Reason and Invention. 849 



equal 6, which was to be proved. If any question is raised as to such result 

 of taking away y, the geometer says the proposition is an axiom, that when 

 equal things are taken from equals, the remaining things are equal, and he 

 cannot prove it any further. But if you cannot see it in the present form of 

 statement, it may be illustrated in some other wa} r . If a boy has five mar- 

 bles in one hand, and an equal number in the other, and you take two 

 from each hand, he will have still an equal number in each hand. This 

 is obviously no reduction of the proposition, but only another view of it, 

 and if the student cannot perceive it in some such form, his cerebral 

 equipment is not adapted to the apprehension of mathematical percep- 

 tions. And this, I take it, is equivalent to saying that the sensation 

 made by one of the equal qualities is not retained in the sensorium long 

 enough to allow the second one to make its sensation during the con- 

 tinuance of the first, and so the duplex, or rather compound sensation 

 which constitutes perception is not experienced. Scarcely any normal 

 human brain would be unequal to so simple a perception. It is almost, 

 if not quite, as simple, and of the same sort as a perception of motion 

 in a moving bodj T . We see the body in a certain place, and before the 

 sensation caused by such view is dulled by the relapse of the erethism 

 of the cells involved, a new stimulation supervenes, of the body in a 

 new position. It is the lapping over, or superposition of each of these 

 slightly differing sensations before its predecessor has faded out, that 

 gives the perception of a new Duality, a continuously changing relation- 

 ship which a single instantaneous sensation could not give. 



This will be more readily appreciated when it is considered that a 

 wheel revolving so rapidly that its spokes cannot be distinguished will, 

 if illuminated by an electric discharge, appear as a wheel standing stock 

 still. In this case the duration of the electric spark is too short to ad- 

 mit of more than one sensation, and there being no succession of sensa- 

 tions there is no perception of motion. It thus appears upon analysis 

 that every perception, and finally every act of reasoning based upon per- 

 ception, may be traced to simple sensation and sense impression. 

 J. Stuart Mill says, ' ' A 11 inference is from particulars to particulars. 

 General propositions are merely registers of such inferences already 

 made and short formulae for making more. " " There is a petitio principii 

 in every syllogism and nothing is really proved by one. Thus to say, 

 all men are mortal, Socrates is a man, therefore Socrates is mortal, 

 the conclusion is assumed in the premise and included in it, 

 not proved by it. " What we really do in such a form of speech is 

 not to piove something, but to predict something. A prediction is 

 simply a recollection in association with an idea of futurity. Our pre- 

 diction that Brown will die is the recollection of the death of other 

 men applied to our idea of Brown. .Our recollections of Brown's life 



