If we have distinguished the series of muscular sensations S + S´ formed by the union of two inverse series, it is because they preserve the totality of our impressions; if now we distinguish the series σ, it is because they preserve certain of our impressions. (When I say that a series of muscular sensations S 'preserves' one of our impressions A, I mean that we ascertain that if we feel the impression A, then the muscular sensations S, we still feel the impression A after these sensations S.)
I have said above it often happens that the series σ do not alter the tactile impressions felt by our first finger; I said often, I did not say always. This it is that we express in our ordinary language by saying that the tactile impressions would not be altered if the finger has not moved, on the condition that neither has the object A, which was in contact with this finger, moved. Before knowing geometry, we could not give this explanation; all we could do is to ascertain that the impression often persists, but not always.
But that the impression often continues is enough to make the series σ appear remarkable to us, to lead us to put in the same class the series Σ and Σ + σ, and hence not regard them as distinct. Under these conditions we have seen that they will engender a physical continuum of three dimensions.
Behold then a space of three dimensions engendered by my first finger. Each of my fingers will create one like it. It remains to consider how we are led to regard them as identical with visual space, as identical with geometric space.
But one reflection before going further; according to the foregoing, we know the points of space, or more generally the final situation of our body, only by the series of muscular sensations revealing to us the movements which have carried us from a certain initial situation to this final situation. But it is clear that this final situation will depend, on the one hand, upon these movements and, on the other hand, upon the initial situation from which we set out. Now these movements are revealed to us by our muscular sensations; but nothing tells us the initial situation; nothing can distinguish it for us from all the other possible situations. This puts well in evidence the essential relativity of space.
4. Identity of the Different Spaces
We are therefore led to compare the two continua C and C´ engendered, for instance, one by my first finger D, the other by my second finger D´. These two physical continua both have three dimensions. To each element of the continuum C, or, if you prefer, to each point of the first tactile space, corresponds a series of muscular sensations Σ, which carry me from a certain initial situation to a certain final situation.[8] Moreover, the same point of this first space will correspond to Σ and Σ + σ, if σ is a series of which we know that it does not make the finger D move.
Similarly to each element of the continuum C´, or to each point of the second tactile space, corresponds a series of sensations Σ´, and the same point will correspond to Σ´ and to Σ´ + σ´, if σ´ is a series which does not make the finger D´ move.
What makes us distinguish the various series designated σ from those called σ´ is that the first do not alter the tactile impressions felt by the finger D and the second preserve those the finger D´ feels.
Now see what we ascertain: in the beginning my finger D´ feels a sensation A´; I make movements which produce muscular sensations S; my finger D feels the impression A; I make movements which produce a series of sensations σ; my finger D continues to feel the impression A, since this is the characteristic property of the series σ; I then make movements which produce the series S´ of muscular sensations, inverse to S in the sense above given to this word. I ascertain then that my finger D´ feels anew the impression A´. (It is of course understood that S has been suitably chosen.)