The double-linkage of field and matter, matter and field, will now be realised. Matter is derived from the fundamental tensor~$g_{\mu\nu}$ by the expression $G_{\mu}^{\nu} - \tfrac{1}{2}g_{\mu}^{\nu}G$; but it is matter so derived which is initially used to measure the fundamental tensor~$g_{\mu\nu}$. We have in this section considered one simple consequence of this cycle---the law of gravitation. It needs a broader analysis to follow out the full consequences, and this will be attempted in Chapter~\ChapNum{VII}, Part~II\@. \Section{67.}{Cylindrical and spherical space-time} According to the foregoing section $\lambda$~does not vanish, and there is a small but finite curvature at every point of space and time. This suggests the consideration of the shape and size of the world as a whole. \index{World!shape of}% Two forms of the world have been suggested--- (1) Einstein's cylindrical world. Here the space-dimensions correspond \index{Cylindrical world}% \index{Einstein's cylindrical world}% to a sphere, but the time-dimension is uncurved. (2) De~Sitter's spherical world. Here all dimensions are spherical; but \index{de Sitter's spherical world}% \index{Sitter@de Sitter's spherical world}% \index{Spherical world}% since it is imaginary time which is homogeneous with the space-coordinates, sections containing real time become hyperbolas instead of circles. \PageSep{156} We must describe these two forms analytically. A point on the surface of a sphere of radius~$R$ is described by two angular variables $\theta$,~$\phi$, such that \[ -ds^{2} = R^{2}(d\theta^{2} + \sin^{2}\theta\, d\phi^{2}). \] Extending this to three dimensions, we have three angular variables such that \[ -ds^{2} = R^{2}\bigl\{d\chi^{2} + \sin^{2}\chi\, (d\theta^{2} + \sin^{2}\theta\, d\phi^{2})\bigr\}. \Tag{(67.11)} \] Accordingly in Einstein's form the interval is given by \[ ds^{2} = -R^{2}\, d\chi^{2} - R^{2} \sin^{2}\chi\, (d\theta^{2} + \sin^{2}\theta\, d\phi^{2}) + dt^{2}. \Tag{(67.12)} \] Of course this form applies only to a survey of the world on the grand scale. Trifling irregularities due to the aggregation of matter into stars and stellar systems are treated as local deviations which can be disregarded. Proceeding from the origin in any direction, $R\chi$~is the distance determined \index{Finiteness of space}% by measurement with rigid scales. But the measured area of a sphere of radius~$R\chi$ is not $4\pi R^{2}\chi^{2}$ but $4\pi R^{2} \sin^{2}\chi$. There is not so much elbow-room in distant parts as Euclid supposed. We reach a ``greatest sphere'' at the distance~$\frac{1}{2}\pi R$; proceeding further, successive spheres contract and decrease to a single point at a distance~$\pi R$---the greatest distance which can exist.