The pure geometry associated with the general formula~\Eq{(2.1)} was studied by Riemann, and is generally called Riemannian geometry. It includes \index{Geometry, Riemannian}% \index{Riemannian geometry}% Euclidean geometry as a special case. \Section{3.}{Measurement of intervals} Consider the operation of proving by measurement that a distance~$AB$ is equal to a distance~$CD$. We take a configuration of events $LMNOP$\dots, viz.\ a measuring-scale, and lay it over~$AB$, and observe that $A$~and $B$ coincide with two particular events $P$,~$Q$ (scale-divisions) of the configuration. We find that the same configuration\footnote {The logical point may be noticed that the measuring-scale in two positions (necessarily at different times) represents the same \emph{configuration} of events, not the same events.} can also be arranged so that $C$~and $D$ coincide with $P$~and $Q$~respectively. Further we apply all possible tests to the measuring-scale to see if it has ``changed'' between the two measurements; and we are only satisfied that the measures are correct if no observable difference can be detected. According to our fundamental axiom, the absence of any observable difference between the two configurations (the structure of the measuring-scale in its two positions) signifies that the intervals are unchanged; in particular the interval between $P$ and~$Q$ is unchanged. It follows that the interval $A$ to~$B$ is equal to the interval $C$ to~$D$. We consider that the experiment proves equality of distance; but it is primarily a test of equality of interval. \PageSep{12} In this experiment time is not involved; and we conclude that in space considered apart from time the test of equality of distance is equality of interval. There is thus a one-to-one correspondence of distances and intervals. We may therefore adopt the same measure-number for the interval as is in general use for the distance, thus settling our plan of affixing measure-numbers to intervals. It follows that, when time is not involved, the interval reduces to the distance. It is for this reason that the quadratic form~\Eq{(2.1)} is needed in order to agree with observation, for it is well known that in three dimensions the square of the distance between two neighbouring points is a quadratic function of their infinitesimal coordinate-differences---a result depending ultimately on the experimental law expressed by Euclid \Vol{I},~47.