A vector may either be a single set of four quantities associated with a special point in space-time, or it may be a set of four functions varying continuously with position. Thus we can have an ``isolated vector'' or a ``vector-field.'' For an illustration of a covariant vector we considered the gradient of an invariant, $\dd\phi/\dd x_{\mu}$; but a covariant vector is not necessarily the gradient of an invariant. The reader will probably be already familiar with the term vector, but the distinction of covariant and contravariant vectors will be new to him. This is because in the elementary analysis only rectangular coordinates are contemplated, and for transformations from one rectangular system to another the laws \Eq{(19.1)} and \Eq{(19.2)} are equivalent to one another. From the geometrical point of view, the \emph{contravariant vector} is the vector with which everyone is familiar; this is because a displacement, or directed distance between two points, is regarded as representing $(dx_{1}, dx_{2}, dx_{3})$\footnote {The customary resolution of a displacement into components in oblique directions assumes this.} which, as we have seen, is contravariant. The covariant vector is a new conception which does not so easily lend itself to graphical illustration. \Section{20.}{The mathematical notion of a vector} The formal definitions in the preceding section do not help much to an understanding of what the notion of a vector really is. We shall try to explain this more fully, taking first the mathematical notion of a vector (with which we are most directly concerned) and leaving the more difficult physical notion to follow. We have a set of four numbers $(A_{1}, A_{2}, A_{3}, A_{4})$ which we associate with some point $(x_{1}, x_{2}, x_{3}, x_{4})$ and with a certain system of coordinates. We make a change of the coordinate-system, and we ask, What will these numbers become in the new coordinates? The question is meaningless; they do not automatically ``become'' anything. Unless we interfere with them they stay as they were. But the mathematician may say ``When I am using the coordinates $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$, I want to talk about the numbers $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$; and when I am using $x_{1}'$, $x_{2}'$, $x_{3}'$, $x_{4}'$, I find that at the corresponding stage of my work I shall want to talk about four different numbers $A_{1}'$, $A_{2}'$, $A_{3}'$, $A_{4}'$. \PageSep{45} %[** TN: Nine fraktur "A"s set \textgoth in the original] So for brevity I propose to call both sets of numbers by the same symbol~$\mf{A}$.'' We reply ``That will be all right, provided that you tell us just what numbers will be denoted by~$\mf{A}$ for \emph{each} of the coordinate-systems you intend to use. Unless you do this we shall not know what you are talking about.''