INDUCTION. 141 



difficulty commence. In entering, any path served him ; 

 in leaving, he must select certain definite paths, and in 

 this selection he must either trust to memory of the way 

 he entered or else make an exhaustive trial of all possible 

 ways. The explorer entering a new country makes sure 

 his line of return by barking the trees. 



The same difficulty arises in many scientific processes. 

 Given any two numbers, we may by a simple and infallible 

 process obtain their product, but it is quite another matter 

 when a large number is given to determine its factors. 

 Can the reader say what two numbers multiplied together 

 will produce the number 8,616,460,799 1 I think it 

 unlikely that any one but myself will ever know ; for 

 they are two large prime numbers, and can only be re- 

 discovered by trying in succession a long series of prime 

 divisors until the right one be fallen upon. The work 

 would probably occupy a good computer for many weeks, 

 but it did not occupy me many minutes to multiply the 

 two factors together. Similarly there is no direct process 

 for discovering whether any number is a prime or not ; 

 it is only by exhaustingly trying all inferior numbers 

 which could be divisors, that we can show there is none, 

 and the labour of the process would be intolerable were it 

 not performed systematically once for all in the process 

 known as the Sieve of Eratosthenes, the results being 

 registered in tables of prime numbers. 



The immense difficulties which are encountered in the 

 solution of algebraic equations are another illustration. 

 Given any algebraic factors, we can easily and infallibly 

 arrive at the product, but given a product it is a matter 

 of infinite difficulty to resolve it into factors. Given any 

 series of quantities however numerous, there is very little 

 trouble in making an equation which shall have those 

 quantities as roots. Let a, 6, c, d, &c., be the quantities ; 

 then (x a) (x b) (x c) (x d) = o 



