ix.] COMBINATIONS AND PERMUTATIONS. 189 



comparative numbers of combinations as displayed in the 

 triangle. 



Connection between the Arithmetical Triangle and the 

 Logical Alphabet. 



There exists a close connection between the arithmetical 

 triangle described in the last section, and the series of 

 combinations of letters called the Logical Alphabet. The 

 one is to mathematical science what the other is to 

 logical science. In fact the figurate numbers, or those 

 exhibited in the triangle, are obtained by summing up the 

 logical combinations. Accordingly, just as the total of the 

 numbers in each line of the triangle is twice as great as 

 that for the preceding line (p. 1 86), so each column of the 

 Alphabet (p. 94) contains twice as many combinations as 

 the preceding one. The like correspondence also exists 

 between the sums of all the lines of figures down to any 

 particular line, and of the combinations down to any 

 particular column. 



By examining any column of the Logical Alphabet we 

 find that the combinations naturally group themselves 

 according to the fisurate numbers. Take the combinations 

 of the letters A, B, C, D ; they consist of all the ways in 

 which I can choose four, three, two, one, or none of the 

 four letters, filling up the vacant spaces with negative 

 terms. 



There is one combination, ABCD, in which all the 

 positive letters are present, ; there are four combinations in 

 each of which three positive letters are present; six in 

 which two are present ; four in which only one is present ; 

 and, finally, there is the single case, abed, in which all 

 positive letters are absent. These numbers, I, 4, 6, 4, i, 

 are those of the fifth line of the arithmetical triangle, and 

 a like correspondence will be found to exist in each 

 column of the Logical Alphabet. 



Numerical abstraction, it has been asserted, consists in 

 overlooking the kind of difference, and retaining only a 

 consciousness of its existence (p. 158). While in logic, 

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

 kind of thing, in arithmetic we distinguish only the classes 

 which depend upon more or less positive terms being 



