The simplest case is when the condition of the world under consideration can be indicated by a single measure-number. Take two such conditions underlying respectively the wave-length~$\lambda$ and period~$T$ of a light-wave. We have the equation \[ \lambda = 3 · 10^{10}\, T. \Tag{(21.1)} \] This equation holds only for one particular plan of assigning measure-numbers (the \CGS\ system). But it may be written in the more general form \[ \lambda = cT, \Tag{(21.2)} \] where $c$~is a velocity having the value $3 · 10^{10}$ in the \CGS\ system. This \PageSep{48} comprises any number of particular equations of the form~\Eq{(21.1)}. For each measure-plan, or system of units, $c$~has a different numerical value. The method of determining the necessary change of~$c$ when a new measure-plan is adopted, is well known; we assign to it the \emph{dimensions} length $\div$ time, and by a simple rule we know how it must be changed when the units of~$\lambda$ and~$T$ are changed. For any general equation the total dimensions of every term ought to be the same. The tensor calculus extends this \emph{principle of dimensions} to changes of \index{Dimensions, principle of}% \index{Principle!of dimensions}% measure-code much more general than mere changes of units. There are \index{Measure-code}% \index{Unit!change of}% conditions of the world which cannot be specified by a single measure-number; some require~$4$, some~$16$, some~$64$, etc., measure-numbers. Their variety is such that they cannot be arranged in a single serial order. Consider then an equation between the measure-numbers of two conditions of the world which require $4$~measure-numbers. The equation, if it is of the necessary general type, must hold for every possible measure-code; this will be the case if, when we transform the measure-code, both sides of the equation are transformed in the same way, i.e.\ if we have to perform the same series of mathematical operations on both sides. We can here make use of the mathematical vector of \SecRef{20}. Let our equation in some measure-code be \[ A_{1}, A_{2}, A_{3}, A_{4} = B_{1}, B_{2}, B_{3}, B_{4}. \Tag{(21.3)} \] Now let us change the code so that the left-hand side becomes \emph{any} four numbers $A_{1}'$, $A_{2}'$, $A_{3}'$, $A_{4}'$. We identify this with the transformation of a covariant vector by associating with the change of measure-code the corresponding transformation of coordinates from~$x_{\mu}$ to~$x_{\mu}'$ as in~\Eq{(19.2)}. But since \Eq{(21.3)}~is to hold in all measure-codes, the transformation of the right-hand side must involve the same set of operations; and the change from $B_{1}$, $B_{2}$, $B_{3}$, $B_{4}$ to $B_{1}'$, $B_{2}'$, $B_{3}'$, $B_{4}'$ will also be the transformation of a covariant vector associated with the \emph{same} transformation of coordinates from~$x_{\mu}$ to~$x_{\mu}'$.