THE FUTURE OF MATHEMATICS. 37 



engineer does not really require the integral in finite 

 terms, he only requires to know the general behaviour 

 of the integral function, or he merely wants a certain 

 figure which would be easily deduced from this in- 

 tegral if we knew it. Ordinarily we do not know 

 it, but we could calculate the figure without it, if we 

 knew just what figure and what degree of exactness 

 the engineer required. 



Formerly an equation was not considered to have 

 been solved until the solution had been expressed 

 by means of a finite number of known functions. 

 But this is impossible in about ninety-nine cases 

 out of a hundred. What we can always do, or rather 

 what we should always try to do, is to solve the 

 problem qualitatively, so to speak — that is, to try to 

 know approximately the general form of the curve 

 which represents the unknown function. 



It then remains to find the exact solution of the 

 problem. But if the unknown cannot be determined 

 by a finite calculation, we can always represent it 

 by an infinite converging series which enables us to 

 calculate it. Can this be regarded as a true solu- 

 tion? The story goes that Newton once communi- 

 cated to Leibnitz an anagram somewhat like the 

 following: aaaaabbbeeeeii, etc. Naturally, Leibnitz 

 did not understand it at all, but we who have the 

 key know that the anagram, translated into modern 

 phraseology, means, " I know how to integrate all 

 differential equations," and we are tempted to make 

 the comment that Newton was either exceedingly 

 fortunate or that he had very singular illusions. 

 What he meant to say was simply that, he could 

 form (by means of indeterminate coefficients) a 



