If, as Weyl assumes, infinitesimal distances which have one end in common are comparable, this must be taken to mean that two small finite distances are capable of a resemblance which may be called "quasi-equality," which grows more nearly complete resemblance as the distance grows smaller. We may assume, as before, that, given a point and a definite route from to , there will be one definite point on this route such that is more nearly equal to than is any other distance by on the route in question. We shall then say that and are "quasi-equal." Take also ... quasi-equal, and , ... quasi-equal. In this way we can construct a co-ordinate mesh with axes , . And we can now construct what will be in effect straight lines through : take all the points which are the corners opposite to of quasi-parallelograms , for different initial points , subject to quasi-equality between and . These points may be regarded as forming the quasi-straight line whose equation is . (Irrationals can be dealt with by the usual methods.) This quasi-straight line will start from in a[Pg 120] certain direction, and may, for differential purposes, be regarded as really a straight line. It is not worth while to proceed further, since it is obvious that we have the necessary material. Degrees of similarity may be, in a sense, measured by quasi-transitiveness. Suppose that , , , ... each have quasi-equality with the next. It may or may not happen that has quasi-equality with . One may presume that this will happen if and are very small and is not very large. Similarly, or rather a fortiori, we cannot infer that has quasi-equality with . The larger the value of for which such an inference remains true, the closer is the resemblance between and or between and . It is to be assumed that, by continually diminishing and the number of steps for which the inference is permitted can be increased without finite limit. If the above is in any degree valid, it would seem that, if space-time is continuous, spatio-temporal measurement depends theoretically upon qualitative similarity, capable of varying degrees, between relations of pairs of points. It is not suggested that the analysis cannot be carried further, but only that this is a valid stage in the process of explaining what is meant by the quantitative character of intervals and by their measurement as numerical multiples of units. [Pg 121] CHAPTER XIII MATTER AND SPACE