COMBINATIONS AND PERMUTATIONS. ' 223 



that there do not exist more than from 3Xio 24 to io 26 

 molecules in a cubic centimetre of a solid or liquid sub- 

 stance^ Assuming these data to be true, for the sake 

 of argument, a simple calculation enables us to show that 

 the almost inconceivably vast sphere of our stellar system 

 if entirely filled with solid matter, would not contain 

 more than about 68 x io 90 atoms, that is to say, a number 

 requiring for its expression 92 places of figures. Now, 

 this number would be immensely less than the fifth order 

 of the powers of two. 



In the variety of logical relations, which may exist 

 between a certain number of logical terms, we also meet 

 a case of higher variations. Two terms, as it has been 

 shewn (p. 154), may form four distinct combinations, 

 but the possible selections from these series of com- 

 binations will be sixteen in number, or, excluding cases 

 of contradiction, seven. Three terms may form eight 

 combinations, allowing 256 selections, or with exclu- 

 sion of contradictory cases, 193. Four terms give sixteen 

 combinations, and no less than 65,536 possible selec- 

 tions from those combinations, the nature of which I 

 naturally abstained from exhaustively examining. Five 

 terms give thirty-two combinations, and 4,294,967,296 

 possible selections ; and for six terms the corresponding 

 numbers are sixty-four and 18,446,744,073,709,551,616. 

 Considering that it is the most common thing in the 

 world to use an argument involving six objects or terms, 

 it may excite some surprise that the complete investiga- 

 tion of the relations in which six such terms may stand 

 to each other, should involve an almost inconceivable 

 number of cases. Yet these numbers of possible logical 

 relations belong only to the second order of combina- 

 tions. 



x 'Nature/ vol. i. p. 553. 



