190 THE PRINCIPLES OF SCIENCE. [CHAP. 



present, and the numbers of these classes immediately 

 produce the numbers of the arithmetical triangle. 



It may here be pointed out that there are two modes in 

 which we can calculate the whole number of combinations 

 of certain tilings. Either we may take the whole number 

 at once as shown in the Logical Alphabet, 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 things, and so 

 on. Hence we arrive at a necessary identity between two 

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

 have 



In a general form of expression we shall have 



2^ = i + 4. n ' ( n ~ ') 4. n(n-0(-2) 



I 1.2 1.2.3 



the terms being continued until they cease to have any 

 yalue. Thus we arrive at a proof of simple cases of the 

 Binomial Theorem, of which each column of the Logical 

 Alphabet is an exemplification. It may be shown that all 

 other mathematical expansions likewise arise out of simple 

 processes of combination, but the more complete considera- 

 tion of this subject must be deferred to another work. 



Possible Variety of Nature and Art. 



We cannot " adequately understand the difficulties 

 which beset us in certain branches of science, unless we 

 have some clear idea of the vast numbers of combinations 

 or permutations which may be possible under certain con- 

 ditions. 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 instruc- 

 tive to consider, in the first place, how immensely great 

 are the numbers of combinations with which we deal in 

 many arts and amusements. 



In dealing a pack of cards, the number of hands, of 

 thirteen cards each, which can be produced is evidently 

 52 x 51 x 50 x ... x 40 divided by I x 2 x 3 . . x 13. 

 or 635,013,559,600. But in whist four hands are simul- 



