Here we have a logically satisfactory theoretical basis for a metric. We may suppose that, as a matter of fact, there are important properties of groups of four points which are "parallelogramical," and that actual physical measurement is an approximate method of discovering which groups have this property. We shall find certain laws approximately fulfilled by rough-and-ready measurements, and fulfilled with increasing accuracy as we introduce refinements into the process of measurement. Consider, for example, Euclid's first axiom: Things which are equal to the same thing are equal to one another. Presumably Euclid regarded this as a logically necessary proposition, and so do people who are engaged in the practice of measurement. If two lengths each equal to a metre are found to be not equal to each other, the plain man assumes that there must be a mistake somewhere. We are therefore continually redefining the actual operations of measurement with a view to verifying Euclid's first axiom as nearly as possible. But with the above-quoted definition of equality of length the first axiom becomes a substantial proposition, namely: If is a parallelogram, and likewise , then is a parallelogram. If this proposition is true, then it is theoretically possible to define measurement in such a way that two lengths each equal to a metre shall always be equal to each other. What is called "accuracy" is, speaking generally, an attempt to obtain a result conformable with some ideal standard supposed to be logical but in fact physical.[Pg 100] What do we mean by saying that a length has been "wrongly" measured? Whatever result we obtain from measuring a given length, the result represents a fact in the world. But in what we call a "wrong" measurement, the fact ascertained is complex and of small universality. If the observer has simply misread a scale, the fact ascertained involves reference to his psychology. If he has neglected a physical correction—e.g. for the temperature of his measure—the fact refers only to a measurement carried out with that particular apparatus on that particular occasion. In relativity theory we have another set of what might be called "inaccurate" measurements—e.g. measurements of the masses of -particles or -particles emitted from radio-active bodies must be corrected for their motion relative to the observer before they acquire any general significance. It is always the search for simple relations which enter into general laws that governs successive refinements. But the existence of such relations (where they do exist) is an empirical fact, so that much that seems prima facie to be logically necessary is really contingent. On the other hand, the number of premisses in a deductive system which has to agree with an empirical science can, by logical skill, be diminished to an extent which may be astonishing. Of this, the theory of relativity is a very remarkable example. The theory is a combination of two diverse