But it is obvious that the axis plane may lie in our space. A point near the plane determines with it a three-dimensional space. When it begins to rotate round the plane it does not move anywhere in this three-dimensional space, but moves out of it. A point can no more rotate round a plane in three-dimensional space than a point can move round a line in two-dimensional space. We will now apply the second of the modes of representation to this case of turning about a plane, building up our analogy step by step from the turning in a plane about a point and that in space about a line, and so on. In order to reduce our considerations to those of the greatest simplicity possible, let us realise how the plane being would think of the motion by which a square is turned round a line. Let, fig. 34, ABCD be a square on his plane, and represent the two dimensions of his space by the axes A_x_ A_y_. Now the motion by which the square is turned over about the line AC involves the third dimension. He cannot represent the motion of the whole square in its turning, but he can represent the motions of parts of it. Let the third axis perpendicular to the plane of the paper be called the axis of _z_. Of the three axes _x_, _y_, _z_, the plane being can represent any two in his space. Let him then draw, in fig. 35, two axes, _x_ and _z_. Here he has in his plane a representation of what exists in the plane which goes off perpendicularly to his space. In this representation the square would not be shown, for in the plane of _xz_ simply the line AB of the square is contained. The plane being then would have before him, in fig. 35, the representation of one line AB of his square and two axes, _x_ and _z_, at right angles. Now it would be obvious to him that, by a turning such as he knows, by a rotation about a point, the line AB can turn round A, and occupying all the intermediate positions, such as AB_{1}, come after half a revolution to lie as A_x_ produced through A. Again, just as he can represent the vertical plane through AB, so he can represent the vertical plane through A´B´, fig. 34, and in a like manner can see that the line A´B´ can turn about the point A´ till it lies in the opposite direction from that which it ran in at first.