To all the good minds who think thus, the preceding statement must have appeared quite extraordinary. But let us see whether they are not subject to an illusion that a more profound analysis would dissipate. What, first of all, are the properties of space, properly so called? I mean of that space which is the object of geometry and which I shall call _geometric space_. The following are some of the most essential: 1º It is continuous; 2º It is infinite; 3º It has three dimensions; 4º It is homogeneous, that is to say, all its points are identical one with another; 5º It is isotropic, that is to say, all the straights which pass through the same point are identical one with another. Compare it now to the frame of our representations and our sensations, which I may call _perceptual space_. VISUAL SPACE.--Consider first a purely visual impression, due to an image formed on the bottom of the retina. A cursory analysis shows us this image as continuous, but as possessing only two dimensions; this already distinguishes from geometric space what we may call _pure visual space_. Besides, this image is enclosed in a limited frame. Finally, there is another difference not less important: _this pure visual space is not homogeneous_. All the points of the retina, aside from the images which may there be formed, do not play the same rôle. The yellow spot can in no way be regarded as identical with a point on the border of the retina. In fact, not only does the same object produce there much more vivid impressions, but in every _limited_ frame the point occupying the center of the frame will never appear as equivalent to a point near one of the borders. No doubt a more profound analysis would show us that this continuity of visual space and its two dimensions are only an illusion; it would separate it therefore still more from geometric space, but we shall not dwell on this remark. Sight, however, enables us to judge of distances and consequently to perceive a third dimension. But every one knows that this perception of the third dimension reduces itself to the sensation of the effort at accommodation it is necessary to make, and to that of the convergence which must be given to the two eyes, to perceive an object distinctly. These are muscular sensations altogether different from the visual sensations which have given us the notion of the first two dimensions. The third dimension therefore will not appear to us as playing the same rôle as the other two. What may be called _complete visual space_ is therefore not an isotropic space. It has, it is true, precisely three dimensions, which means that the elements of our visual sensations (those at least which combine to form the notion of extension) will be completely defined when three of them are known; to use the language of mathematics, they will be functions of three independent variables.