COMBINATIONS AND PERMUTATIONS. 



215 



with negative terms. I may arrange the combinations as 

 follows : 



ABCD . Four out of four . . i combination. 

 ABCc/ 



- Three out of four . 4 combinations. 

 A5CD 



aBCD 



Two out of four . . 6 combinations. 



aECd 

 aBcD 

 a&CD 



&amp;gt; One out of four ... 4 combinations. 



J 

 abed None out of four . . 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 been 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 



