175. To sum up: If points are defined simply by relations to other points, i.e., if all position is relative, every point must have to every other point one, and only one, relation independent of the rest of space. This relation is the distance between the two points. Now a relation between two points can only be defined by a line joining them—nay further, it may be contended that a relation can only be a line joining them. Hence a unique relation involves a unique line, i.e., a line determined by any two of its points. Only in a space which admits of such a line is linear magnitude a logically possible conception. But when once we have established the possibility, in general, of drawing such lines, and therefore of measuring linear magnitudes, we may find that a certain magnitude has a peculiar relation to the constitution of space. The straight line may turn out to be of finite length, and in this case its length will give a certain peculiar magnitude, the space-constant. Two antipodal points, that is, points which bisect the entire straight line, will then have a relation of magnitude which, though unaltered by motion, is rendered peculiar by a certain constant relation to the rest of space. This peculiarity presupposes a measure of linear magnitude in general, and cannot, therefore, upset the apriority of the axiom of the straight line. But it destroys, for points having the peculiar antipodal relation to each other, the argument which proved that the relation between two points could not, since it was unchanged by motion, have reference to the rest of space. Thus it is intelligible that, for such special points, the axiom breaks down, and an infinite number of straight lines are possible between them; but unless we had started with assuming the general validity of the axiom, we could never have reached a position in which antipodal points could have been known to be peculiar, or, indeed, a position which would have enabled us to give any quantitative definition whatever of particular points.
Distance and the straight line, as relations uniquely determined by two points, are thus à priori necessary to metrical Geometry. But further, they are properties which must belong to any form of externality. Since their necessity for Geometry was deduced from homogeneity and the relativity of position, and since these are necessary properties of any form of externality, the same argument proves both conclusions. We thus obtain, as in the case of Free Mobility, a double apriority: The axiom of Distance, and its implication, the axiom of the Straight Line, are, on the one hand, presupposed in the possibility of spatial magnitude, and cannot, therefore, be contradicted by any experience resulting from the measurement of space; while they are consequences, on the other hand, of the necessary properties of any form of externality which is to render possible experience of an external world.
176. In connection with the straight line, it will be convenient to discuss the conditions of a metrical coordinate system. The projective coordinate system, as we have seen, aims only at a convenient nomenclature for different points, and can be set up without introducing the notion of spatial quantity. But a metrical coordinate system does much more than this. It defines every point quantitatively, by its quantitative spatial relations to a certain coordinate figure. Only when the system of coordinates is thus metrical, i.e., when every coordinate represents some spatial magnitude, which is itself a relation of the point defined to some other point or figure—can operations with coordinates lead to a metrical result. When, as in projective Geometry, the coordinates are not spatial magnitudes, no amount of transformation can give a metrical result. I wish to prove, here, that a metrical coordinate system necessarily involves the straight line, and cannot, without a logical fallacy, be set up on any other basis. The projective system of coordinates, as we saw, is entirely based on the straight line; but the metrical system is more important, since its quantities embody actual information as to spatial magnitudes, which, in projective Geometry, is not the case.
In the first place, a point's metrical coordinates constitute a complete quantitative definition of it; now a point can only be defined, as we have seen, by its relations to other points, and these relations can only be defined by means of the straight line. Consequently, any metrical system of coordinates must involve the straight line, as the basis of its definitions of points.
This à priori argument, however, though I believe it to be quite sound, is not likely to carry conviction to any one persuaded of the opposite. Let us, therefore, examine metrical coordinate systems in detail, and show, in each case, their dependence on the straight line.
