In the first place, what do we mean by measurement? Clearly we do not mean any method of assigning numbers to a collection of objects; there must be properties of importance connected with the numbers assigned. We do not say that the books in the British Museum are "measured" by their press-marks. Given any collection whose cardinal number is less than or equal to , we can assign some or all of the real numbers as "press-marks" of the several members of the collection. Given any collection of terms, it can be arranged in a Euclidean or non-Euclidean space of any known sort with any finite number of dimensions, and when so arranged it will be amenable to the whole of metrical geometry. But the "distance" between two terms of the collection, when it is defined in this way, will, in general, be quite unimportant, in the sense that it will have only such properties as follow tautologically from its definition, not such further empirical properties as would make the definition valuable. So long as this is the case, there is no reason to prefer one to another of the various incompatible systems of distances which pure mathematics would allow us to assign. Let us take an illustration. In projective geometry we start from a set of axioms which say nothing about quantity, and do not even obviously involve order. But it is found that they do lead to an order, and that, by means of the order,[Pg 110] co-ordinates can be assigned to points. These co-ordinates have a definite projective meaning: they represent the series of quadrilateral constructions required to reach the point in question from certain given initial points, (I omit complications concerning limits; these are dealt with in the chapter "Projective Geometry" in The Principles of Mathematics.) In this case, it may seem doubtful whether we have measurement or not. We have assigned co-ordinates in a manner which preserves the order-relations of points, and it turns out that the ordinary distance between two points is a simple function of their projective co-ordinates, though the function is somewhat different according as space is Euclidean, hyperbolic, or elliptic. It is just because of this difference that we shall not say we have "measured" distances when we have introduced projective co-ordinates. These co-ordinates, for example, will not tell us, even approximately, how long it would take to walk from one place to another, and this is the sort of thing that measurement ought to tell us. What, then, is meant when it is said that, in the theory of relativity, there is a metrical relation of interval? Let us take up the matter at the point where Eddington leaves it. He suggests that all that is needed is "comparability" between two point-pairs, or, as he says, between two "displacements." (We may leave aside for the moment the question whether this is only to hold for point-pairs which are very near together.) This language seems somewhat vague; let us try to give it precision.