112 THE PRINCIPLES OF SCIENCE. 



be written down on the unoccupied part of the slate, and 

 the proper series of combinations being chosen, the contra 

 dictory combinations can be struck out with the pencil. 

 I have used a slate of this kind, which I call a Logical 

 Slate, for more than ten years, and it has saved me much 

 trouble. It is hardly possible to apply this process to 

 problems of more than six terms, owing to the large num 

 ber of combinations which would require examination ; 

 thus seven terms would give 128 combinations, eight 

 terms 256, nine terms 512, ten terms 1024, eleven terms 

 2048, twelve terms 4096, and so on in geometrical pro 

 gression. 



Abstraction of Indifferent Circumstances. 



There is a simple but highly important process of 

 inference which enables us to abstract, eliminate or disre 

 gard all circumstances indifferently present and absent. 

 Thus if I were to state that a triangle is a figure of 

 three sides, with or without equal angles/ the latter 

 qualification would be superfluous, because by a law of 

 thought I know that angles must be either equal or 

 unequal. To add the qualification gives no new know 

 ledge since the existence of the two alternatives will be 

 understood in the absence of any information to the 

 contrary. Accordingly, when two alternatives differ only 

 as regards a single component term which is positive in 

 one and negative in the other, we may always reduce 

 them to one term by striking out their indifferent part. 

 It is really a process of substitution which enables us 

 to do this ; for having any proposition of the form 



A = ABC -|- ABc, (i) 



we know by the Law of Duality that 



B = BC -|- Be. (2) 



Hence AB = ABC | ABc. (3) 



