In restricting myself to red sensations, I therefore impose upon myself an artificial limitation and I neglect systematically one whole side of the question; but it is only by this artifice that I am able to analyze visual space without mingling any motor sensation.
Imagine a line traced on the retina and dividing in two its surface; and set apart the red sensations affecting a point of this line, or those differing from them too little to be distinguished from them. The aggregate of these sensations will form a sort of cut that I shall call C, and it is clear that this cut suffices to divide the manifold of possible red sensations, and that if I take two red sensations affecting two points situated on one side and the other of the line, I can not pass from one of these sensations to the other in a continuous way without passing at a certain moment through a sensation belonging to the cut.
If, therefore, the cut has n dimensions, the total manifold of my red sensations, or if you wish, the whole visual space, will have n + 1.
Now, I distinguish the red sensations affecting a point of the cut C. The assemblage of these sensations will form a new cut C´. It is clear that this will divide the cut C, always giving to the word divide the same meaning.
If, therefore, the cut C´ has n dimensions, the cut C will have n + 1 and the whole of visual space n + 2.
If all the red sensations affecting the same point of the retina were regarded as identical, the cut C´ reducing to a single element would have 0 dimensions, and visual space would have 2.
And yet most often it is said that the eye gives us the sense of a third dimension, and enables us in a certain measure to recognize the distance of objects. When we seek to analyze this feeling, we ascertain that it reduces either to the consciousness of the convergence of the eyes, or to that of the effort of accommodation which the ciliary muscle makes to focus the image.
Two red sensations affecting the same point of the retina will therefore be regarded as identical only if they are accompanied by the same sensation of convergence and also by the same sensation of effort of accommodation or at least by sensations of convergence and accommodation so slightly different as to be indistinguishable.
On this account the cut C´ is itself a continuum and the cut C has more than one dimension.
But it happens precisely that experience teaches us that when two visual sensations are accompanied by the same sensation of convergence, they are likewise accompanied by the same sensation of accommodation. If then we form a new cut C´´ with all those of the sensations of the cut C´, which are accompanied by a certain sensation of convergence, in accordance with the preceding law they will all be indistinguishable and may be regarded as identical. Therefore C´´ will not be a continuum and will have 0 dimension; and as C´´ divides C´ it will thence result that C´ has one, C two and the whole visual space three dimensions.