It will be noticed that our rectangular coordinates $(x_{1}, x_{2}, x_{3}, x_{4})$ in this and the previous section approximate to Euclidean, not Galilean, coordinates. Consequently $x_{4}$~is imaginary time, and $G_{(1)}$~is not in any real direction in the world. There is no radius of curvature in a real timelike direction. This does not mean that our discussion is limited to three dimensions; it includes all directions in the four-dimensional world outside the light-cone, \PageSep{155} and applies to the space-dimensions of material structures moving with any speed up to the speed of light. The real quadric of curvature terminates at the light-cone, and the mathematical continuation of it lies not inside the cone but in directions of imaginary time which do not concern us. By consideration of extension in timelike directions we obtain a confirmation of these views, which is, I think, not entirely fantastic. We have said that an electron would not know how large it ought to be unless there existed independent lengths in space for it to measure itself against. Similarly it would not know how long it ought to exist unless there existed a length in time for it to measure itself against. But there is no radius of curvature in a \Chg{time-like}{timelike} direction; so the electron does \emph{not} know how long it ought to exist. Therefore it just goes on existing indefinitely. The alternative laws of gravitation discussed in \SecRef{62} would be obtained if the radius of the unit of material structure adjusted itself as a definite fraction not of the radius of curvature, but of other directed lengths (of a more complex origin) characteristic of empty space-time. In \SecRef{56} it was necessary to postulate that the gravitational field due to an ultimate particle of matter has symmetrical properties. This has now been \index{Particle!symmetry of}% justified. We have introduced a new and far-reaching principle into the relativity theory, viz.\ that symmetry itself can only be relative; and the \index{Symmetry!a relative attribute}% \index{Symmetry!of a particle}% particle, which so far as mechanics is concerned is to be identified with its gravitational field, is the standard of symmetry. We reach the same result if we attempt to define symmetry by the propagation of light, so that the cone $ds = 0$ is taken as the standard of symmetry. It is clear that if the locus $ds = 0$ has complete symmetry about an axis (taken as the axis of~$t$) $ds^{2}$~must be expressible by the formula~\Eq{(38.12)}.