110 THE PRINCIPLES OF SCIENCE. 



In a theoretical point of view we may conceive that the 

 abecedarium is always extended indefinitely. Every new 

 quality or circumstance which can belong to an object, 

 subdivides each combination or class, so that the number 

 of such combinations when unrestricted by logical con- 

 ditions is represented by an indefinitely high power of 

 two. The extremely rapid increase in the number of sub- 

 divisions obliges us to confine our attention to a few 

 circumstances at a time. 



When contemplating the properties of this abecedarium, 

 I am often inclined to think that Pythagoras perceived 

 the deep logical importance of duality ; for while unity 

 was the symbol of identity and harmony, he described the 

 number two as the origin of contrasts, or the symbol of 

 diversity, division and separation. The number four or 

 the Tetractys was also regarded by him as one of the chief 

 elements of existence, for it represented the generating 

 virtue whence come all combinations. 



In one of the golden verses ascribed to Pythagoras, he 

 conjures his pupil to be virtuous 6 : 



'By him who stampt T/ie Four upon the Mind, 

 The Four, the fount of Nature's endless stream.' 



Now four and the higher powers of duality do represent 

 in this logical system the variety of combinations which 

 can be generated in the absence of logical restrictions. The 

 followers of Pythagoras may have shrouded their master's 

 doctrines in mysterious and superstitious notions, but in 

 many points these doctrines seem to have some basis in 

 logical philosophy. 



The Logical Slate. 



To a person who has once comprehended the extreme 

 significance and utility of the Logical Abecedarium, the 



e Whewell, 'History of the Inductive Sciences,' vol. i. p. 222. 



