If, in pursuance of our habit of thought, we now supplement the propositions of Euclidean geometry by the single proposition that two points on a practically rigid body always correspond to the same distance (line-interval), independently of any changes in position to which we may subject the body, the propositions of Euclidean geometry then resolve themselves into propositions on the possible relative position of practically rigid bodies.[1] Geometry which has been supplemented in this way is then to be treated as a branch of physics. We can now legitimately ask as to the “truth” of geometrical propositions interpreted in this way, since we are justified in asking whether these propositions are satisfied for those real things we have associated with the geometrical ideas. In less exact terms we can express this by saying that by the “truth” of a geometrical proposition in this sense we understand its validity for a construction with rule and compasses. [1] It follows that a natural object is associated also with a straight line. Three points _A, B_ and _C_ on a rigid body thus lie in a straight line when the points _A_ and _C_ being given, _B_ is chosen such that the sum of the distances _AB_ and _BC_ is as short as possible. This incomplete suggestion will suffice for the present purpose. Of course the conviction of the “truth” of geometrical propositions in this sense is founded exclusively on rather incomplete experience. For the present we shall assume the “truth” of the geometrical propositions, then at a later stage (in the general theory of relativity) we shall see that this “truth” is limited, and we shall consider the extent of its limitation. II. THE SYSTEM OF CO-ORDINATES On the basis of the physical interpretation of distance which has been indicated, we are also in a position to establish the distance between two points on a rigid body by means of measurements. For this purpose we require a “distance” (rod _S_) which is to be used once and for all, and which we employ as a standard measure. If, now, _A_ and _B_ are two points on a rigid body, we can construct the line joining them according to the rules of geometry; then, starting from _A_, we can mark off the distance _S_ time after time until we reach _B_. The number of these operations required is the numerical measure of the distance _AB_. This is the basis of all measurement of length.[2] [2] Here we have assumed that there is nothing left over _i.e._ that the measurement gives a whole number. This difficulty is got over by the use of divided measuring-rods, the introduction of which does not demand any fundamentally new method.