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. We shall of course consider two of these transitions as two operations of the same nature when they are associated with the same muscular sensations. Nothing then prevents us from imagining that these operations combine according to any law we choose, for example, so as to form a group with the same structure as that of the movements of a rigid solid of four dimensions. Here there is nothing unpicturable, and yet these sensations are precisely those which would be felt by a being possessed of a two-dimensional retina who could move in space of four dimensions. In this sense we may say the fourth dimension is imaginable. CONCLUSIONS.--We see that experience plays an indispensable rôle in the genesis of geometry; but it would be an error thence to conclude that geometry is, even in part, an experimental science. If it were experimental, it would be only approximative and provisional. And what rough approximation! Geometry would be only the study of the movements of solids; but in reality it is not occupied with natural solids, it has for object certain ideal solids, absolutely rigid, which are only a simplified and very remote image of natural solids. The notion of these ideal solids is drawn from all parts of our mind, and experience is only an occasion which induces us to bring it forth from them. The object of geometry is the study of a particular 'group'; but the general group concept pre-exists, at least potentially, in our minds. It is imposed on us, not as form of our sense, but as form of our understanding. Only, from among all the possible groups, that must be chosen which will be, so to speak, the _standard_ to which we shall refer natural phenomena. Experience guides us in this choice without forcing it upon us; it tells us not which is the truest geometry, but which is the most _convenient_. Notice that I have been able to describe the fantastic worlds above imagined _without ceasing to employ the language of ordinary geometry_. And, in fact, we should not have to change it if transported thither. Beings educated there would doubtless find it more convenient to create a geometry different from ours, and better adapted to their impressions. As for us, in face of the _same_ impressions, it is certain we should find it more convenient not to change our habits. CHAPTER V EXPERIENCE AND GEOMETRY 1. Already in the preceding pages I have several times tried to show that the principles of geometry are not experimental facts and that in particular Euclid's postulate can not be proven experimentally.