160 THE PRINCIPLES OF SCIENCE. 



development. We may also have ' all A's are all B's, 

 and all B's are C's,' r even 'all A's are all B's, and all 

 B's are all CV All such premises admit of variations, 

 greater or less in number, the logical distinctness of which 

 can only be determined by trial in detail. Disjunctive 

 propositions either singly or in pairs were also treated, 

 but were often found to be equivalent to other propo- 

 sitions of a simpler form ; thus A =. ABC -|- Kbc is exactly 

 the same in meaning as AB = AC. 



This mode of exhaustive trial bears some analogy to 

 that ancient mathematical process called the Sieve of 

 Eratosthenes. Having taken a long series of the natural 

 numbers, Eratosthenes is said to have calculated out in 

 succession all the multiples of every number, and to have 

 marked them off, so that at last the prime numbers alone 

 remained, and the factors of every number were exhaus- 

 tively discovered. My problem of 2 56 series of combinations 

 is the logical analogue, the chief points of difference being 

 that there is a limit to the number of cases, and that prime 

 numbers have no analogue in logic, since every series of 

 combinations corresponds to a law or group of conditions. 

 But the analogy is perfect in the point that they are 

 both inverse processes. There is no mode of ascertaining 

 that a number is prime but by showing that it is not 

 the product of any assignable factors. So there is no 

 mode of ascertaining what laws are embodied in any 

 series of combinations but trying exhaustively the laws 

 which would give them. Just as the results of Erato- 

 sthenes' method have been worked out to a great extent 

 and registered in tables for the convenience of other 

 mathematicians, I have endeavoured to work out the 

 inverse logical problem to the utmost extent which is 

 at present practicable or useful. 



I have thus found that there are altogether fifteen 

 conditions or series of conditions which may govern 



