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?
Consider the two series σ and S + σ + S´ = σ´. I have said that often, but not always, the series σ preserves the tactile impression A felt by the finger D; and similarly it often happens, but not always, that the series σ´ preserves the tactile impression A´ felt by the finger D´. Now I ascertain that it happens very often (that is, much more often than what I have just called 'often') that when the series σ has preserved the impression A of the finger D, the series σ´ preserves at the same time the impression A´ of the finger D´; and, inversely, that if the first impression is altered, the second is likewise. That happens very often, but not always.