Let us call one of these aggregates of sensations an element. That will be something analogous to the point of the mathematicians; it will not be altogether the same thing however. We can not say our element is without extension, since we can not distinguish it from neighboring elements and it is thus surrounded by a sort of haze. If the astronomical comparison may be allowed, our 'elements' would be like nebulae, whereas the mathematical points would be like stars.
That being granted, a system of elements will form a continuum if we can pass from any one of them to any other, by a series of consecutive elements such that each is indistinguishable from the preceding. This linear series is to the line of the mathematician what an isolated element was to the point.
Before going farther, I must explain what is meant by a cut. Consider a continuum C and remove from it certain of its elements which for an instant we shall regard as no longer belonging to this continuum. The aggregate of the elements so removed will be called a cut. It may happen that, thanks to this cut, C may be subdivided into several distinct continua, the aggregate of the remaining elements ceasing to form a unique continuum.
There will then be on C two elements, A and B, that must be regarded as belonging to two distinct continua, and this will be recognized because it will be impossible to find a linear series of consecutive elements of C, each of these elements indistinguishable from the preceding, the first being A and the last B, without one of the elements of this series being indistinguishable from one of the elements of the cut.
On the contrary, it may happen that the cut made is insufficient to subdivide the continuum C. To classify the physical continua, we will examine precisely what are the cuts which must be made to subdivide them.
If a physical continuum C can be subdivided by a cut reducing to a finite number of elements all distinguishable from one another (and consequently forming neither a continuum, nor several continua), we shall say C is a one-dimensional continuum.
If, on the contrary, C can be subdivided only by cuts which are themselves continua, we shall say C has several dimensions. If cuts which are continua of one dimension suffice, we shall say C has two dimensions; if cuts of two dimensions suffice, we shall say C has three dimensions, and so on.
Thus is defined the notion of the physical continuum of several dimensions, thanks to this very simple fact that two aggregates of sensations are distinguishable or indistinguishable.
The Mathematical Continuum of Several Dimensions.—Thence the notion of the mathematical continuum of n dimensions has sprung quite naturally by a process very like that we discussed at the beginning of this chapter. A point of such a continuum, you know, appears to us as defined by a system of n distinct magnitudes called its coordinates.
These magnitudes need not always be measurable; there is, for instance, a branch of geometry independent of the measurement of these magnitudes, in which it is only a question of knowing, for example, whether on a curve ABC, the point B is between the points A and C, and not of knowing whether the arc AB is equal to the arc BC or twice as great. This is what is called Analysis Situs.