684 SCIENCE AND HYPOTHESIS 



failed. The obstacle which prevented us from proving the law of 

 acceleration is that we have no definition of force. This obstacle 

 subsists in its entirety, since the principle invoked has not furnished 

 us with the missing definition. The principle of relative motion is 

 none the less very interesting, and deserves to be considered for its 

 own sake. Let us try to enunciate it in an accurate manner. We have 

 said above that the accelerations of the different bodies which form 

 part of an isolated system only depend on their velocities and their 

 relative positions, and not on their velocities and their absolute posi- 

 tions, provided that the movable axes to which the relative motion is 

 referred move uniformly in a straight line ; or, if it is preferred, their 

 accelerations depend only on the differences of their velocities and the 

 differences of their co-ordinates, and not on the absolute values of 

 these velocities and co-ordinates. If this principle is true for relative 

 accelerations, or rather for differences of acceleration, by combining 

 it with the law of reaction we shall deduce that it is true for 

 absolute accelerations. It remains to be seen how we can prove 

 that differences of acceleration depend only on differences of 

 velocities and co-ordinates; or, to speak in mathematical language, 

 that these differences of co-ordinates satisfy differential equations of 

 the second order. Can this proof be deduced from experiment or 

 from a priori conditions? Remembering what we have said before, 

 the reader will give his own answer. Thus enunciated, in fact, the 

 principle of relative motion curiously resembles what I called above 

 the generalized principle of inertia; it is not quite the same thing, 

 since it is a question of differences of co-ordinates, and not of the co- 

 ordinates themselves. The new principle teaches us something more 

 than the old, but the same discussion applies to it, and would lead 

 to the same conclusions. We need not recur to it. 



Newton's Argument. Here we find a very important and even 

 slightly disturbing question. I have said that the principle of relative 

 motion was not for us simply a result of experiment ; and that a priori 

 every contrary hypothesis would be repugnant to the mind. But, then, 

 why is the principle only true if the motion of the movable axes is 

 uniform and in a straight line? It seems that it should be imposed 

 upon us with the same force if the motion is accelerated, or at any rate 

 if it reduces to a uniform rotation. In these two cases, in fact, the 

 principle is not true. I need not dwell on the case in which the mo- 

 tion of the axes is in a straight line and not uniform. The paradox 

 does not bear a moment's examination. If I am in a railway carriage, 

 and if the train, striking against any obstacle whatever, is suddenly 

 stopped, I shall be projected on to the opposite side, although I have 

 not been directly acted upon by any force. There is nothing mysteri- 

 ous in that, and if I have not been subject to the action of any exter- 

 nal force, the train has experienced an external impact. There can be 



