and use. 



44 



dimension and form are dependent upon position. Thus 

 in an ordinary convex mirror objects appear smaller as 

 they recede laterally from the centre of the picture ; 

 straight lines become curved ; objects infinitely distant 

 in front of the mirror appear at a distance only equal 

 to the focal length behind. And by suitable modifications 

 in the curvature of the mirror, representations could 

 similarly be obtained of space of various configurations. 

 Its meaning The diversity in kind of these spaces is of course 

 infinite ; they vary with the mode in which we generalize 

 our conceptions of ordinary space ; but upon each as a 

 basis it is possible to construct a consistent system of 

 geometry, whose laws, as a matter of strict reasoning, 

 have a validity and truth not inferior to those with 

 which we are habitually familiar. Such systems having 

 been actually constructed, the question has not un- 

 naturally been asked, whether there is anything in 

 nature or in the outer world to which they correspond ; 

 whether, admitting that for our limited experience ordi- 

 nary geometry amply suffices, we may understand that 

 for powers more extensive in range or more minute in 

 definition some more general scheme would be requi- 

 site ? Thus, for example, although the one may serve for 

 the solar system, is it legitimate to suppose that it may 

 fail to apply at distances reaching to the fixed stars, or 

 to regions beyond ? Or again, if our vision could discern 

 the minute configuration of portions of space, which to 

 our ordinary powers appear infinitesimally small, should 

 we expect to find that all our usual Geometry is but a 

 special case, sufficient indeed for daily use, but after all 

 only a rough approximation to a truer although perhaps 

 more complicated scheme? Traces of these questions 

 are in fact to be found in the writings of some of our 

 greatest and most original Mathematicians. Gauss, 



