If we were not able to attach any concrete meaning to the expression for $D$ the value of all this would be materially lessened. Consider, for instance, the continuum of music notes. There is no distance between different notes. There is of course significance in talking about the difference in pitch, in intensity, in duration, in timbre, between two notes; but there is none in a mode of speech that implies a composite expression indicating how far one note escapes being identical with another in all four respects at once. The trouble, of course, is that the four dimensions of the music-note continuum are not measurable in terms of a common unit. If they were, we should expect to measure their combination more or less absolutely in terms of this same unit. We can make measurements in all three dimensions of Euclidean space with the same unit, with the same measuring rod in fact. [This presents a peculiarity of our three-space which is not possessed by all three-dimensional manifolds. Riemann has given another illustration in the system of all possible colors, composed of arbitrary proportions of the three primaries, red, green and violet. This system forms a three-dimensional continuum; but we cannot measure the "distance" or difference between two colors in terms of the difference between two others.]130 Accordingly, in spite of the fact that the Euclidean three-space gives us a formal representation of the color continuum, and in spite of the fact that the hypothetical four-dimensional Euclidean space would perform a like office for the music-note continuum, this representation would be without significance. We should not say that the geometry of these two manifolds is Euclidean. We should realize that any set of numerical elements can be plotted in a Euclidean space of the appropriate dimensionality; and that accordingly, before allowing such a plot to influence us to classify the geometry of the given manifold as Euclidean, we must pause long enough to ask whether the rest of the Euclidean system fits into the picture. If the square root of the sum of the squares of the coordinate-differences between two elements possesses significance in the given continuum, and if it is invariant between observers of that continuum who employ different bases of reference, then and only then may we allege the Euclidean character of the given continuum. If under this test the given continuum fails of Euclideanism, it is in order to ask what type of geometry it does present. If it is of such character that the "distance" between two elements possesses significance, we should answer this question by investigating that distance in the hope of discovering a non-Euclidean expression for it which will be invariant. If it is not of such character, we should seek some other characteristic of single elements or groups of elements, of real physical significance and of such sort that the numerical expression for it would be invariant.