We have already seen that the notion of distance is impossible without the straight line. We cannot, therefore, define our coordinates in any of the ordinary ways, as the distances from three planes, lines, points, spheres, or what not. Polar coordinates are impossible, since,—waiving the straightness of the radius vector—the length of the radius vector becomes unmeaning. Triangular coordinates involve not only angles, which must in the limit be rectilinear, but straight lines, or at any rate some well-defined curves. Now curves can only be metrically defined in two ways: Either by relation to the straight line, as, e.g., by the curvature at any point, or by purely analytical equations, which presuppose an intelligible system of metrical coordinates. What methods remain for assigning these arbitrary values to different points? Nay, how are we to get any estimate of the difference—to avoid the more special notion of distance—between two points? The very notion of a point has become illusory. When we have a coordinate system, we may define a point by its three coordinates; in the absence of such a system, we may define the notion of point in general as the intersection of three surfaces or of two curves. Here we take surfaces and curves as notions which intuition makes plain, but if we wish them to give us a precise numerical definition of particular points, we must specify the kind of surface or curve to be used. Now this, as we have seen, is only possible when we presuppose either the straight line, or a coordinate system. It follows that every coordinate system presupposes the straight line, and is logically impossible without it.
177. The above three axioms, we have seen, are à priori necessary to metrical Geometry. No others can be necessary, since metrical systems, logically as unassailable as Euclid's, and dealing with spaces equally homogeneous and equally relational, have been constructed by the metageometers, without the help of any other axioms. The remaining axioms of Euclidean Geometry—the axiom of parallels, the axiom that the number of dimensions is three, and Euclid's form of the axiom of the straight line (two straight lines cannot enclose a space)—are not essential to the possibility of metrical Geometry, i.e., are not deducible from the fact that a science of spatial magnitudes is possible. They are rather to be regarded as empirical laws, obtained, like the empirical laws of other sciences, by actual investigation of the given subject-matter—in this instance, experienced space.
178. In summing up the distinctive argument of this Section, we may give it a more general form, and discuss the conditions of measurement in any continuous manifold, i.e., the qualities necessary to the manifold, in order that quantities in it may be determinable, not only as to the more or less, but as to the precise how much.
Measurement, we may say, is the application of number to continua, or, if we prefer it, the transformation of mere quantity into number of units. Using quantity to denote the vague more or less, and magnitude to denote the precise number of units, the problem of measurement may be defined as the transformation of quantity into magnitude.
Now a number, to begin with, is a whole consisting of smaller units, all of these units being qualitatively alike. In order, therefore, that a continuous quantity may be expressible as a number, it must, on the one hand, be itself a whole, and must, on the other hand, be divisible into qualitatively similar parts. In the aspect of a whole, the quantity is intensive; in the aspect of an aggregate of parts, it is extensive. A purely intensive quantity, therefore, is not numerable—a purely extensive quantity, if any such could be imagined, would not be a single quantity at all, since it would have to consist of wholly unsynthesized particulars. A measurable quantity, therefore, is a whole divisible into similar parts. But a continuous quantity, if divisible at all, must be infinitely divisible. For otherwise the points at which it could be divided would form natural barriers, and so destroy its continuity. But further, it is not sufficient that there should be a possibility of division into mutually external parts; while the parts, to be perceptible as parts, must be mutually external, they must also, to be knowable as equal parts, be capable of overcoming their mutual externality. For this, as we have seen, we require superposition, which involves Free Mobility and homogeneity—the absence of Free Mobility in time, where all other requisites of measurement are fulfilled, renders direct measurement of time impossible. Hence infinite divisibility, free mobility, and homogeneity are necessary for the possibility of measurement in any continuous manifold, and these, as we have seen, are equivalent to our three axioms. These axioms are necessary, therefore, not only for spatial measurement, but for all measurement. The only manifold given in experience, in which these conditions are satisfied, is space. All other exact measurement—as could be proved, I believe, for every separate case—is effected, as we saw in the case of time, by reduction to a spatial correlative. This explains the paramount importance, to exact science, of the mechanical view of nature, which reduces all phenomena to motions in time and space. For number is, of all conceptions, the easiest to operate with, and science seeks everywhere for an opportunity to apply it, but finds this opportunity only by means of spatial equivalents to phenomena[177].