COMBINATIONS AND PERMUTATIONS. 



215 



with negative terms. I may arrange the combinations as 

 follows : 



ABCD . Four out of four . . i combination. 



ABCd 

 ABcD 



AZ>CD 

 aBCD 



KBcd 



* Three out of four . . 4 combinations. 



AlCd 



aBcD 



t 

 Two out of four . . 6 combinations. 



Abed 

 aBcd 

 abCd 

 abcD 



One out of four . . 

 abed None out of four . 



4 combinations. 

 i combination. 



The numbers, it will be noticed, are exactly the same 

 as those in the fifth line of the arithmetical triangle, and 

 an exactly similar correspondence would be found to 

 exist in the case of each other column of the Abece- 

 darium. 



Numerical abstraction, it has bean asserted, consists in 

 overlooking the kind of difference, and retaining only a 

 consciousness of its existence (p. 177). While in logic, 

 then, we have to deal with each combination as a separate 

 kind of thing, in arithmetic we can distinguish only the 

 classes which depend upon more or less positive terms 

 being present, and the numbers of these classes imme- 

 diately produce the numbers of the arithmetical triangle. 



It may here be pointed out that there are two modes 



