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.
The images of external objects are painted on the retina, which is a two-dimensional canvas; they are perspectives.
But, as eye and objects are movable, we see in succession various perspectives of the same body, taken from different points of view.
At the same time, we find that the transition from one perspective to another is often accompanied by muscular sensations.
If the transition from the perspective A to the perspective B, and that from the perspective A´ to the perspective B´ are accompanied by the same muscular sensations, we liken them one to the other as operations of the same nature.
Studying then the laws according to which these operations combine, we recognize that they form a group, which has the same structure as that of the movements of rigid solids.
Now, we have seen that it is from the properties of this group we have