30 ORIGIN AND SIGNIFICANCE OF 



relations to experience, with the question whether it 

 is logically possible to replace them by others. 



Seeing that the researches in question are more 

 immediately designed to furnish proofs for experts in 

 a region which, more than almost any other, requires 

 a higher power of abstraction, and that they are vir- 

 tually inaccessible to the non-mathematician, I will 

 endeavour to explain to such a one the question at 

 issue. I need scarcely remark that my explanation 

 will give no proof of the correctness of the new views. 

 He who seeks this proof must take the trouble to 

 study the original researches. 



Anyone who has entered the gates of the first ele- 

 mentary axioms of geometry, that is, the mathematical 

 doctrine of space, finds on his path that unbroken 

 chain of conclusions of which I just spoke, by which 

 the ever more varied and more complicated figures 

 are brought within the domain of law. But even in 

 their first elements certain principles are laid down, 

 with respect to which geometry confesses that she 

 cannot prove them, and can only assume that anyone 

 who understands the essence of these principles will 

 at once admit their correctness. These are the so- 

 called axioms. 



For example, the proposition that if the shortest 

 line drawn between two points is called a straight line, 

 there can be only one such straight line. Again, it is 



