A movable object will then become smaller and smaller in proportion as it approaches the limit-sphere.
Note first that, though this world is limited from the point of view of our ordinary geometry, it will appear infinite to its inhabitants.
In fact, when these try to approach the limit-sphere, they cool off and become smaller and smaller. Therefore the steps they take are also smaller and smaller, so that they can never reach the limiting sphere.
If, for us, geometry is only the study of the laws according to which rigid solids move, for these imaginary beings it will be the study of the laws of motion of solids distorted by the differences of temperature just spoken of.
No doubt, in our world, natural solids likewise undergo variations of form and volume due to warming or cooling. But we neglect these variations in laying the foundations of geometry, because, besides their being very slight, they are irregular and consequently seem to us accidental.
In our hypothetical world, this would no longer be the case, and these variations would follow regular and very simple laws.
Moreover, the various solid pieces of which the bodies of its inhabitants would be composed would undergo the same variations of form and volume.
I will make still another hypothesis; I will suppose light traverses media diversely refractive and such that the index of refraction is inversely proportional to R2 − r2. It is easy to see that, under these conditions, the rays of light would not be rectilinear, but circular.
To justify what precedes, it remains for me to show that certain changes in the position of external objects can be corrected by correlative movements of the sentient beings inhabiting this imaginary world, and that in such a way as to restore the primitive aggregate of impressions experienced by these sentient beings.
Suppose in fact that an object is displaced, undergoing deformation, not as a rigid solid, but as a solid subjected to unequal dilatations in exact conformity to the law of temperature above supposed. Permit me for brevity to call such a movement a non-Euclidean displacement.