ABSTRACT
The fact is that motion, as the word is used by geometers, has a meaning entirely different from that which it has in daily life, just as a variable, in mathematics, is not something which changes, but is usually, on the contrary, something incapable of change. So it is with motion. Motion is a certain class of one-one relations, each of which has every point of space for its extension, and each of which has a converse also belonging to the class. That is, a motion is a one-one relation, in which the referent and the relatum are both points, and in which every point may appear as referent and again as relatum. A motion is not this only: on the contrary, it has this further characteristic, that the metrical properties of any class of referents are identical with those of the corresponding class of relata. This characteristic, together with the other, defines a motion as used in Geometry, or rather, it defines a motion or a reflexion; but this point need not be elucidated at present. What is clear is, that a motion presupposes the existence, in different parts of space, of figures having the same metrical properties, and cannot be used to define those properties. And it is this sense of the word motion, not the usual material sense, which is relevant to Euclid’s use of superposition.
