The first equation of~\Eq{(5.3)} may be written \[ x'/\beta = x + ut, \] \PageSep{26} which shows that~$S$, besides making the allowance~$ut$ for the motion of his origin, divides by~$\beta$ all lengths in the $x$-direction measured by~$S'$. On the other hand the equation $y' = y$ shows that $S$~accepts $S'$'s~measures in directions transverse to their relative motion. Let $S'$~take his standard metre (at rest relative to him, and therefore moving relative to~$S$) and point it first in the transverse direction~$y'$ and then in the longitudinal direction~$x'$. For~$S'$ its length is $1$~metre in each position, since it is his standard; for~$S$ the length is $1$~metre in the transverse position and $1/\beta$~metres in the longitudinal position. Thus $S$~finds that a moving rod contracts when turned from the transverse to the longitudinal position. The question remains, How does the length of this moving rod compare with the length of a similarly constituted rod at rest relative to~$S$? The answer is that the transverse dimensions are the same whilst the longitudinal dimensions are contracted. We can prove this by a \Foreign{reductio ad absurdum}. For suppose that a rod moving transversely were longer than a similar rod at rest. Take two similar transverse rods $A$ and~$A'$ at rest relatively to~$S$ and $S'$~respectively. Then $S$~must regard $A'$ as the longer, since it is moving relatively to him; and $S'$~must regard $A$ as the longer, since it is moving relatively to him. But this is impossible since, according to the equation $y = y'$, $S$~and $S'$ agree as to transverse measures. We see that the Lorentz transformation~\Eq{(5.1)} requires that $(x, y, z, t)$ and $(x', y', z', t')$ should be measured with standards of identical material constitution, but moving respectively with $S$ and~$S'$. This was really implicit in our deduction of the transformation, because the property of the two systems is that they give the same formula~\Eq{(5.2)} for the interval; and the test of complete similarity of the standards is equality of all corresponding intervals occurring in them. The fourth equation of~\Eq{(5.1)} is \index{Retardation of moving clocks}% \[ t = \beta (t' - ux'/c^{2}). \] Consider a clock recording the time~$t'$, which accordingly is at rest in $S'$'s system ($x = \text{const.}$). Then for any time-lapse by this clock, we have \[ \delta t = \beta\, \delta t', \] since $\delta x' = 0$. That is to say, $S$~does not accept the time as recorded by this moving clock, but multiplies its readings by~$\beta$, as though the clock were going slow. This agrees with the result already found in~\Eq{(4.9)}.