The whole volume of space (determined by rigid scales) is finite and equal to \[ \int_{0}^{\pi} 4\pi R^{2} \sin^{2}\chi · R\, d\chi = 2\pi^{2} R^{3}. \Tag{(67.2)} \] Although the volume of space is finite, there is no boundary; nor is there any centre of spherical space. Every point stands in the same relation to the rest of space as every other point. To obtain de~Sitter's form, we generalise~\Eq{(67.11)} to four dimensions (i.e.\ a spherical four-dimensional surface drawn in Euclidean space of five dimensions). We have four angular variables $\omega$, $\zeta$, $\theta$,~$\phi$, and \[ -ds^{2} = R^{2} \bigl[d\omega^{2} + \sin^{2}\omega \bigl\{d\zeta^{2} + \sin^{2}\zeta (d\theta + \sin^{2}\theta\, d\phi^{2})\bigr\}\bigr]. \Tag{(67.31)} \] In order to obtain a coordinate-system whose physical interpretation is more easily recognisable, we make the transformation \begin{align*} \cos\omega &= \cos\chi \cos it, \\ \cot\zeta &= \cot\chi \sin it, \end{align*} which gives \[ \left. \begin{aligned} \sin\chi &= \sin\zeta \sin\omega\Add{,} \\ \tan it &= \cos\zeta \tan\omega. \end{aligned} \right\} \Tag{(67.32)} \] Working out the results of this substitution, we obtain \[ ds^{2} = -R^{2}\, d\chi^{2} - R^{2} \sin^{2}\chi\, (d\theta^{2} + \sin^{2}\theta\, d\phi^{2}) + R^{2} \cos^{2}\chi · dt^{2}. \Tag{(67.33)} \] So far as space $(\chi, \theta, \phi)$ is concerned, this agrees with Einstein's form~\Eq{(67.12)}; but the variable~$t$, which will be regarded as the ``time''\footnote {The velocity of light at the origin is now~$R$. In the usual units the time would be~$Rt$.} in this world, has different properties. For a clock at rest ($\chi$, $\theta$, $\phi = \text{const.}$) we have \[ ds = R\cos\chi\, dt, \Tag{(67.4)} \] \PageSep{157} \index{Atom, time of vibration of!in de Sitter's world}% \index{Horizon of world}% \index{Recession of spiral nebulae}% so that the ``time'' of any cycle is proportional to~$\sec\chi$. The clock-beats become longer and longer as we recede from the origin; in particular the vibrations of an atom become slower. Moreover we can detect by practical measurement this slowing down of atomic vibrations, because it is preserved in the transmission of the light to us. The coordinates \Eq{(67.33)} form a statical system, the velocity of light being independent of~$t$; hence the light-pulses are all delayed in transmission by the same ``time'' and reach us at the same intervals of~$t$ as they were emitted. Spectral lines emanating from distant \index{Displacement of spectral lines to red, in sun!in nebulae}% \index{Red-shift!in nebulae}% \index{Spectral lines, displacement!in nebulae}% sources at rest should consequently appear displaced towards the red.