Page:Popular Science Monthly Volume 70.djvu/87

Rh 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.

That fixed, in order that $$\Sigma$$ and $$\Sigma'$$ should correspond to the same point $$M=M'$$ of the first space, it is necessary and sufficient for us to have $$\Sigma'=\Sigma+\sigma$$. We shall then have: $$S+\Sigma'+\Sigma'=S+\Sigma+\sigma+ S'=8+\Sigma+S'+S+\sigma+S'$$.

But we have just ascertained that $$S+\sigma+8'$$ was one of the series $$\sigma$$. We shall therefore have: $$S+\Sigma'+S'=S+\Sigma+S'+\sigma'$$, which means that the series $$S+\Sigma'+S'$$ and $$S+\Sigma+S'$$ 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