Accordingly the mathematician begins by giving us a list of the numbers that $\mf{A}$~will signify in the different coordinate-systems. We here denote these numbers by letters. $\mf{A}$~will mean\footnote {For convenience I take a three-dimensional illustration.} \begin{align*} &\text{$X$, $Y$, $Z$ for certain rectangular coordinates $x$, $y$, $z$,} \\ &\text{$R$, $\Theta$, $\Phi$ for certain polar coordinates $r$, $\theta$, $\phi$,} \\ &\text{$\Lambda$, $M$, $N$ for certain ellipsoidal coordinates $\lambda$, $\mu$, $\nu$.} \end{align*} ``But,'' says the mathematician, ``I shall never finish at this rate. There are an infinite number of coordinate-systems which I want to use. I see that I must alter my plan. I will give you a general rule to find the new values of~$\mf{A}$ when you pass from one coordinate-system to another; so that it is only necessary for me to give you one set of values and you can find all the others for yourselves.'' In mentioning a \emph{rule} the mathematician gives up his arbitrary power of making~$\mf{A}$ mean anything according to his fancy at the moment. He binds himself down to some kind of regularity. Indeed we might have suspected that our orderly-minded friend would have some principle in his assignment of different meanings to~$\mf{A}$. But even so, can we make any guess at the rule he is likely to adopt unless we have some idea of the problem he is working at in which~$\mf{A}$ occurs? I think we can; it is not necessary to know anything about the nature of his problem, whether it relates to the world of physics or to something purely conceptual; it is sufficient that we know a little about the nature of a mathematician. What kind of rule could he adopt? Let us examine the quantities which can enter into it. There are first the two sets of numbers to be connected, say, $X$, $Y$, $Z$ and $R$, $\Theta$, $\Phi$. Nothing has been said as to these being analytical functions of any kind; so far as we know they are isolated numbers. Therefore there can be no question of introducing their derivatives. They are regarded as located at some point of space $(x, y, z)$ and $(r, \theta, \phi)$, otherwise the question of coordinates could scarcely arise. They are changed because the coordinate-system has changed \emph{at this point}, and that change is defined by quantities like $\dfrac{\dd r}{\dd x}$, $\dfrac{\dd^{2}\theta}{\dd x\, \dd y}$, and so on. The integral coordinates themselves, $x$,~$y$,~$z$, $r$,~$\theta$,~$\phi$, cannot be involved; because they express relations to a distant origin, whereas we are concerned only with changes at the spot where $(X, Y, Z)$ is located. Thus the rule must involve only the numbers $X$,~$Y$,~$Z$, $R$,~$\Theta$,~$\Phi$ combined with the mutual derivatives of $x$,~$y$,~$z$, $r$,~$\theta$,~$\phi$. \PageSep{46}