Chapter V (part 4)

57 i.e., upon the surface, it is impossible to define the position of the solid, the side of which constitutes the given surface, so in the geometry of three dimensions, in three-dimensional space, it is impossible to define a four-dimensional space. Briefly speaking, as planimetry is insufficient for the investigation of the problems of stereometry, so is stereometry insufficient for four-dimensional space. As a conclusion from all of the above we may repeat that every point of our space is the section of a line in higher space, or as B. Riemann expressed it: the material atom is the entrance of the fourth dimension into three-dimensional space. ______ For a nearer approach to the problem of higher dimensions and of higher space it is necessary first of all to understand the constitution and properties of the higher dimensional region in comparison with the region of three dimensions. Then only will appear the possibility of a more exact investigation of this region, and a classification of the laws governing it. What is it that it is necessary to understand? It seems to me that first of all it is necessary to understand that we are considering not two regions spatially different, and not two regions of which one (again spatially, "geometrically") constitutes a part of the other, but two methods of receptivity of one and the same unique world of a space which is unique. Furthermore it is necessary to understand that all objects known to us exist not only in those categories in which they are perceived by us, but in an infinite number of others in which we do not and cannot sense them. And we must learn first to think things in other categories, and then so far as we are able, to imagine them therein. Only after doing this can we possibly develop the faculty to apprehend them in higher space—and to sense "higher" space itself. Or perhaps the first necessity is the direct perception of everything in the outside world which does not fit into the frame of three dimensions, which exists independently of the categories of time and space—everything that for this reason we are accustomed to consider as non-existent. If variability is an indication of the three-dimensional world, then let us search for the constant and p. 58 thereby approach to an understanding of the four-dimensional world. We have become accustomed to count as really existing only that which is measurable in terms of length, breadth and height; but as has been shown it is necessary to expand the limits of the really existing . Mensurability is too rough an indication of existence, because mensurability itself is too conditioned a conception. We may say that for any approach to the exact investigation of the higher dimensional region the certainty obtained by the immediate sensation is probably indispensable; that much that is immeasurable exists just as really as, and even more really than, much that is measurable. Footnotes 55:1 C. H. Hinton, "A New Era of Thought," p. 44.