vii.] INDUCTION. 123 



is given it is quite another matter to determine its factors. 

 Can the reader say what two numbers multiplied together 

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

 unlikely that anyone 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 exhaustively 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 per- 

 formed 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 afford 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, b, c, d, &c., be the quantities ; 



then (x - a) (x - fy (x - c) (a - rf) = O 



is the equation required, and we only need to multiply out 

 the expression on the left hand by ordinary rules. But 

 having given a complex algebraic expression equated to 

 zero, it is a matter of exceeding difficulty to discover all 

 the roots. Mathematicians have exhausted their highest 

 powers in carrying the complete solution up to the fourth 

 degree. In every other mathematical operation the inverse 

 process is far more difficult than the direct process, sub- 

 traction than addition, division than multiplication, evo- 

 lution than involution ; but the difficulty increases vastly 

 as the process becomes more complex. Differentiation, 

 the direct process, is always capable of performance by 

 fixed rules, but as these rules produce considerable variety 

 of results, the inverse process of integration presents im- 

 mense difficulties, and in an infinite majority of cases 

 surpasses the present resources of mathematicians. There 



