We see then that unless the mathematician disregards the need for self-consistency in his rule, he must inevitably make his quantity~$\mf{A}$ either a contravariant or a covariant vector. The choice between these is entirely at his discretion. He might obtain a wider choice by disregarding the property of self-consistency---by selecting a particular coordinate-system, $x$,~$y$,~$z$, and insisting that values in other coordinate-systems must always be obtained by \PageSep{47} applying the rule immediately to $X$,~$Y$,~$Z$, and not permitting intermediate transformations. In practice he does not do this, perhaps because he can never make up his mind that any particular coordinates are deserving of this special distinction. We see now that a mathematical vector is a common name for an infinite \index{Vector!physical notion of}% number of sets of quantities, each set being associated with one of an infinite number of systems of coordinates. The arbitrariness in the association is removed by postulating that some method is followed, and that no one system of coordinates is singled out for special distinction. In technical language the transformations must form a \emph{Group. The quantity $(R, \Theta, \Phi)$ \index{Group}% is in no sense the same quantity as $(X, Y, Z)$}; they have a common name and a certain analytical connection, but the idea of anything like identity is entirely excluded from the mathematical notion of a vector. \Section{21.}{The physical notion of a vector} The components of a force $(X, Y, Z)$, $(X', Y', Z')$, etc. in different systems of Cartesian coordinates, rectangular or oblique, form a contravariant vector. This is evident because in elementary mechanics a force is resolved into components according to the parallelogram law just as a displacement~$dx_{\mu}$ is resolved, and we have seen that $dx_{\mu}$~is a contravariant vector. So far as the mathematical notion of the vector is concerned, the quantities $(X, Y, Z)$ and $(X', Y', Z')$ are not to be regarded as in any way identical; but in physics we conceive that both quantities express some kind of condition or relation of the world, and this condition is the same whether expressed by $(X, Y, Z)$ or by $(X', Y', Z')$. The physical vector is this vaguely conceived entity, which is independent of the coordinate-system, and is at the back of our measurements of force. A world-condition cannot appear directly in a mathematical equation; only the \emph{measure} of the world-condition can appear. Any number or set of numbers which can serve to specify uniquely a condition of the world may \index{Condition of the world}% be called a measure of it. In using the phrase ``condition of the world'' I intend to be as non-committal as possible; whatever in the external world determines the values of the physical quantities which we observe, will be included in the phrase.