Note 144 — Introductory material

We require, then, at the very outset, some criterion of spatial equality: without such a criterion metrical Geometry would become wholly impossible. It might appear, at first sight, as though this need not be an axiom, but might be a mere definition. In part this is true, but not wholly. The part which is merely a definition is given in Euclid's eighth axiom: "Magnitudes which exactly coincide are equal." But this gives a sufficient criterion only when the magnitudes to be compared already occupy the same position. When, as will normally be the case, the two spatial magnitudes are external to one another—as, indeed, must be the case, if they are distinct, and not whole and part—the two magnitudes can only be made to coincide by a motion of one or both of them. In order, therefore, that our definition of spatial magnitude may give unambiguous results, coincidence when superposed, if it can ever occur, must occur always, whatever path be pursued in bringing it about. Hence, if mere motion could alter shapes, our criterion of equality would break down. It follows that the application of the conception of magnitude to figures in space involves the following axiom[150]: Spatial magnitudes can be moved from place to place without distortion; or, as it may be put, Shapes do not in any way depend upon absolute position in space.