This means that the series S + σ + S´, preserving the tactile impressions of the finger D´, is one of the series I have called σ´. Inversely, if one takes any series σ´, S´ + σ´ + S will be one of the series that we call σ´.
Thus if S is suitably chosen, S + σ + S´ will be a series σ´, and by making σ vary in all possible ways, we shall obtain all the possible series σ´.
Not yet knowing geometry, we limit ourselves to verifying all that, but here is how those who know geometry would explain the fact. In the beginning my finger D´ is at the point M, in contact with the object a, which makes it feel the impression A´. I make the movements corresponding to the series S; I have said that this series should be suitably chosen, I should so make this choice that these movements carry the finger D to the point originally occupied by the finger D´, that is, to the point M; this finger D will thus be in contact with the object a, which will make it feel the impression A.
I then make the movements corresponding to the series σ; in these movements, by hypothesis, the position of the finger D does not change, this finger therefore remains in contact with the object a and continues to feel the impression A. Finally I make the movements corresponding to the series S´. As S´ is inverse to S, these movements carry the finger D´ to the point previously occupied by the finger D, that is, to the point M. If, as may be supposed, the object a has not budged, this finger D´ will be in contact with this object and will feel anew the impression A´.... Q.E.D.
Let us see the consequences. I consider a series of muscular sensations Σ. To this series will correspond a point M of the first tactile space. Now take again the two series S and S´, inverses of one another, of which we have just spoken. To the series S + Σ + S´ will correspond a point N of the second tactile space, since to any series of muscular sensations corresponds, as we have said, a point, whether in the first space or in the second.
I am going to consider the two points N and M, thus defined, as corresponding. What authorizes me so to do? For this correspondence to be admissible, it is necessary that if two points M and M´, corresponding in the first space to two series Σ and Σ´, are identical, so also are the two corresponding points of the second space N and N´, that is, the two points which correspond to the two series S + Σ + S´ and S + Σ´ + S´. Now we shall see that this condition is fulfilled.
First a remark. As S and S´ are inverses of one another, we shall have S + S´ = 0, and consequently S + S´ + Σ = Σ + S + S´ = Σ, or again Σ + S + S´ + Σ´ = Σ + Σ´; but it does not follow that we have S + Σ + S´ = Σ; because, though we have used the addition sign to represent the succession of our sensations, it is clear that the order of this succession is not indifferent: we can not, therefore, as in ordinary addition, invert the order of the terms; to use abridged language, our operations are associative, but not commutative.
That fixed, in order that Σ and Σ´ should correspond to the same point M = M´ of the first space, it is necessary and sufficient for us to have Σ´ = Σ + σ. We shall then have: S + Σ´ + S´ = S + Σ + σ + S´ = S + Σ + S´ + S + σ + S´.
But we have just ascertained that S + σ + S´ was one of the series σ´. We shall therefore have: S + Σ´ + S´ = S + Σ + S´ + σ´, which means that the series S + Σ´ + S´ and S + Σ + S´ correspond to the same point N = N´ of the second space. Q.E.D.
Our two spaces therefore correspond point for point; they can be 'transformed' one into the other; they are isomorphic. How are we led to conclude thence that they are identical?