THE FOURTH DIMENSION. 85 



"infinitely great" and "unlimited," are confounded. All that is at 

 variance with our practical conceptions is that space can anywhere 

 have a boundary; not that it may possibly be of tremendous but 

 finite magnitude. 



It will now be asked if we cannot determine by actual observa- 

 tion whether the measure of curvature of experiential space is ex- 

 actly zero or slightly different therefrom. The theorem of the sum 

 of the angles of a triangle and the conclusions which follow from 

 this theorem do indeed supply us with a means of ascertaining this 

 fact. And the results of observation have been, that the measure of 

 curvature of space is in all probability exactly equal to zero or if it is 

 slightly different from zero it is so little so that the technical means of 

 observation at our command and especially our telescopes are not compe- 

 tent to determine the amount of the deviation. More, we cannot with 

 any certainty say. 



All these reflections, to which the criticism of the hypotheses 

 that underlie geometry long ago led investigators, compel us to in- 

 stitute a comparison between the space of experience and other 

 three-dimensional aggregates of points (spaces), which we cannot 

 mentally represent but can in thought and word accurately define 

 and investigate. As soon, however, as we are fully implicated in the 

 task of accurately investigating the properties of three-dimensional 

 aggregates of points, we find ourselves similarly forced to regard 

 such aggregates as the component elements of a manifoldness of 

 more than three dimensions. In this way the exact criticism of 

 even ordinary geometry leads us to the abstract assumption of a 

 space of more than three dimensions. And as the extension of every 

 idea gives a clearer and more translucent form to the idea as it orig- 

 inally stood, here too the idea of multi-dimensioned aggregates 

 of points and the investigation of their properties has thrown a new 

 light on the truths of ordinary geometry and placed its properties 

 in clearer relief. Among the numerous examples which show how 

 the notion of a space of multiple dimensions has been of great ser- 

 vice to science in the investigation of three-dimensional space, we 

 shall give one a place here which is within the comprehension of 

 non -mathematicians. 



