Now, as it appears certain that what geometers are accustomed to call "dimension" is both relative and interchangeable in meaning--the one becoming the other according as it is viewed--the conclusion very naturally follows that neither constructive nor symbolic geometry is based upon dimension as commensurable quality. The real basis of the non-Euclidean geometry is dimension as direction. For whatever else may be said of the fourth dimension so-called it is certainly unthinkable, even to the metageometricians, when it is absolved from direction although no specific direction can be assigned to it. It is agreed perhaps among all non-Euclidean publicists that the fourth dimension must lie in a "direction which is at right angles to all the three dimensions." But if they are asked how this direction may be ascertained or even imagined they are nonplused because they simply do not know. The difficulty in this connection seems to hinge about the question of identifying the conditions of the world of phantasy with those of the world of sense. There are distortions, ramifications, submersibles, duplex convolutions and other mathetic acrobatics which can be performed in the realm of the conceptual the execution of which could never be actualized in the objective world. Because these antics are possible in the premises of the mathematical imagination is scarce justification for the attempts at reproduction in an actualized and phenomenal universe. One of the proudest boasts of the fourth dimensionist is that hyperspace offers the possibility of a new species of rotation, namely, _rotation about a plane_. He refers to the fact that in the so-called one-space, rotation can take place only about a point. For instance in Figure 7, the line _ab_ represents a one-space in which rotation can take place only about one of the two points _a_ and _b_. In Figure 8 which represents a two-space, rotation may take place about the line _ab_ or the line _cd_, etc., or, in other words, the plane _abcd_ can be rotated on the axial line _ab_ in the direction of the third dimension. In tridimensional space only two kinds of rotation are possible, namely, rotation about a point and about a line. In the fourth dimension it is claimed that rotation can take place about a plane. For example, the cube in Figure 9, by manipulation in the direction of the fourth dimension, can be made to rotate about the side _abgf_. A very ingenious argument is used to show how rotation about a plane is thinkable and possible in hyperspace. But with this, as with the entire fabric of hyperspace speculations, dependence is placed almost entirely upon analogous and symbolic conceptions for evidence as to the consistency and rationality of the conclusions arrived at.