This selection first appeared as an article, "L'Espace et la géométrie," in Revue de métaphysique et de morale, 1895 t.iii, pp. 631-46.

Beings with a mind made like ours, with the same senses that we have, but without any prior education, could receive from a properly selected world impressions such that they would be led to construct a geometry different from Euclid's and to localize the phenomena of that exterior world in a non-Euclidean space or even in a four-dimensional space.

For us, educated in this present world, if we were suddenly transported into that new world , we would have no difficulty in relating its phenomena to our Euclidean space.

A man who devoted his life to it could perhaps succeed in picturing to himself a fourth dimension…

In the same way as a non-Euclidean world, we can represent a world having four dimensions. The sense of sight, even with a single eye, joined to muscular sensations relative to movements of the eyeball could suffice to our knowing a three-dimensional world.

The images of exterior objects come and paint themselves on the retina, which is a two-dimensional tabula; these are perspectives. But, since these objects are mobile, and since the same is true of our eye, we see successively different perspectives of the same body, taken from several different points of view. We notice at the same time that the passage from one perspective to another is often accompanied by muscular sensations.

If the passage from perspective A to perspective B and that from perspective A' to perspective B' are accompanied by the same muscular sensations, we consider them to be operations of the same kind. Studying afterward the laws according to which these operations combine, we recognize that they form a group having the same structure as that of the movements of invariable solids.

Now, we have seen that it is from the properties of this group that we have deduced the notions of geometric and of three dimensions. Thus we understand how the idea of a three-dimensional space was able to arise from the spectacle of these perspectives, even though each of them has only two dimensions, because they succeed one another in accordance with certain laws. In the same way that we can make on a plane the perspective of a figure having three dimensions, we can make that of a four-dimensional figure on a tabula with three (or two) dimensions. It is only a game for the geometrician. We can even take from a single figure several perspectives from several different viewpoints. We can easily represent these perspectives since they have only three dimensions.

Let us imagine that the diverse perspectives of a single object succeed one another; that the passage from one to the other is accompanied by muscular sensations. We will naturally consider two of these passages as two operations of the same kind when they are associated with the same muscular sensations. Then nothing prevents our imagining that these operations might combine following any law that we wish, for example in such a way as to form a group having the same structure as that of an invariable four-dimensional solid.

In all this there is nothing we cannot represent and nevertheless these sensations are precisely those that a being furnished with a two-dimensional retina would feel if be could move in four-dimensional space. It is in this sense that we may say that we can represent the fourth dimension.

We say that experience plays an indispensable role in the genesis of geometry; but it would be an error to conclude that geometry is an experimental science, even in part.

If it were experimental it would be only approximate and provisional. And what a crude approximation!

Geometry would only be the study of the movements of solids; but in reality it is not concerned with natural solids, it has as its object certain ideal solids, absolutely invariable, which are only a simplified and very distant image of the natural ones.

The notion of ideal bodies is drawn entirely from our minds and experience is only an occasion that invites us to construct such a notion.

The object of geometry is the study of a particular "group"; but the general concept of group pre-exists in our minds, at least potentially. It is imposed on us, not as a form of sensibility, but as a form of understanding.

Still, among all possible groups, we must choose that one which will be, so to speak, the standard to which we will refer natural phenomena.

Experience guides us in this choice which it does not impose on us; it makes us recognize not what is the truest geometry, but what is the most convenient. It will be noticed that I have been able to describe the fantastic worlds that I imagined above without ceasing to use the language of ordinary geometry. Indeed, we should not have to change anything if we were to be transported to such a world.

Beings who were educated there would no doubt find it more convenient to create a geometry different from ours, better adapted to their impressions. As for us, in the face of the same impressions, it is certain that we would find it more convenient not to change our habits. (1)