SPACE 661 



taken place in a medium in which those laws would be so different, 

 might have a very different geometry from ours. 



Suppose, for example, a world enclosed in a large sphere and 

 subject to the following laws: The temperature is not uniform; it 

 is greatest at the centre, and gradually decreases as we move towards 

 the circumference of the sphere, where it is absolute zero. The law 

 of this temperature is as follows : If E be the radius of the sphere, 

 and r the distance of the point considered from the centre, the abso- 

 lute temperature will be proportional to R 2 r 2 . Further, I shall sup- 

 pose that in this world all bodies have the same co-efficient of dilata- 

 tion, so that the linear dilatation of any body is proportional to its ab- 

 solute temperature. Finally, I shall assume that a body transported 

 from one point to another of different temperature is instantaneously 

 in thermal equilibrium with its new environment. There is nothing 

 in these hypotheses either contradictory or unimaginable. A moving 

 object will become smaller and smaller as it approaches the circum- 

 ference of the sphere. Let us observe, in the first place, that although 

 from the point of view of our ordinary geometry this world is finite, 

 to its inhabitants it will appear infinite. As they approach the sur- 

 face of the sphere they become colder, and at the same time smaller 

 and smaller. The steps they take are therefore also smaller and 

 smaller, so that they can never reach the boundary of the sphere. If 

 to us geometry is only the study of the laws according to which in- 

 variable solids move, to these imaginary beings it will be the study 

 of the laws of motion of solids deformed by the differences of tem- 

 perature alluded to. 



No doubt, in our world, natural solids also experience variations of 

 form and volume due to differences of temperature. But in laying 

 the foundations of geometry we neglect these variations; for besides 

 being but small they are irregular, and consequently appear to us to 

 be accidental. In our hypothetical world this will no longer be the 

 case, the variations will obey very simple and regular laws. On the 

 other hand, the different solid parts of which the bodies of these 

 inhabitants are composed will undergo the same variations of form 

 and volume. 



Let me make another hypothesis: suppose that light passes through 

 media of different refractive indices, such that the index of refrac- 

 tion is inversely proportional to ft* r 2 . Under these conditions it is 

 clear that the rays of light will no longer be rectilinear but circular. 

 To justify what lias been said, we have to prove that certain changes 

 in the position of external objects may be corrected by correlative 

 movements of the beings which inhabit this imaginary world; and in 

 such a way as to restore the primitive aggregate of the impressions 

 experienced by these sentient beings. Suppose, for example, that an 

 object is displaced and deformed, not like an invariable solid, but like 



