Page:Popular Science Monthly Volume 70.djvu/85

Rh We who do not yet know geometry can not reason in this way; we can only verify. But then a question arises; how, before knowing geometry, have we been led to distinguish from the others these series a-where the finger does not budge? It is, in fact, only after having made this distinction that we could be led to regard $$\Sigma$$ and $$\Sigma+\sigma$$ as identical, and it is on this condition alone, as we have just seen, that we can arrive at space of three dimensions.

We are led to distinguish the series $$\sigma$$, because it often happens that when we have executed the movements which correspond to these series o-of muscular sensations, the tactile sensations which are transmitted to us by the nerve of the finger that we have called the first finger, persist and are not altered by these movements. Experience alone tells us that and it alone could tell us.

If we have distinguished the series of muscular sensations $$s+s'$$ formed by the union of two inverse series, it is because they preserve the totality of our impressions; if now we distinguish the series $$\sigma$$, it is because they preserve certain of our impressions. (When I say that a series of muscular sensations $$s$$ 'preserves' one of our impressions $$A$$, I mean that we ascertain that if we feel the impression $$A$$, then the muscular sensations $$s$$, we still feel the impression $$A$$ after these sensations $$s$$.)

I have said above it often happens that the series $$\sigma$$ do not alter the tactile impressions felt by our first finger; I said often, I did not say $$always.$$ This it is that we express in our ordinary language by saying that the tactile impressions would not be altered if the finger has not moved, on the condition that neither has the object $$A$$, which was in contact with this finger, moved. Before knowing geometry, we could not give this explanation; all we could do is to ascertain that the impression often persists, but not always.

But that the impression often continues is enough to make the series o-appear remarkable to us, to lead us to put in the same class the series $$\Sigma$$ and $$\Sigma+\sigma$$, and hence not regard them as distinct. Under these conditions we have seen that they will engender a physical continuum of three dimensions.

Behold then a space of three dimensions engendered by my first finger. Each of my fingers will create one like it. It remains to consider how we are led to regard them as identical with visual space, as identical with geometric space.

But one reflection before going further; according to the foregoing, we know the points of space, or more generally the final situation of our body, only by the series of muscular sensations revealing to us the movements which have carried us from a certain initial situation to this final situation. But it is clear that this final situation will depend, on the one hand, upon these movements and, on the other hand, upon the initial situation from which we set out. Now these movements are