“intensity” of the ordinal characteristic, and are ordered according to the variation of that intensity.
17.1. It is very difficult to state more than this about the ordinal characteristic without a loss of generality; the definition of the characteristic in each special case will, as we said before, depend upon the nature of the aggregate in question, and the determination of the manner of variation of this characteristic will be a matter of practical physics, of practical possibilities and necessities, which will give rise to a certain method of ‘measurement’. One suggestion which offers itself at this point arises from the fact of the sameness of the ordinal characteristic for all elements of the aggregate: in this respect the ordinal characteristic invites comparison with the extensional one, and it is suggested, that if the two are not identical in all cases, it should be possible for them to be the same at least in some cases. An investigation of the relation of the ordinal to the extensional characteristic offers an interesting field of research, into which it is, however, not our intention to enter in the present essay.
17.2. Neither shall we make it a part of our present task to study the equally interesting and more important problem of measurement, which is concerned with quantitative determinations of the variation of the ordinal characteristic; it will here suffice if we say that the possibility of measurement of a given continuous aggregate is dependent upon the existence of a certain relation between every two elements of the continuum, which relation it is possible to express numerically, and to compare as to equality or inequality between various pairs of elements; this relation we denote by the term “interval,” and spatial “distance” is a particular case of it.
Dimensionality.—18. The ordinal characteristic of a given aggregate may be simple or complex, as its variation between two arbitrary elements of the aggregate can be determined by a single comparison or by a number of comparisons mutually independent; thus, two colours may be compared as to their wave lengths and as to their intensity (energy), and no amount of measurement of the one will give us any idea of the magnitude of the other. The number of such mutually independent comparisons necessary to determine the relation between two arbitrary elements in the continuum is called the dimensionality of the aggregate: thus, where one determination is sufficient, the aggregate is said to be one-dimensional, where two are necessary, two-dimensional, three, three-dimensional, etc.
Co-ordinates.—19. An important consideration which offers
