Suppose that an observer has chosen a definite system of space-coordinates and of time-reckoning $(x_{1}, x_{2}, x_{3}, x_{4})$ and that the geometry of these is given by \[ ds^{2} = g_{11}\, dx_{1}^{2} + g_{22}\, dx_{2}^{2} + \dots + 2g_{12}\, dx_{1}\, dx_{2}^{2} + \cdots. \Tag{(16.1)} \] Let him be under the mistaken impression that the geometry is \[ ds_{0}^{2} = -dx_{1}^{2} - dx_{2}^{2} - dx_{3}^{2} + dx_{4}^{2}, \Tag{(16.2)} \] that being the geometry with which he is most familiar in pure mathematics. We use~$ds_{0}$ to distinguish his mistaken value of the interval. Since intervals can be compared by experimental methods, he ought soon to discover that his~$ds$ cannot be reconciled with observational results, and so realise his mistake. But the mind does not so readily get rid of an obsession. It is more likely that our observer will continue in his opinion, and attribute the discrepancy of the observations to some influence which is present and affects the behaviour of his test-bodies. He will, so to speak, introduce a supernatural agency which he can blame for the consequences of his mistake. Let us examine what name he would apply to this agency. Of the four test-bodies considered the moving particle is in general the most sensitive to small changes of geometry, and it would be by this test that the observer would first discover discrepancies. The path laid down for it by our observer is \[ \int ds_{0} \text{ is stationary,} \] \PageSep{38} i.e.\ a straight line in the coordinates $(x_{1}, x_{2}, x_{3}, x_{4})$. The particle, of course, pays no heed to this, and moves in the different track \[ \int ds \text{ is stationary.} \] Although apparently undisturbed it deviates from ``uniform motion in a straight line.'' The name given to any agency which causes deviation from uniform motion in a straight line is \emph{force} according to the Newtonian definition of force. Hence the agency invoked through our observer's mistake is described as a ``field of force.'' The field of force is not always introduced by inadvertence as in the foregoing illustration. It is sometimes introduced deliberately by the mathematician, e.g.\ when he introduces the centrifugal force. There would be little \index{Centrifugal force}% advantage and many disadvantages in banishing the phrase ``field of force'' from our vocabulary. We shall therefore regularise the procedure which our observer has adopted. We call~\Eq{(16.2)} the \emph{abstract geometry} of the system of coordinates $(x_{1}, x_{2}, x_{3}, x_{4})$; it may be chosen arbitrarily by the observer. The \index{Natural coordinates!geometry}% \emph{natural geometry} is~\Eq{(16.1)}. \emph{A field of force represents the discrepancy between the natural geometry of a coordinate-system and the abstract geometry arbitrarily ascribed to it.}