Suppose that we have a small region of the world throughout which the $g$'s can be treated as constants\footnotemark.\footnotetext {It will be shown in \SecRef{36} that it is always possible to transform the coordinates so that the first derivatives of the~$g$'s vanish at a selected point. We shall suppose that this preliminary transformation has already been made, in order that the constancy of the~$g$'s may be a valid approximation through as large a region as possible round the selected point.} In that case the right-hand side of~\Eq{(2.1)} can be broken up into the sum of four squares, admitting imaginary coefficients if necessary. Thus writing \begin{align*} y_{1} &= a_{1} x_{1} + a_{2} x_{2} + a_{3} x_{3} + a_{4} x_{4}, \\ y_{2} &= b_{1} x_{1} + b_{2} x_{2} + b_{3} x_{3} + b_{4} x_{4},\quad\text{etc.,} \\ \intertext{so that} dy_{1} &= a_{1}\, dx_{1} + a_{2}\, dx_{2} + a_{3}\, dx_{3} + a_{4}\, dx_{4},\quad\text{etc.,} \end{align*} we can choose the constants $a_{1}$, $b_{1}$,~\dots\ so that \Eq{(2.1)}~becomes \[ ds^{2} = dy_{1}^{2} + dy_{2}^{2} + dy_{3}^{2} + dy_{4}^{2}. \Tag{(4.1)} \] For, substituting for the $dy$'s and comparing coefficients with~\Eq{(2.1)}, we have only $10$~equations to be satisfied by the $16$~constants. There are thus many ways of making the reduction. Note, however, that the reduction to the sum of four squares of complete differentials is not in general possible for a \emph{large} region, where the $g$'s have to be treated as functions, not constants. Consider all the events for which $y_{4}$~has some specified value. These will form a three-dimensional world. Since $dy_{4}$~is zero for every pair of these events, their mutual intervals are given by \[ ds^{2} = dy_{1}^{2} + dy_{2}^{2} + dy_{3}^{2}. \Tag{(4.2)} \] But this is exactly like familiar space in which the interval (which we have shown to be the same as the distance for space without time) is given by \[ ds^{2} = dx^{2} + dy^{2} + dz^{2}, \Tag{(4.3)} \] where $x$, $y$, $z$ are rectangular coordinates. Hence a section of the world by $y_{4} = \text{const.}$ will appear to us as space, and $y_{1}$,~$y_{2}$,~$y_{3}$ will appear to us as rectangular coordinates. The coordinate-frames $y_{1}$,~$y_{2}$,~$y_{3}$, and $x$,~$y$,~$z$, are examples of the systems $S$ and~$S'$ of \SecRef{1}, for which the intervals between corresponding pairs of mesh-corners are equal. The two systems are therefore exactly alike observationally; and if one appears to us to be a rectangular frame in space, so also must the other. One proviso must be noted; the coordinates $y_{1}$,~$y_{2}$,~$y_{3}$ for real events must be real, as in familiar space, otherwise the resemblance would be only formal.