[dupuis] presidential ADDRESS 7 



were changed from a circle to some elliptic or irregular form of closed 

 curve, the vermal inhabitant would have still an infinite space, as far as 

 its intelligence could determine ; but as the curvature would be different 

 at different points, the worm, in accommodating itself to these curva- 

 tures would experience different sensations at the various points, which, 

 although really due to the character of the s^^ace in which it moved, 

 would by it be referred to some physical change in its own nature or in 

 the universe of which it was cognizant. 



Carrying his illustration one stage further, Clifford supjDOses a fish, 

 intelligent, but of infinitesimal thickness, that is a portion of a geometrical 

 surface, moving in a sea of infinitesimal depth. Such a being could have 

 knowledge of two dimensions only. If its sea were a plane, the fish's 

 space would be infinite, of two dimensions and same. Also, if the sea 

 were a portion of the surface of a sphere, the two-dimensional space, as 

 far as the determination of its form or limits by the intelligence of the 

 fish could go, would be still infinite and same ; since a surface figure 

 which would coincide with the surface of a sphere at one point will 

 coincide at every point. 



But if the sphere were transformed into an ellipsoid, or into any 

 otber surface closed or open, with varying curvature from point to 

 jjoint, the space of the fish would be still apparently infinite, as far as its 

 knowledge could go, but as it would have to undergo change of form at 

 different points, in order to accommodate itself to changes in curvature of 

 its sea, it would become cognizant of different sensations at these points ; 

 but instead of referring these changes of sensation to the proper source, 

 the curvature of the space in which it was constrained to move, it would 

 naturally refer them to some change in its own physical constitution, or 

 in that of the universe of which it had any knowledge. 



Here illustration must cease, and the argument must be founded 

 upon analogy alone, for it is not possible to illustrate that which in itself 

 is inconceivable, 



Eeasoning from analogy, then, the new geonaeter says that it is pos- 

 sible, just as in the case of the worm in its circular tube, and of the fish 

 upon its spherical surface sea, that our three dimensional space may 

 have some propert}^ which we might call a bend in it, so that although it 

 seems to us to be infinite in extent, it may, in reality, be finite. In which 

 case, of course, it could not possibly admit of the Euclidean straight line, 

 for that is necessarily infinite in length. Looking at such a possibility, 

 then, the new geometer argues that it is not wise to state dogmatically 

 that Euclidean geometry is the absolute geometr^^, or that Euclidean 

 space is the only possible space, or that the universe is infinite in extent, 

 although, of course, it is unlimited, just in the same way, all necessary 

 changes being made, as the power to keep on moving without coming to 

 a necessary termination, is, for the imaginary worm and fish, unlimited. 



