216 THE PRINCIPLES OF SCIENCE. 



in which we can calculate the whole number of com 

 binations of certain things. Either we may take the 

 whole number at once as shown in the Abecedarium, in 

 which case the number will be some power of two, or 

 else we may calculate successively, by aid of permutations, 

 the number of combinations of none, one, two, three, and 

 so on. Hence we arrive at a necessary identity between 

 two series of numbers. In the case of four things we 

 shall have 



2 4 = i + - + 4 3 + 4 3 2 + 4 -3-2.1 



I 1.2 1.2.3 I . 2 . 3 . 4 



In a general form of expression we shall have 



n n n . (ni) n (n-i\ (nz\ 

 2 =i+- + - / 4. 1 LL _J- + &e 



I 1.2 1.2-3 



the terms being continued until they cease to have any 

 value. Thus we have arrived at a proof of simple cases 

 of the Binomial Theorem, of which each column of the 

 Abecedarium is an exemplification. It may be shown 

 that all other mathematical expansions likewise arise out 

 of simple processes of combination, but the more complete 

 consideration of this subject must be deferred. 



Possible Variety of Nature and Art. 



We cannot adequately understand the difficulties which 

 beset us in certain branches of science, unless we gain 

 a clear idea of the vast number of combinations or per 

 mutations which may be possible under certain conditions. 

 Thus only can we learn how hopeless it would be to 

 attempt to treat nature in detail, and exhaust the whole 

 number of events which might arise. It is instructive to 

 consider, in the first place, how immensely great are the 

 numbers of combinations with which we deal in many 

 arts and amusements. 



