392 THE PRINCIPLES OF SCIENCE. 



I will go a step further, and say that we must have 

 places in our scientific classifications for purely imaginary 

 existences. A very large proportion of the mathematical 

 functions which are conceivable have no application to the 

 circumstances of this world. Physicists certainly do in- 

 vestigate the nature and consequences of forces which 

 nowhere exist. Newton's * Principia ' is full of such inves- 

 tigations. In one chapter of his 'Mecanique Celeste' 

 Laplace indulges in a remarkable speculation as to what 

 the laws of motion would have been if momentum instead 

 of varying simply as the velocity had been a more com- 

 plicated function of it. I have already mentioned (vol. i. 

 p. 256) that Sir George Airy contemplated the existence 

 of a world in which the laws of force should be such that 

 a perpetual motion would be possible, and the Law of 

 Conservation of Energy would not hold true. 



Thought is not bound down to the limits of what is mate- 

 rially existent, but is circumscribed only by those Funda- 

 mental Laws of Identity, Contradiction and Duality, which 

 were laid down at the outset. This is the point at which 

 I should differ from Mr. Herbert Spencer. He appears to 

 suppose that a classification is complete if it has a place 

 for every existing object, and this may perhaps seem to 

 be practically sufficient ; but it is subject to two profound 

 objections. Firstly, we do not know all that exists, and 

 therefore in limiting our classes we are erroneously omitting 

 multitudes of objects of unknown form and nature which 

 may exist either on this earth or in other parts of space. 

 Secondly, as I have explained, the powers of thought are 

 not limited by material existences, and we may or, for some 

 purposes, must imagine objects which probably do not 

 exist, and if we imagine them we ought (strictly speak- 

 ing) to find appropriate places for them in the classifi- 

 cations of science. 



The chief difficulty of this subject, however, consists in 



