BOOK II, 



NUMBER, VAEIETY, AND PROBABILITY. 



CHAPTER VIII. 



PRINCIPLES OF NUMBER. 



NOT without reason did Pythagoras represent the world 

 as ruled by number. Into almost all our acts of thought 

 number enters, and in proportion as we can define numeri- 

 cally we enjoy exact and useful knowledge of the Universe. 

 The science of numbers, too, has hitherto presented the 

 widest and most practicable training in logic. So free and 

 energetic has been the study of mathematical forms, com- 

 pared with the forms of logic, that mathematicians have 

 passed far in advance of pure logicians. Occasionally, in 

 recent times, they have condescended to apply their 

 algebraic instrument to a reflex treatment of the primary 

 logical science. It is thus that we owe to profound mathe- 

 maticians, such as John Herschel, Whewell, De Morgan, or 

 Boole, the regeneration of logic in the present century. I 

 entertain no doubt that it is in maintaining a close alliance 

 with quantitative reasoning that we must look for further 

 progress in our comprehension of qualitative inference. 



I cannot assent, indeed, to the common notion that 

 certainty begins and ends with numerical determination. 

 Nothing is more certain than logical truth. The laws of 

 identity and difference are the tests of all that is certain 



