Suppose that between two point-pairs there is sometimes, but not always, a symmetrical transitive relation . Then we can define as "the distance between and " the class of all point-pairs having the relation to (). If now instead of ) we write , we shall have: [Pg 111] From these two it follows that every pair of objects , in the field of is such that This seems to be as much as is strictly implied by Eddington's words, but it is certainly not all that we need. Nor does it become sufficient if we add: There must be a connection between distances and ordinal relations, there must be ways of adding distances, and there must be ways of inferring new distances from a certain number of data, as in . If all these conditions are fulfilled, we can then proceed to ask whether our distances have any further important physical properties. The sort of relation that will not do is illustrated if we take to mean that and have the same apparent dimensions in the visual field of a certain observer—e.g. the diameters of the sun and moon will approximately have this relation, which is symmetrical and transitive, but physically unimportant. Let us see what is necessary in order to get a definition of distance which will have as many as possible of the properties possessed by distance in elementary geometry. [Pg 112] If we confine ourselves to three dimensions, we can at once define a plane: it will consist of all points equidistant from two given points. The points in this plane which are equidistant from two given points in it lie on a straight line; we may take this as the definition of a straight line. Thus given two points, , , we can define the middle point of it is the point on which is equidistant from and . We shall need an axiom to the effect that this point always exists and is always unique. Thus we can halve distances and double them: we shall of course define as half of . From this point onwards, the assignment of numerical measures to our distances offers no difficulty. It is therefore only necessary to scrutinize what has already been said.