Page:Popular Science Monthly Volume 69.djvu/552

548 the position $$A$$ to the position $$B$$ I may take several routes. To the first of these routes will correspond a series $$S$$ of muscular sensations; to a second route will correspond another series $$S''$$ of muscular sensations which generally will be completely different, since other muscles will be used.

How am I led to regard these two series $$S$$ and $$S''$$ as corresponding to the same displacement $$AB$$? It is because these two series are capable of correcting the same external change. Apart from that, they have nothing in common.

Let us now consider two external changes: $$\alpha$$ and $$\beta$$, which shall be, for instance, the rotation of a sphere half blue, half red, and that of a sphere half yellow, half green; these two changes have nothing in common, since the one is for us the passing of blue into red and the other the passing of yellow into green. Consider, on the other hand, two series of internal changes $$S$$ and $$S''$$; like the others, they will have nothing in common. And yet I say that $$\alpha$$ and $$\beta$$ correspond to the same displacement, and that $$S$$ and $$S''$$ correspond also to the same displacement. Why? Simply because $$S$$ can correct $$\beta$$ as well as $$\alpha$$ and because $$\alpha$$ can be corrected by $$S''$$ as well as by $$S$$. And then a question suggests itself: If I have ascertained that $$S$$ corrects $$\alpha$$ and $$\beta$$ and that $$S$$ corrects $$\alpha$$, am I certain that $$S$$ likewise corrects $$\beta$$? Experiment alone can teach us whether this law is verified. If it were not verified, at least approximately, there would be no geometry, there would be no space, because we should have no more interest in classifying the internal and external changes as I have just done, and, for instance, in distinguishing changes of state from changes of position.

It is interesting to see what has been the rôle of experience in all this. It has shown me that a certain law is approximately verified. It has not told me wherefore space is, and that it satisfies the condition in question. I knew in fact, before all experience, that space satisfied this condition or that it would not be; nor have I any right to say that experience told me that geometry is possible; I very well see that geometry is possible, since it does not imply contradiction; experience only tells me that geometry is useful.

Although motor impressions have had, as I have just explained, an altogether preponderant influence in the genesis of the notion of space, which never would have taken birth without them, it will not be without interest to examine also the role of visual impressions and to investigate how many dimensions 'visual space' has, and for that purpose to apply to these impressions the definition of § 3.

A first difficulty presents itself: consider a red color sensation affecting a certain point of the retina; and on the other hand a blue color sensation affecting the same point of the retina. It is necessary