42 SCIENCE AND METHOD. 



Geometry. 



It would seem that geometry can contain nothing 

 that is not already contained in algebra or analysis, and 

 that geometric facts are nothing but the facts of algebra 

 or analysis expressed in another language. It might 

 be supposed, then, that after the review that has just 

 been made, there would be nothing left to say having 

 any special bearing on geometry. But this would 

 imply a failure to recognize the great importance of a 

 well-formed language, or to understand what is added 

 to things themselves by the method of expressing, and 

 consequently of grouping, those things. 



To begin with, geometric considerations lead us to 

 set ourselves new problems. These are certainly, if 

 you will, analytical problems, but they are problems 

 we should never have set ourselves on the score of 

 analysis. Analysis, however, profits by them, as it 

 profits by those it is obliged to solve in order to 

 satisfy the requirements of physics. 



One great advantage of geometry lies precisely in 

 the fact that the senses can come to the assistance of 

 the intellect, and help to determine the road to be 

 followed, and many minds prefer to reduce the 

 problems of analysis to geometric form. Unfortu- 

 nately our senses cannot carry us very far, and they 

 leave us in the lurch as soon as we wish to pass 

 outside the three classical dimensions. Does this 

 mean that when we have left this restricted domain 

 in which they would seem to wish to imprison us, we 

 must no longer count on anything but pure analysis, 

 and that all geometry of more than three dimensions 

 is vain and without object? In the generation which 



