To put the contrast in another form, in our common outlook the idea of \index{Extension and location}% \index{Location and extension}% position or \emph{location} seems to be fundamental. From it we derive distance or \emph{extension} as a subsidiary notion, which covers part but not all of the conceptions which we associate with location. Position is looked upon as the physical fact---a coincidence with what is vaguely conceived of as an identifiable point of space---whereas distance is looked upon as an abstraction or a computational result calculable when the positions are known. The view which we are going to adopt reverses this. Extension (distance, interval) is now fundamental; and the location of an object is a computational result summarising the physical fact that it is at certain intervals from the other objects in the world. Any idea contained in the concept location which is not \PageSep{10} expressible by reference to distances from other objects, must be dismissed from our minds. Our ultimate analysis of space leads us not to a ``here'' and a ``there,'' but to an extension such as that which relates ``here'' and ``there.'' To put the conclusion rather crudely---space is not a lot of points close together; it is a lot of distances interlocked. Accordingly our fundamental hypothesis is that--- \emph{Everything connected with location which enters into observational knowledge---everything we can know about the configuration of events---is contained \index{Configuration of events}% in a relation of extension between pairs of events.} This relation is called the \emph{interval}, and its measure is denoted by~$ds$.% \index{Interval}% If we have a system~$S$ consisting of events $A$,~$B$, $C$, $D$,~\dots, and a system~$S'$ consisting of events $A'$,~$B'$, $C'$, $D'$,~\dots, then the fundamental hypothesis implies that the two systems will be exactly alike observationally if, and only if, all pairs of corresponding intervals in the two systems are equal, $AB = A'B'$, $AC = A'C'$,~\dots. In that case if $S$~and $S'$ are material systems they will appear to us as precisely similar bodies or mechanisms; or if $S$~and $S'$ correspond to the same material body at different times, it will appear that the body has not undergone any change detectable by observation. But the position, motion, or orientation of the body may be different; that is a change detectable by observation, not of the system~$S$, but of a wider system comprising $S$ and surrounding bodies. Again let the systems $S$ and $S'$ be abstract coordinate-frames of reference, the events being the corners of the meshes; if all corresponding intervals in the two systems are equal, we shall recognise that the coordinate-frames are of precisely the same kind---rectangular, polar, unaccelerated, rotating, etc. \Section{2.}{The fundamental quadratic form}