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 [sigma]; 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 [Sigma]. 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_ + [Sigma] + _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 [Sigma] and [Sigma]', 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_ + [Sigma] + _S'_ and _S_ + [Sigma]' + _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'_ + [Sigma] = [Sigma] + _S_ + _S'_ = [Sigma], or again [Sigma] + _S_ + _S'_ + [Sigma]' = [Sigma] + [Sigma]'; but it does not follow that we have _S_ + [Sigma] + _S'_ = [Sigma]; 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.