Finally, I shall suppose that a body transported from one point to another of different temperature is put immediately into thermal equilibrium with its new environment. Nothing in these hypotheses is contradictory or unimaginable. 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 _R_^{2} - _r_^{2}. 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_. If a sentient being happens to be in the neighborhood, his impressions will be modified by the displacement of the object, but he can reestablish them by moving in a suitable manner. It suffices if finally the aggregate of the object and the sentient being, considered as forming a single body, has undergone one of those particular displacements I have just called non-Euclidean. This is possible if it be supposed that the limbs of these beings dilate according to the same law as the other bodies of the world they inhabit.