body part by part. Let us apply this method to the turning of a cube so as to become the mirror image of itself. In our space we can construct three independent axes, _x_, _y_, _z_, shown in fig. 36. Suppose that there is a fourth axis, _w_, at right angles to each and every one of them. We cannot, keeping all three axes, _x_, _y_, _z_, represent _w_ in our space; but if we relinquish one of our three axes we can let the fourth axis take its place, and we can represent what lies in the space, determined by the two axes we retain and the fourth axis. [Illustration: Fig. 37.] Let us suppose that we let the _y_ axis drop, and that we represent the _w_ axis as occupying its direction. We have in fig. 37 a drawing of what we should then see of the cube. The square ABCD, remains unchanged, for that is in the plane of _xz_, and we still have that plane. But from this plane the cube stretches out in the direction of the _y_ axis. Now the _y_ axis is gone, and so we have no more of the cube than the face ABCD. Considering now this face ABCD, we see that it is free to turn about the line AB. It can rotate in the _x_ to _w_ direction about this line. In fig. 38 it is shown on its way, and it can evidently continue this rotation till it lies on the other side of the _z_ axis in the plane of _xz_. We can also take a section parallel to the face ABCD, and then letting drop all of our space except the plane of that section, introduce the _w_ axis, running in the old _y_ direction. This section can be represented by the same drawing, fig. 38, and we see that it can rotate about the line on its left until it swings half way round and runs in the opposite direction to that which it ran in before. These turnings of the different sections are not inconsistent, and taken all together they will bring the cube from the position shown in fig. 36 to that shown in fig. 41. [Illustration: Fig. 38.] Since we have three axes at our disposal in our space, we are not obliged to represent the _w_ axis by any particular one. We may let any axis we like disappear, and let the fourth axis take its place. [Illustration: Fig. 39.]