Page:Popular Science Monthly Volume 69.djvu/555

can say is that experience has taught us that it is convenient to attribute three dimensions to space.

But visual space is only one part of space, and in even the notion of this space there is something artificial, as I have explained at the beginning. The real space is motor space and this it is that we shall examine in the following chapter.

 

Chapter IV. Space and its Three Dimensions

§ 1. The Group of Displacements

Let us sum up briefly the results obtained. We proposed to investigate what was meant in saying that space has three dimensions and we have asked first what is a physical continuum and when it may be said to have n dimensions. If we consider different systems of impressions and compare them with one another, we often recognize that two of these systems of impressions are indistinguishable (which is ordinarily expressed in saying that they are too close to one another, and that our senses are too crude, for us to distinguish them) and we ascertain besides that two of these systems can sometimes be discriminated from one another though indistinguishable from a third system. In that case we say the manifold of these systems of impressions forms a physical continuum $C$. And each of these systems is called an element of the continuum $C$.

How many dimensions has this continuum? Take first two elements $A$ and $B$ of $C,$ and suppose there exists a series $\Sigma$ of elements, all belonging to the continuum $C,$ of such a sort that $A$ and $B$ are the two extreme terms of this series and that each term of the series is indistinguishable from the preceding. If such a series $\Sigma$ can be found, we say that $A$ and $B$ are joined to one another; and if any two elements of $C$ are joined to one another, we say that $C$ is all of one piece.

Now take on the continuum $C$ a certain number of elements in a way altogether arbitrary. The aggregate of these elements will be called a cut. Among the various series $\Sigma$ which join $A$ to $B$, we shall distinguish those of which an element is indistinguishable from one of the elements of the cut (we shall say that these are they which cut the cut) and those of which all the elements are distinguishable from all those of the cut. If all the series $\Sigma$ which join $A$ to $B$ cut the cut, we shall say that $A$ and $B$ are separated by the cut, and that the cut divides $C$. If we can not find on $C$ two elements which are separated by the cut, we shall say that the cut does not divide $C$.

These definitions laid down, if the continuum $C$ can be divided by cuts which do not themselves form a continuum, this continuum $C$ has only one dimension; in the contrary case it has several. If a cut forming a continuum of 1 dimension suffices to divide $C,C$ will have 2 dimensions; if a cut forming a continuum of 2 dimensions suffices, $C$ will have 3 dimensions, etc. Thanks to these definitions, we can always