chosen, I should so make this choice that these movements carry the finger to the point originally occupied by the finger , that is, to the point ; this finger will thus be in contact with the object , which will make it feel the impression .
I then make the movements corresponding to the series ; in these movements, by hypothesis, the position of the finger does not change, this finger therefore remains in contact with the object and continues to feel the impression . Finally I make the movements corresponding to the series . As is inverse to , these movements carry the finger to the point previously occupied by the finger , that is, to the point . If, as may be supposed, the object a has not budged, this finger will be in contact with this object and will feel anew the impression . . . .
Let us see the consequences. I consider a series of muscular sensations . To this series will correspond a point of the first tactile space. Now take again the two series and , inverses of one another, of which we have just spoken. To the series will correspond a point 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 and , thus defined, as corresponding. What authorizes me so to do? For this correspondence to be admissible, it is necessary that if two points and , corresponding in the first space to two series and , are identical, so also are the two corresponding points of the second space and , that is the two points which correspond to the two series and . Now we shall see that this condition is fulfilled.
First a remark. As and are inverses of one another, we shall have , and consequently , or again ; but it does not follow that we have ; 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 and should correspond to the same point of the first space, it is necessary and sufficient for us to have . We shall then have: .
But we have just ascertained that was one of the series . We shall therefore have: , which means that the series and correspond to the same point of the second space.
Our two spaces therefore correspond point for point; they can be