94 MATHEMATICS 



learn from physical observation. In one point, 

 namely in connection with what is usually known 

 as the axiom of parallels, this disability has proved 

 to be of the greatest importance in the develop- 

 ment of the Science. Euclid's axiom of parallels, 

 being a statement of what happens in unbounded 

 space, is essentially incapable of direct verification. 

 Indirect verification, for example, by observation 

 of the sum of the angles of a triangle, is indecisive, 

 on account of the essential inexactitude of our 

 measurements, and on account of the fact that 

 the size of the triangles that can be observed, 

 even by the Astronomer, is limited. All the 

 numerous attempts that have been made to prove 

 the truth of the axiom, that is to show that it is 

 logically deducible from the other axioms and 

 postulates, proved a failure in face of the most 

 determined efforts to throw light upon the matter ; 

 it has now been demonstrated that this failure was 

 inevitable. It is absolutely necessary for the 

 development of rational geometry, either to postu- 

 late, as Euclid does, in some form or other, the 

 truth of the axiom of parallels, or else to substitute 

 for it some postulate of a divergent character, but 

 still not inconsistent with our intuitions of actual 

 spatial relations. During the last century, the 

 actual failure to prove the truth of Euclid's axiom 

 within the rational scheme, led to the investigation 

 of the results of making a different postulation, and 

 at least two systems, that of Bolyai-Lobachewsky, 



