214 THE PRINCIPLES OF SCIENCE. 







By taking a proper line of the triangle, an answer 

 may be had under any more natural supposition. This 

 theory of comparative frequency of divergence from an 

 average, was first adequately noticed by M. Quetelet, and 

 has lately been employed in a very interesting and bold 

 manner by Mr. Galton, in his work on ' Hereditary Genius.' 

 We shall afterwards find that the theory of error, to which 

 is made the ultimate appeal in cases of quantitative in- 

 vestigation, is founded upon the comparative numbers of 

 combinations as displayed in the triangle. 



Connection between the Arithmetical Triangle and the 

 Logical A becedarium. 



There exists a close connection between the arith- 

 metical triangle described in the last section, and the 

 series of combinations of letters called the Logical Abece- 

 darium. 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 

 was twice as great as that for the preceding line (p. 210), 

 so each column of the Abecedarium (p. 109) contained 

 twice as many combinations as the preceding one. The 

 like correspondence would also exist 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 one column of the Abecedarium, we 

 shall also find that the combinations naturally group 

 themselves according to the figurate 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 



