\First{The} subject of this mathematical treatise is not pure mathematics but \index{Mathematics contrasted with physics}% physics. The vocabulary of the physicist comprises a number of words such as length, angle, velocity, force, work, potential, current, etc., which we shall \index{Length!definition of}% call briefly ``physical quantities.'' Some of these terms occur in pure mathematics \index{Physical quantities}% also; in that subject they may have a generalised meaning which does not concern us here. The pure mathematician deals with ideal quantities defined as having the properties which he deliberately assigns to them. But in an experimental science we have to discover properties not to assign them; and physical quantities are defined primarily according to the way in which we recognise them when confronted by them in our observation of the world around us. Consider, for example, a length or distance between two points. It is a numerical quantity associated with the two points; and we all know the procedure followed in practice in assigning this numerical quantity to two points in nature. A definition of distance will be obtained by stating the exact procedure; that clearly must be the primary definition if we are to make sure of using the word in the sense familiar to everybody. The pure mathematician proceeds differently; he defines distance as an attribute of the two points which obeys certain laws---the axioms of the geometry which he happens to have chosen---and he is not concerned with the question how this ``distance'' would exhibit itself in practical observation. So far as his own investigations are concerned, he takes care to use the word self-consistently; but it does not necessarily denote the thing which the rest of mankind are accustomed to recognise as the distance of the two points.