142 THE PRINCIPLES OF SCIENCE. 



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 dis 

 cover 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, subtraction than addition, division 

 than multiplication, evolution than involution ; but the 

 difficulty increases vastly as the process becomes more 

 complex. The differentiation, the direct process, is always 

 capable of performance by certain fixed rules, but as these 

 produce considerable variety of results, the inverse process 

 of integration presents immense difficulties, and in an 

 infinite majority of cases surpasses the present resources 

 of mathematicians. There are no infallible and general 

 rules for its accomplishment ; it must be done by trial, 

 by guesswork, by remembering the results of differentia 

 tion, and using them as a guide. 



Coming more nearly to our own immediate subject, 

 exactly the same difficulty exists in determining the law 

 which certain numbers obey. Given a general mathe 

 matical expression, we can infallibly ascertain its value 

 for any required value of the variable. But I am not 

 aware that mathematicians have ever attempted to lay 

 down the rules of a process by which, having given cer 

 tain numbers, one might discover a rational or precise 

 formula from which they proceed. The problem is always 

 indeterminate, because an infinite number of formulae 

 agreeing with certain numbers, might always be dis 

 covered with sufficient trouble. 



The reader may test his power of detecting a law, by 

 contemplation of its results, if he, not being a mathema 

 tician, will attempt to point out the law obeyed by the 



