32 OKIGIN AND SIGNIFICANCE OF 



strives to succeed where all before him have failed. 

 And it is quite right that each should make the trial 

 afresh ; for, as the question has hitherto stood, it is 

 only by the fruitlessness of one's own efforts that one 

 can be convinced of the impossibility of finding a 

 proof. Meanwhile solitary inquirers are always from 

 time to time appearing who become so deeply en- 

 tangled in complicated trains of reasoning that they 

 can no longer discover their mistakes and believe they 

 have solved the problem. The axiom of parallels 

 especially has called forth a great number of seeming 

 demonstrations. 



The main difficulty in these inquiries is, and always 

 has been, the readiness with which results of everyday 

 experience become mixed up as apparent necessities of 

 thought with the logical processes, so long as Euclid's 

 method of constructive intuition is exclusively followed 

 in geometry. It is in particular extremely difficult, on 

 this method, to be quite sure that in the steps, pre- 

 scribed for the demonstration we have not involun- 

 tarily and unconsciously drawn in some most general 

 results of experience, which the power of executing 

 certain parts of the operation has already taught us 

 practically. In drawing any subsidiary line for the 

 sake of his demonstration, the well-trained geometer 

 always asks if it is possible to draw such a line. It is 

 well known that problems of construction play an essen- 



