We thus arrive at the result that in an equation which is independent of the measure-plan both sides must be covariant or both contravariant vectors. We shall extend this later to conditions expressed by $16$, $64$,~\dots, measure-numbers; the general rule is that both sides of the equation must have the same elements of covariance and contravariance. Covariance and contravariance are a kind of generalised dimension, showing how the measure of one condition of the world is changed when the measure of another condition is changed. The ordinary theory of change of units is merely an elementary case of this. Coordinates are the identification-numbers of the points of space-time. There is no fundamental distinction between measure-numbers and identification-numbers, so that we may regard the change of coordinates as part of the general change applied to all measure-numbers. The change of coordinates \PageSep{49} no longer leads the way, as it did in \SecRef{20}; it is placed on the same level with the other changes of measure. When we applied a change of measure-code to~\Eq{(21.3)} we associated with it a change of coordinates; but it is to be noted that the change of coordinates \index{Coordinates!representation of displacement}% was then ambiguous, since the two sides of the equation might have been taken as both contravariant instead of both covariant; and further the change did not refer explicitly to coordinates in the world---it was a mere entry in the mathematician's note-book in order that he might have the satisfaction of calling $A_{\mu}$ and~$B_{\mu}$ vectors consistently with his definition. Now if the measure-plan of a condition~$A_{\mu}$ is changed the measures of other conditions and relations associated with it will be changed. Among these is a certain relation of two events which we may call the \emph{aspect}\footnote {The relation of \emph{aspect} (or in its graphical conception \emph{displacement}) with four measure-numbers \index{Aspect, relation of}% \index{Displacement}% seems to be derived from the relation of \emph{interval} with one measure-number, by taking account not only of the mutual interval between the two events but also of their intervals from all surrounding events.} of one from the other; and this relation requires four measure-numbers to specify it. Somewhat arbitrarily we decide to make the aspect a contravariant vector, and the measure-numbers assigned to it are denoted by~$(dx)^{\mu}$. That settles the ambiguity once for all. For obscure psychological reasons the mind has singled out this transcendental relation of aspect for graphical representation, so that it is conceived by us as a \emph{displacement} or difference of location in a frame of space-time. Its measure-numbers $(dx)^{\mu}$ are represented graphically as coordinate-differences~$dx_{\mu}$, and so for each measure-code of aspect we get a corresponding coordinate-frame of location. This ``real'' coordinate-frame can now