When time is involved other appliances are used for measuring intervals. If we have a mechanism capable of cyclic motion, its cycles will measure equal intervals provided the mechanism, its laws of behaviour, and all relevant surrounding circumstances, remain precisely similar. For the phrase ``precisely similar'' means that no observable differences can be detected in the mechanism or its behaviour; and that, as we have seen, requires that all corresponding intervals should be equal. In particular the interval between the events marking the beginning and end of the cycle is unaltered. Thus a clock primarily measures equal intervals; it is only under more restricted conditions that it also measures the time-coordinate~$t$. In general any repetition of an operation under similar conditions, but for a different time, place, orientation and velocity (attendant circumstances which have a relative but not an absolute significance\footnotemark),\footnotetext {They express relations to events which are not concerned in the test, e.g.\ to the sun and stars.} tests, equality of interval. It is obvious from common experience that intervals which can be measured with a clock cannot be measured with a scale, and \Foreign{vice versa}. We have thus two varieties of intervals, which are provided for in the formula~\Eq{(2.1)}, since $ds^{2}$~may be positive or negative and the measure of the interval will accordingly be expressed by a real or an imaginary number. The \index{Imaginary intervals}% abbreviated phrase ``imaginary interval'' must not be allowed to mislead; there is nothing imaginary in the corresponding relation; it is merely that in our arbitrary code an imaginary number is assigned as its measure-number. We might have adopted a different code, and have taken, for example, the antilogarithm of~$ds^{2}$ as the measure of the interval; in that case space-intervals would have received code-numbers from~$1$ to~$\infty$, and time-intervals numbers from~$0$ to~$1$. When we encounter~$\sqrt{-1}$ in our investigations, we must remember that it has been introduced by our choice of measure-code and must not think of it as occurring with some mystical significance in the external world. \PageSep{13} \Section{4.}{Rectangular coordinates and time} \index{Rectangular coordinates and time}% \index{Coordinate-systems!rectangular}%