14 MONISM IN ARITHMETIC. 



differences exactly as we do with ordinary differences. Now it is 

 just this practical shift of imparting meanings to combinations, which 

 logically applied deduces naturally the whole system of arithmetic 

 from the idea of counting and of addition, and which we may char- 

 acterise, therefore, as the foundation-principle of its whole construc- 

 tion. This principle, which Hankel once called the principle of per- 

 manence, but which I prefer to call the PRINCIPLE OF NO EXCEPTION, 

 may be stated in general terms as follows : 



In the construction of arithmetic every combination of two previously 

 defined numbers by a sign for a previously defined operation-^plus, minus, 

 times, etc.*) shall be invested with meaning, even where the original defi- 

 nition of the operation used excludes such a combination; and the mean- 

 ing imparted is to be such that the combination considered shall obey the 

 same formula of definition as a combination having from the outset a sig- 

 nification, so that the old laws of reckoning shall still hold good and may 

 still be applied to it. 



A person who is competent to apply this principle rigorously 

 and logically will arrive at combinations of numbers whose results 

 are termed irrational or imaginary with the same necessity and fa- 

 cility as at the combinations above discussed, whose results are 

 termed negative numbers and zero. To think of such combinations 

 as results and to call the products reached also "numbers " is a mis- 

 use of language. It were better if we used the phrase forms of num- 

 bers for all numbers that are not the results of counting. But usus 

 tyrannus! 



It will now be my task to show how all numbers at which arith- 

 metic ever has arrived or ever can arrive naturally flow from the 

 simple application of the principle of no exception. 



Owing to the commutative and associative laws for addition it 

 is wholly indifferent for the result of a series of additive processes 

 in what order the numbers to be summed are added. For example, 



a + (b + c + </) -f (e +/) ^ja+*+<) + ( ! i+ e} + /. 

 The necessary consequence of this is that we may neglect the con- 

 sideration of the order of the numbers and give heed only to what 

 the quantities are that are to be summed, and, when they are equal, 

 take note of only two things, namely, of what the quantity which is 



