GEOMETRY 93 



is well known. A much more recent example of 

 modification of philosophic thought, as the result 

 of mathematical research, will here be referred to, 

 viz. the effect of modern theories of Geometry 

 upon our conceptions of spatial relations. 



The properties and relations assigned to the 

 ideal objects of rational Geometry were suggested 

 by observations of spatial relations in the physical 

 world; there are however noteworthy limitations 

 to the amount of leading that can in this regard 

 be obtained from our actual perceptions. In the 

 first place, all our spatial perceptions are affected 

 by an essential element of inexactitude, the amount 

 of which varies with the degree of precision of the 

 instruments we use for measurement, but which 

 can never be wholly eliminated. In the second 

 place, all our actual observations have reference to 

 some more or less bounded, and certainly finite, 

 portion of what we call physical space, although 

 we have the intuition that it is always possible 

 to pass beyond the space at any time observed. 

 In rational Geometry, we have in the scheme of 

 axioms, definitions, and postulates, not only to 

 make precise statements as to the possession of 

 certain entities of certain precise properties, whereas 

 the corresponding physical objects possess those 

 properties only roughly, but we have also to make 

 statements which refer to what happens in every 

 part of unbounded space; and to do this we have 

 to pass in certain respects beyond anything we can 



