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.
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.
Although from the point of view of our ordinary geometry there is a deformation of the bodies in this displacement and their various parts are no longer in the same relative position, nevertheless we shall see that the impressions of the sentient being have once more become the same.
In fact, though the mutual distances of the various parts may have varied, yet the parts originally in contact are again in contact. Therefore the tactile impressions have not changed.
On the other hand, taking into account the hypothesis made above in regard to the refraction and the curvature of the rays of light, the visual impressions will also have remained the same.
These imaginary beings will therefore like ourselves be led to classify the phenomena they witness and to distinguish among them the 'changes of position' susceptible of correction by a correlative voluntary movement.
If they construct a geometry, it will not be, as ours is, the study of the movements of our rigid solids; it will be the study of the changes of position which they will thus have distinguished and which are none other than the 'non-Euclidean displacements'; it will be non-Euclidean geometry.
Thus beings like ourselves, educated in such a world, would not have the same geometry as ours.
The World of Four Dimensions.—We can represent to ourselves a four-dimensional world just as well as a non-Euclidean.
The sense of sight, even with a single eye, together with the muscular sensations relative to the movements of the eyeball, would suffice to teach us space of three dimensions.