XXX.] CLASSIFICATION. 707 



would have been if momentum, instead of varying simply 

 as the velocity, had been a more complicated function of 

 it. 1 have already mentioned (p. 223) that Airy contem-- 

 plated the existence of a world in wliich 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 

 materially existent, but is circumscribed only by those 

 Fundamental Laws of Identity, Contradiction and Duality, 

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

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

 suppose that a classiiication 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 to find places for 

 tliem in the classilications of science. 



The chief diificulty of this subject, however, consists in 

 the fact that mathematical or other certain laws may 

 entirely forbid the existence of some combinations. The 

 circle may be defined as a plane curve of equal curvature, 

 and it is a property of the circle that it contains the greatest 

 area within the least possible perimeter. May we then 

 contemplate mentally a circle not a figure of greatest pos- 

 sible area? Or, to take a still simpler example, a paralleh)- 

 gram pos.sesses the property of having the oj)posite angles 

 equal. May we then mentally divide parallelograms into 

 two classes acc(jrding as they do or do not have their 0])po- 

 site angles equal ? it might seem absurd to do so, because 

 we know that one of the two species of parallelogram 

 would be non-existent. But, then, unless the student had 

 previously contemplated the existence of both species as 

 possible, what is the meaning of the thirty-fourth jiroposi- 

 tion of Euclid's first book ? We cannot deny or disprove 

 the existence of a certain combination without thereby in 



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