3S SCIENCE AND METHOD. 



series of powers formally satisfying the equation 

 presented. 



To-day a similar solution would no longer satisfy 

 us, for two reasons — because the convergence is too 

 slow, and because the terms succeed one another 

 without obeying any law. On the other hand the 

 series 6 appears to us to leave nothing to be desired, 

 first, because it converges very rapidly (this is for 

 the practical man who wants his number as quickly 

 as possible), and secondly, because we perceive at a 

 glance the law of the terms, which satisfies the 

 aesthetic requirements of the theorist. 



There are, therefore, no longer some problems 

 solved and others unsolved, there are only problems 

 more or less solved, according as this is accomplished 

 by a series of more or less rapid convergence or 

 regulated by a more or less harmonious law. Never- 

 theless an imperfect solution may happen to lead 

 us towards a better one. 



Sometimes the series is of such slow convergence 

 that the calculation is impracticable, and we have 

 only succeeded in demonstrating the possibility of 

 the problem. The engineer considers this absurd, 

 and he is right, since it will not help him to com- 

 plete his construction within the time allowed. He 

 doesn't trouble himself with the question whether it 

 will be of use to the engineers of the twent)'-second 

 century. We think differently, and we are sometimes 

 more pleased at having economized a day's work 

 for our grandchildren than an hour for our contem- 

 poraries. 



Sometimes by groping, so to speak, empirically, 

 we arrive at a formula that is sufficiently convergent. 



