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without change of size or alteration of form, entirely 

 falls away, or becomes only approximate in a space 

 wherein dimension and form are dependent upon position. 

 For instance, if we consider merely the case of figures 

 traced on a flattened globe like 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 generalizations are not restricted 

 to the case of figures traced upon a surface ; they may 

 apply also to solid figures in a space whose very con- 

 figuration 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 tempe- 

 rature, and our scale as a rod instantaneously susceptible 

 of expansion 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 instru- 

 ments to accept the structure of the locality in which 

 they are applied. These ideas are doubtless diflicult of 

 apprehension, at all events at the outset ; but Helmholtz 

 has poiated 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 correct representation of ordinary space, in 

 which, subject to the usual laws of perspective, every 

 object appears in the same form and of the same di- 

 mensions 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 





