24 report— 1878. 



the earth's surface, or upon an eggshell, such figures cannot be made to 

 slide upon the surface without change of form, as is the case with figures 

 traced upon a plane or even upon a sphere. But, further still, these 

 generalisations are not restricted to the case of figures traced upon a sur- 

 face ; they may apply also to solid figures in a space whose very configu- 

 ration varies from point to point. We may, for instance, imagine a space 

 in which our rule or scale of measurement varies as it extends, or as it 

 moves about, in one direction or another ; a space, in fact, whose geometric 

 density is not uniformly distributed. Thus we might picture to ourselves 

 such a space as a field having a more or less complicated distribution of 

 temperature, and our scale as a rod instantaneously susceptible of expan- 

 sion or contraction under the influence of heat: or we might suppose 

 space to be even crystalline in its geometric formation, and our scale and 

 measuring instruments to accept the structure of the locality in which 

 they are applied. These ideas are doubtless difficult of apprehension, at 

 all events at the outset; but Helmholtz has pointed out a very familiar 

 phenomenon which may be regarded as a diagram of such a kind of space. 

 The picture formed by reflexion from a plane mirror may be taken as a 

 carrect representation of ordinary space, in which, subject to the usual 

 laws of perspective, every object appears in the same form and of the same 

 dimensions whatever be its position. In like manner the picture formed 

 by reflexion from a curved mirror may be regarded as the representation 

 •of a space wherein 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 jficture ; 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. 



The diversity in kind of these spaces is of course infinite ; they vary 

 with the mode in which we generalise 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 unnaturally 

 been asked, whether there is anything in nature or in the outer world to 

 which they correspond ; whether, admitting that for our limited experi- 

 ence ordinary geometry amply suffices, we may understand that for powers 

 more extensive in range or more minute in definition some more general 

 scheme would be requisite ? 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, suffi- 



