Note 124 — Introductory material

Now when we consider what is involved in such absolute qualitative equivalence, we find at once, as its most obvious prerequisite, the perfect homogeneity of space. For it is assumed that a figure can be completely defined by its internal relations, and that the external relations, which constitute its position, though they suffice to distinguish it from other figures, in no way affect its internal properties, which are regarded as qualitatively identical with those of figures with quite different external relations. If this were not the case, anything analogous to projective transformation would be impossible. For such transformation always alters the position, i.e. the external relations, of a figure, and could not, therefore, if figures were dependent on their relations to other figures or to empty space, be studied without reference to other figures, or to the absolute position of the original figure. We require for our principle, in short, what may be called the mutual passivity and reciprocal independence of two parts or figures of space.