J?02 On an Impromptu Demonstration of Mr. Cayley. 



the group 



1, 1, 1, 1 



1, 1, I 



1, 1 

 i. e, 4, 3, 2, say B. 



In A the number of parts is 4. In B the greatest part is 4 ; 

 the others might be (although they happen not in this particular 

 instance to be) 4, but cannot be greater than 4. And so every 

 A in which the number of parts is 4 will give rise to a B in 

 which 4 is one of the parts, and every other part is 4 or less, 

 and evidently (although, as above remarked, this is immaterial to 

 the demonstration) every such B gives reciprocally the same A 

 from which it is itself derived ; hence the number of A's and B's 

 is equal. This is the theorem which, for the sake of distinction, 

 I have called the Corollary to Euler's. Euler's own is proved by 

 the same diagram ; for if we define A as a grouping where the 

 number of parts does not exceed 4, we get a definition of B as a 

 grouping where the greatest part does not exceed 4, and so in 

 general. We see that this theorem may be varied also by affirm- 

 ing that the number of ways in which n may be broken up, so 

 that there shall never be less than (m) parts, is the same as the 

 number of ways in which it may be broken up into parts, the 

 greatest of which in any one way is not less than (m) . So, again, 

 a similar diagram makes it apparent, that if we break up each of 

 i numbers into parts so that the sum of the greatest parts shall 

 not exceed (or be less than) m, the number of ways in which this 

 can be done will be the same as the number of ways in which 

 these i numbers can be simultaneously partitioned so that the 

 total number of parts in any simultaneous partitionment shall 

 never exceed (or never be less than) m ; and doubtless an exten- 

 sive range of analogous general theorems relative to the parti- 

 tioning of numbers may be struck out by aid of the same dia- 

 gram, by no means easily demonstrable unless this simple mode 

 of conversion happen to be thought of, but in that event becoming 

 intuitively apparent. This mode of conversion is precisely that 

 (only applied to a more general state of things) whereby, in ele- 

 mentary arithmetic, it is established that (m) times (w) is the 

 same as (n) times (w). A consideration of the process by which 

 the mind satisfies itself of the universality of this law, has been 

 always sufficient to convince me of the absurdity of ascribing to 

 an inductive process the capacity of the human mind for forming 

 general ideas concerning necessary relations. 



7 New Square, Lincoln's Inn, 

 January 28, 1863. 



