Granting this proviso, we have reduced the general expression to \[ ds^{2} = dx^{2} + dy^{2} + dz^{2} + dy_{4}^{2}, \Tag{(4.4)} \] where $x$, $y$, $z$ will be recognised by us as rectangular coordinates in space. Clearly $y_{4}$~must involve the time, otherwise our location of events by the four coordinates would be incomplete; but we must not too hastily identify it with the time~$t$. \PageSep{14} I suppose that the following would be generally accepted as a satisfactory (pre-relativity) definition of equal time-intervals:---if we have a mechanism \index{Time!definition of}% capable of cyclic motion, its cycles will measure equal durations of time \emph{anywhere} and \emph{anywhen}, provided the mechanism, its laws of behaviour, and all outside influences remain precisely similar. To this the relativist would add the condition that the mechanism (as a whole) must be at rest in the space-time frame considered, because it is now known that a clock in motion goes slow in comparison with a fixed clock. The non-relativist does not disagree in fact, though he takes a slightly different view; he regards the proviso that the mechanism must be at rest as already included in his enunciation, because for him motion involves progress through the aether, which (he considers) directly affects the behaviour of the clock, and is one of those ``outside influences'' which have to be kept ``precisely similar.'' Since then it is agreed that the mechanism as a whole is to be at rest, and the moving parts return to the same positions after a complete cycle, we shall have for the two events marking the beginning and end of the cycle \[ dx,\ dy,\ dz = 0. \] Accordingly \Eq{(4.4)} gives for this case \[ ds^{2} = dy_{4}^{2}. \] We have seen in \SecRef{3} that the cycles of the mechanism in all cases correspond to equal intervals~$ds$; hence they correspond to equal values of~$dy_{4}^{2}$. But by the above definition of time they also correspond to equal lapses of time~$dt$; hence we must have $dy_{4}$~proportional to~$dt$, and we express this proportionality by writing \[ dy_{4} = ic\, dt, \Tag{(4.5)} \] where $i = \sqrt{-1}$, and $c$~is a constant. It is, of course, possible that $c$~may be an imaginary number, but provisionally we shall suppose it real. Then \Eq{(4.4)} becomes \[ ds^{2} = dx^{2} + dy^{2} + dz^{2} - c^{2}\, dt^{2}. \Tag{(4.6)} \]