Page:Popular Science Monthly Volume 69.djvu/560

556 between the instants $$\alpha$$ and $$\alpha'$$, nor the object $$B$$ between the instants $$\beta$$ and $$\beta'$$; assume this. At the instant $$\alpha$$, the object $$A$$ occupied a certain point $$M$$ of space. Now at this instant it touched my first finger, and as touch does not operate at a distance, my first finger was likewise at the point $$M$$. I afterward made the series S of movements and at the end of this series, at the instant $$\alpha'$$, I ascertained that the object $$A$$ touched my second finger. I thence conclude that this second finger was then at $$M$$, that is, that the movements $$S$$ had the result of bringing the second finger to the place of the first. At the instant $$\beta$$ the object $$B$$ has come in contact with my second finger: as I have not budged, this second finger has remained at $$M$$; therefore the object $$B$$ has come to $$M$$; by hypothesis it does not budge up to the instant $$\beta'$$. But between the instants $$\beta$$ and $$\beta'$$ I have made the movements $$S'$$; as these movements are the inverse of the movements $$S$$, they must have for effect bringing the first finger in the place of the second. At the instant $$\beta'$$ this first finger will, therefore, be at $$M$$; and as the object $$B$$, is likewise at $$M$$, this object B will touch my first finger. To the question put, the answer should, therefore, be yes.

We who do not yet know geometry can not reason thus; but we ascertain that this anticipation is ordinarily realized; and we can always explain the exceptions by saying that the object $$A$$ has moved between the instants $$\alpha$$ and $$\alpha'$$, or the object $$B$$ between the instants $$\beta$$ and $$\beta'$$.

But could not experience have given a contrary result? Would this contrary result have been absurd in itself? Evidently not. What should we have done then if experience had given this contrary result? Would all geometry thus have become impossible? Not the least in the world. We should have contented ourselves with concluding that touch can operate at a distance.

When I say, touch does not operate at a distance, but sight operates at a distance, this assertion has only one meaning, which is as follows: To recognize whether $$B$$ occupies at the instant $$\beta$$ the point occupied by $$A$$ at the instant $$\alpha'$$, I can use a multitude of different criteria. In one my eye intervenes, in another my first finger, in another my second finger, etc. Well, it is sufficient for the criterion relative to one of my fingers to be satisfied in order that all the others should be satisfied, but it is not sufficient that the criterion relative to the eye should be. This is the sense of my assertion, I content myself with affirming an experimental fact which is ordinarily verified.

At the end of the preceding chapter we analyzed visual space; we saw that to engender this space it is necessary to bring in the retinal sensations, the sensation of convergence and the sensation of accommodation; that if these last two were not always in accord, visual space would have four dimensions in place of three; we also saw that if we brought in only the retinal sensations, we should obtain 'simple visual space,' of only two dimensions. On the other hand, consider tactile