Chapter II (part 2)

By a remarkable analysis the great geometers above mentioned have shown that space is not limited as ordinary experience would seem to inform us, but that we are quite capable of conceiving different kinds of space. ( A New Era of Thought .) Hinton invented a complicated system for the education and development of the space sense by means of exercises with groups the cubes of different colors. The books above mentioned are devoted to the exposition of this system. In my opinion Hinton's exercises are interesting from a theoretical standpoint, but they are practically valuable only for such as have the same turn of mind as Hinton's own. Exercises of the mind according to his system must first of all lead to the development of the ability to imagine objects , not as the eye sees them, i.e., in perspective, but as they are geometrically—to learn to imagine the cube, for example, simultaneously from all sides. Moreover such a development of the imagination p. 25 as overcomes the illusions of perspective results in the expansion of the limits of consciousness, thus creating new conceptions and augmenting the faculty for perceiving analogies . ______ Kant established the fact that the development of knowledge under the existing conditions of receptivity will not bring us any closer to things in themselves. But there are theories asserting that it is possible, if desired, to change the very conditions of receptivity, and thus to approach the true substance of things. In the books above referred to, Hinton tries to unite the scientific foundations of such theories. Our space as we ordinarily think of it is conceived as limited—not in extent, but in a certain way which can only be realized when we think of our ways of measuring space objects. It is found that there are only three independent directions in which a body can be measured—it must have height, length and breadth, but it has no more than these dimensions, if any other measurement be taken in it, this new measurement will be found to be compounded of the old measurements. It is impossible to find a point in the body which could not be arrived at by travelling in combinations of the three directions already taken. But why should space be limited to three independent directions? Geometers have found that there is no reason why bodies which we can measure should thus be limited. As a matter of fact all the bodies which we can measure are thus limited. So we come to this conclusion, that the space which we use for conceiving ordinary objects in the world is limited to three dimensions. But it might be possible for there to be beings living in a world such that they would conceive a space of four dimensions . 1 It is possible to say a great deal about space of higher dimensions than our own, and to work out analytically many problems which suggest themselves. But can we conceive four-dimensional space in the same way in which we can conceive our own space? Can we think of a body in four dimensions as a unit having properties in the same way as we think of a body having a definite shape in the space with which we are familiar? There is really no more difficulty in conceiving four-dimensional shapes, when we go about it in the right way, than in conceiving the idea of solid shapes, nor is there any mystery at all about it. When the faculty to apprehend in four dimensions is acquired—or rather when it is brought into consciousness—for it exists in every p. 26 one in imperfect form—a new horizon opens. The mind acquires a development of power, and in this use of ampler space as a mode of thought, a path is opened by using that very truth which, when first stated by Kant, seemed to close the mind within such fast limits. Our perception is subject to the condition of being in space. But space is not limited as we at first think. The next step after having formed this power of conception in ampler space, is to investigate nature and see what phenomena are to be explained by four-dimensional relations.