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 derived the notion of geometric space and that of three dimensions.
We understand thus how the idea of a space of three dimensions could take birth from the pageant of these perspectives, though each of them is of only two dimensions, since they follow one another according to certain laws.
Well, just as the perspective of a three-dimensional figure can be made on a plane, we can make that of a four-dimensional figure on a picture of three (or of two) dimensions. To a geometer this is only child's play.
We can even take of the same figure several perspectives from several different points of view.
We can easily represent to ourselves these perspectives, since they are of only three dimensions.
Imagine that the various perspectives of the same object succeed one another, and that the transition from one to the other is accompanied by muscular sensations.