Page:Popular Science Monthly Volume 69.djvu/557

Rh change. Two of these internal changes $$\alpha'$$ and $$\beta'$$ shall be regarded as indistinguishable: (1) if they are so naturally, that is, if they are too close to one another; (2) if $$\alpha'$$ is capable of correcting the same external change as a third internal change naturally indistinguishable from $$\beta'$$. In this second case, they will be, so to speak, indistinguishable by convention, I mean by agreeing to disregard circumstances which might distinguish them.

Our continuum is now entirely defined, since we know its elements and have fixed under what conditions they may be regarded as indistinguishable. We thus have all that is necessary to apply our definition and determine how many dimensions this continuum has. We shall recognize that it has six. The continuum of displacements is, therefore, not equivalent to space, since the number of dimensions is not the same; it is only related to space. Now how do we know that this continuum of displacements has six dimensions? We know it by experience.

It would be easy to describe the experiments by which we could arrive at this result. It would be seen that in this continuum cuts can be made which divide it and which are continua; that these cuts themselves can be divided by other cuts of the second order which yet are continua, and that this would stop only after cuts of the sixth order which would no longer be continua. From our definitions that would mean that the group of displacements has six dimensions.

That would be easy, I have said, but that would be rather long; and would it not be a little superficial? This group of displacements, we have seen, is related to space, and space could be deduced from it, but it is not equivalent to space, since it has not the same number of dimensions; and when we shall have shown how the notion of this continuum can be formed and how that of space may be deduced from it, it might always be asked why space of three dimensions is much more familiar to us than this continuum of six dimensions, and consequently doubted whether it was by this detour that the notion of space was formed in the human mind.

What is a point? How do we know whether two points of space are identical or different? Or, in other words, when I say: The object $$A$$ occupied at the instant $$\alpha''$$ the point which the object $$B$$ occupies at the instant $$\beta$$, what does that mean?

Such is the problem we set ourselves in the preceding chapter, § 4. As I have explained it, it is not a question of comparing the positions of the objects $$A$$ and $$B$$ in absolute space; the question then would manifestly have no meaning. It is a question of comparing the positions of these two objects with regard to axes invariably bound to my body, supposing always this body replaced in the same attitude.