Page:Popular Science Monthly Volume 70.djvu/86

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THE POPULAR SCIENCE MONTHLY

vealed to us by our muscular sensations; but nothing tells us the initial situation; nothing can distinguish it for us from all the other possible situations. This puts well in evidence the essential relativity of space.

§ 4. Identity of the Different Spaces

We are therefore led to compare the two continua ${\displaystyle C}$ and ${\displaystyle C'}$ engendered, for instance, one by my first finger ${\displaystyle D}$, the other by my second finger ${\displaystyle D'}$. These two physical continua both have three dimensions. To each element of the continuum ${\displaystyle C}$, or, if you prefer, to each point of the first tactile space, corresponds a series of muscular sensations ${\displaystyle \Sigma }$, which carry me from a certain initial situation to a certain final situation.[1] Moreover, the same point of this first space will correspond to ${\displaystyle \Sigma }$ and to ${\displaystyle \Sigma +\sigma }$, if ${\displaystyle \sigma }$ is a series of which we know that it does not make the finger ${\displaystyle D}$ move.

Similarly to each element of the continuum ${\displaystyle C'}$, or to each point of the second tactile space, corresponds a series of sensations ${\displaystyle \Sigma '}$, and the same point will correspond to ${\displaystyle \Sigma '}$ and to ${\displaystyle \Sigma '+\sigma '}$, if ${\displaystyle \sigma '}$ is a series which does not make the finger ${\displaystyle D'}$ move.

What makes us distinguish the various series designated ${\displaystyle \sigma }$ from those called ${\displaystyle \sigma '}$ is that the first do not alter the tactile impressions felt by the finger ${\displaystyle D}$ and the second preserve those the finger ${\displaystyle D'}$ feels.

Now see what we ascertain: in the beginning my finger ${\displaystyle D'}$ feels a sensation ${\displaystyle A'}$; I make movements which produce muscular sensations ${\displaystyle S}$; my finger ${\displaystyle D}$ feels the impression ${\displaystyle A}$; I make movements which produce a series of sensations ${\displaystyle \sigma }$; my finger ${\displaystyle D}$ continues to feel the impression ${\displaystyle A}$, since this is the characteristic property of the series ${\displaystyle \sigma }$; I then make movements which produce the series ${\displaystyle S'}$ of muscular sensations, inverse to ${\displaystyle S}$ in the sense above given to this word. I ascertain then that my finger ${\displaystyle D'}$ feels anew the impression ${\displaystyle A'}$. (It is of course understood that ${\displaystyle S}$ has been suitably chosen.)

This means that the series ${\displaystyle s+\sigma +s'}$, preserving the tactile impressions of the finger ${\displaystyle D'}$, is one of the series I have called ${\displaystyle \sigma '}$. Inversely, if one takes any series ${\displaystyle \sigma '}$, ${\displaystyle s'+\sigma '+s}$ will be one of the series that we call ${\displaystyle \sigma }$.

Thus if ${\displaystyle s}$ is suitably chosen, ${\displaystyle s+\sigma +s'}$ will be a series ${\displaystyle \sigma '}$, and by making ${\displaystyle \sigma }$ vary in all possible ways, we shall obtain all the possible series ${\displaystyle \sigma '}$.

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 ${\displaystyle D'}$ is at the point ${\displaystyle M}$, in contact with the object ${\displaystyle a}$, which makes it feel the impression ${\displaystyle A'}$. I make the movements corresponding to the series ${\displaystyle S}$; I have said that this series should be suitably

1. In place of saying that we refer space to axes rigidly bound to our body, perhaps it would be better to say, in conformity to what precedes, that we refer it to axes rigidly bound to the initial situation of our body.