# Page:Popular Science Monthly Volume 70.djvu/87

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THE VALUE OF SCIENCE

chosen, I should so make this choice that these movements carry the finger ${\displaystyle D}$ to the point originally occupied by the finger ${\displaystyle D'}$, that is, to the point ${\displaystyle M}$; this finger ${\displaystyle D}$ will thus be in contact with the object ${\displaystyle a}$, which will make it feel the impression ${\displaystyle A}$.

I then make the movements corresponding to the series ${\displaystyle \sigma }$; in these movements, by hypothesis, the position of the finger ${\displaystyle D}$ does not change, this finger therefore remains in contact with the object ${\displaystyle a}$ and continues to feel the impression ${\displaystyle A}$. Finally I make the movements corresponding to the series ${\displaystyle S'}$. As ${\displaystyle S'}$ is inverse to ${\displaystyle S}$, these movements carry the finger ${\displaystyle D'}$ to the point previously occupied by the finger ${\displaystyle D}$, that is, to the point ${\displaystyle M}$. If, as may be supposed, the object a has not budged, this finger ${\displaystyle D'}$ will be in contact with this object and will feel anew the impression ${\displaystyle A'}$. . . . ${\displaystyle Q.E.D.}$

Let us see the consequences. I consider a series of muscular sensations ${\displaystyle \Sigma }$. To this series will correspond a point ${\displaystyle M}$ of the first tactile space. Now take again the two series ${\displaystyle s}$ and ${\displaystyle s'}$, inverses of one another, of which we have just spoken. To the series ${\displaystyle s+\Sigma +s'}$ will correspond a point ${\displaystyle N}$ 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 ${\displaystyle N}$ and ${\displaystyle M}$, thus defined, as corresponding. What authorizes me so to do? For this correspondence to be admissible, it is necessary that if two points ${\displaystyle M}$ and ${\displaystyle M'}$, corresponding in the first space to two series ${\displaystyle \Sigma }$ and ${\displaystyle \Sigma '}$, are identical, so also are the two corresponding points of the second space ${\displaystyle N}$ and ${\displaystyle N'}$, that is the two points which correspond to the two series ${\displaystyle s+\Sigma +s'}$ and ${\displaystyle s+\Sigma '+s'}$. Now we shall see that this condition is fulfilled.

First a remark. As ${\displaystyle S}$ and ${\displaystyle S'}$ are inverses of one another, we shall have ${\displaystyle S+S'=0}$, and consequently ${\displaystyle S+S'+\Sigma =\Sigma +S+S'=\Sigma }$, or again ${\displaystyle \Sigma +S+S'+\Sigma '=\Sigma +\Sigma '}$; but it does not follow that we have ${\displaystyle S+\Sigma +S'=\Sigma }$; 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 ${\displaystyle \Sigma }$ and ${\displaystyle \Sigma '}$ should correspond to the same point ${\displaystyle M=M'}$ of the first space, it is necessary and sufficient for us to have ${\displaystyle \Sigma '=\Sigma +\sigma }$. We shall then have: ${\displaystyle S+\Sigma '+\Sigma '=S+\Sigma +\sigma +S'=8+\Sigma +S'+S+\sigma +S'}$.

But we have just ascertained that ${\displaystyle S+\sigma +8'}$ was one of the series ${\displaystyle \sigma }$. We shall therefore have: ${\displaystyle S+\Sigma '+S'=S+\Sigma +S'+\sigma '}$, which means that the series ${\displaystyle S+\Sigma '+S'}$ and ${\displaystyle S+\Sigma +S'}$ correspond to the same point ${\displaystyle N=N'}$ of the second space. ${\displaystyle Q.E.D.}$

Our two spaces therefore correspond point for point; they can be