8 J. G, MacGEBGOE on THE 



iug ou the particles of the system, iu which case the forces acting on the particles of the 

 system become pairs of equal and opposite forces acting between them and the body 

 which is being used as point of reference. 



We now come to the question of the independence of the laws of motion. Are they 

 independent of one another and of other more simple axioms, or are they capable of de- 

 duction ? It is obviously impossible to give a final answer to this question. All that is 

 possible is to discuss various efforts which have been made to deduce them and to reach 

 a conclusion as to whether or not they have been successful. 



"With regard to the first law. Maxwell ' has maintained that " the denial of it is in 

 contradiction to the only system of consistent doctrine about space and time which the 

 human mind has been able to form." If this be so, it must be possible to deduce the law 

 from the doctrine of space and time, and the law cannot therefore be held to be hypothe- 

 tical in character. Maxwell's argument is as follows : — " If the velocity [of a body freed 

 from the action of force] does not remain constant, let us suppose it to vary. The change 

 of velocity must have a definite direction and magnitude. By the maxim of Art. xix. 

 [that the same causes will always produce the same effects], this variation must be the 

 same, whatever be the time or place of the experiment. The direction of the change of 

 motion must therefore be determined either by the direction of the motion itself or by 

 some direction fixed in the body. Let us, iu the first place, suppose the law to be that 

 the velocity diminishes at a certain rate The velocity referred to in this hypotheti- 

 cal law can only be the velocity referred to a point absolutely at rest. For if it is a rela- 

 tive velocity, its direction as well as its magnitude depends on the velocity of the point of 



reference Hence the hypothetical law is without meaning unless we admit the 



possibility of defining absolute rest and absolute velocity." 



Tait- says of this argument that it " is a good example of a valuable application of a 

 principle which, iu its widest scope, is inconsistent with the true foundations of physical 

 science. It is in fact the exceedingly dangerous ' principle of sufficient reason ' — which 

 requires for its legitimate use the utmost talent and knowledge on the part of the user." 

 Fortunately the utmost talent and knowledge are not requisite on the part of the critic of 

 its use ; and I hope, therefore, it may not be regarded as presumptuous in me to endeavour 

 to prove the argument to be fallacious. 



One is led to think it must be so by noticing that it may be used to prove Newton's 

 law, as well as the law assumed for the sake of argument to replace it, to be without 

 meaning. For this purpose all that is necessary is to substitute déplacement for velocity 

 or motion, wherever these words occur in the above quotation, and changes for diminishes. 

 The argument is thus transformed into one equally good or bad, in favour of the cessation 

 of motion on the cessation of the action of force, as against Newton's law. 



The fallacy seems to me to lie in the incomplete recognition of the relativity of a law 

 of the kind under consideration. When it is recognized that a velocity whether uniform 

 or variable can be specified only relatively to some point or points it becomes obvious 

 that not only Maxwell's hypothetical law^, but Newton's law also, and indeed, any law 

 giving the variation of the velocity of a body under any circumstances, must be without 



' ' Matter and Motion,' Art. xli. 



■ Ency. Brit, 9th Ed., Art. Mechanics, ? 298. 



