In a more thorough treatment of non-Euclidean geometry other non-metrical methods of presentation are sometimes adhered to. One of these is based on the theory of groups and projective geometry; another is due to the discoveries of Levi-Civita and Weyl and depends on the fundamental concept of an infinitesimal parallel displacement; it opens the way to Weyl’s still more general geometry. Still another mathematical method of exploring space is that of continuous tracks which was investigated by Eisenhart and Veblen. In this book we shall refrain from discussing these more difficult methods, for they present too technical an aspect;[10] but it would be a great mistake to assume that these imaginative flights of the pure mathematicians were of no utility to physical science. Apart from the deep philosophical light which they throw on the entire problem of space, we know from past experience that what was at its inception a pure mathematical dream has more than once become at some later date the image of physical reality. Non-Euclidean geometry, and possibly Weyl’s still stranger geometry, are cases in point. We have now to consider a number of questions pertaining to location and motion in space. Mathematical space, as we have seen, is -dimensional and amorphous. But in order to bring it into closer contact with the physical space of experience we may assume the necessary postulates of order and contiguity to have been specified. We thus obtain three-dimensional mathematical space. Now mathematical space, in view of its amorphous nature, is essentially relative. The very homogeneity of space, its sameness “here” as “there,” precludes our being able to differentiate one absolute position from another. It will follow that absolute motion, i.e., a variation of absolute position in empty amorphous mathematical space, can have no significance. Our sole means of giving significance to position and motion will be by selecting some three-dimensional frame of reference as standard. For instance, we might consider three mutually perpendicular axes [Pg 45] meeting at a point. A more concrete illustration would be afforded by considering the two adjoining walls and the floor of our room. Then, after having selected the standard rods which we intend to employ, and also a clock, we should proceed to measure the position and change of position of an object with respect to our room. This would yield us the relative position and motion of the object (relative to our room).