vii.] INDUCTION. 145 



ABcd 



It will be seen that the four proximates are respectively 



obverse to the four ultimates, and that the mediates form 



three pairs of obverses. Every proximate or ultimate is 



distant i and 3 respectively from such a pair of mediates. 



Aided by this system of nomenclature Professor Clifford 



proceeds to an exhaustive enumeration of types, in which 



it is impossible to follow him. The results are as follows : 



i -fold statements I type 



2 ,, 4 types 



3 n 6 



4 ,, 19 } 159 



5 27 



6 47 



7 55 

 8-fold statements 78 



Now as each seven-fold or less-than-seven-fold statement 

 is complementary to a nine-fold or more-than-nine-fold 

 statement, it follows that the complete number of types 

 will be 159 x 2 + 78 = 396. 



It appears then that the types of statement concerning 

 four classes are only about 26 times as numerous as those 

 concerning three classes, fifteen in number, although the 

 number of possible combinations is 256 times as great. 



Professor Clifford informs me that the knowledge of the 

 possible groupings of subdivisions of classes which he 

 obtained by this inquiry has been of service to him in 

 some applications of hyper-elliptic functions to which he 

 has subsequently been led. Professor Cayley has since 

 expressed his opinion that this line of investigation should 

 be followed out, owing to the bearing of the theory of 

 compound combinations upon the higher geometry. 1 It 

 seems likely that many unexpected points of connection 



1 Proceedings of the Manchester Literary and Philosophical Society, 

 6th February, 1877, vol. xvi., p. 113. 



L 



