To take a special example, an observer at a station on the earth has found a particular set of coordinates $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$ best suited to his needs. He calls them $x$, $y$, $z$, $t$ in the belief that they are actually rectangular coordinates and time, and his terminology---straight line, circle, density, uniform velocity, etc.---follows from this identification. But, as shown in \SecRef{4}, this nomenclature can only agree with the measures made by clocks and scales provided \Eq{(16.2)}~is satisfied; and if \Eq{(16.2)}~is satisfied, the tracks of undisturbed particles must be straight lines. Experiment immediately shows that this is not the case; the tracks of undisturbed particles are parabolas. But instead of accepting the verdict of experiment and admitting that $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$ are not what he supposed they were, our observer introduces a field of force to explain why his test is not fulfilled. A certain part of this field of force might have been avoided if he had taken originally a different set of coordinates (not rotating with the earth); and in so far as the field of force arises on this account it is generally recognised that it is a mathematical fiction---the centrifugal force. But there is a residuum which cannot be got rid of by any choice of coordinates; there exists no extensive coordinate-system having the simple properties which were ascribed to $x$, $y$, $z$,~$t$. The intrinsic nature of space-time near the earth is not of the kind which admits coordinates with Galilean geometry. This irreducible field of force constitutes the field of terrestrial gravitation. The statement that space-time round the earth is ``curved''---that is to say, that it is not of the kind which admits Galilean coordinates---is not an hypothesis; it is an equivalent expression of the observed fact that an irreducible field of force is present, having regard to the Newtonian definition of force. It is this fact of observation which demands the introduction of non-Galilean space-time and non-Euclidean space into the theory. \Section{17.}{The Principle of Equivalence} In \SecRef{15} we have stated the laws of motion of undisturbed material particles and of light-pulses in a form independent of the coordinates chosen. Since a great deal will depend upon the truth of these laws it is desirable to \PageSep{40} consider what justification there is for believing them to be both accurate and universal. Three courses are open: \Item{(a)} It will be shown in Chapters \ChapNum{IV} and \ChapNum{VI} that these laws follow rigorously from a more fundamental discussion of the nature of matter and of electromagnetic fields; that is to say, the hypotheses underlying them may be pushed a stage further back.