Note 146 — Introductory material

B. Geometrical Argument. Let us see next what sort of Geometry we could construct without this axiom. The ultimate standard of comparison of spatial magnitudes must, as we saw in introducing the axiom, be equality when superposed; but need we, from this equality, infer equality when separated? It has been urged by Erdmann that, for the more immediate purposes of Geometry, this would be unnecessary[153]. We might construct a new Geometry, he thinks, in which sizes varied with motion on any definite law. Such a view, as I shall show below, involves a logical error as to the nature of magnitude. But before pointing this out, let us discuss the geometrical consequences of assuming its truth. Suppose the length of an infinitesimal arc in some standard position were ds; then in any other position p its length would be ds.f(p), where the form of the function f(p) must be supposed known. But how are we to determine the position p? For this purpose, we require p's coordinates, i.e., some measurement of distance from the origin. But the distance from the origin could only be measured if we assumed our law f(p) to measure it by. For suppose the origin to be O, and Op to be a straight line whose length is required. If we have a measuring rod with which we travel along the line and measure successive infinitesimal arcs, the measuring rod will change its size as we move, so that an arc which appears by the measure to be ds will really be f(s).ds, where s is the previously traversed distance. If, on the other hand, we move our line Op slowly through the origin, and measure each piece as it passes through, our measure, it is true, will not alter, but now we have no means of discovering the law by which any element has changed its length in coming to the origin. Hence, until we assume our function f(p), we have no means of determining p, for we have just seen that distances from the origin can only be estimated by means of the law f(p). It follows that experience can neither prove nor disprove the constancy of shapes throughout motion, since, if shapes were not constant, we should have to assume a law of their variation before measurement became possible, and therefore measurement could not itself reveal that variation to us[154].