Let us outline briefly how perhaps the basis of Euclidean geom etry may be gained from the concepts of distance. We start from the equality of distances (axiom of the equality of distances). Suppose that of two unequal distances one is always greater than the other. The same axioms are to hold for the inequality of dis tances as hold for the inequality of numbers. Three distances AB', BC', CA' may, if CA' be suitably chosen, have their marks BB', CC', AA' superposed on one another in such a way that a triangle ABC results. The distance CA' has an upper limit for which this construction is still just possible. The points A, (BB') and C then lie in a "straight line" (definition). This leads to the concepts: producing a distance by an amount equal to itself ; dividing a distance into equal parts ; expressing a distance in terms of a number by means of a measuring-rod (definition of the space-interval between two points).
When the concept of the interval between two points or the length of a distance has been gained in this way we require only the following axiom (Pythagoras' theorem) in order to arrive at Euclidean geometry analytically. To every point of space (body of reference) three numbers (co-ordinates) x, y, z may be as signed—and conversely—in such a way that for each pair of points A (xi, and B (x2, y2, the theorem holds: measure-number AB= (x2— + (Y2— • All further concepts and propositions of Euclidean geometry can then be built up purely logically on this basis, in particular also the propositions about the straight line and the plane. These remarks are not, of course, intended to replace the strictly axiomatic construction of Euclidean geometry. We merely wish to indicate plausibly how all conceptions of geometry may be traced back to that of distance. We might equally well have epitomised the whole basis of Euclidean geometry in the last theorem above. The relation to the foundations of experience would then be furnished by means of a supplementary theorem.
The co-ordinate may and must be chosen so that two pairs of points separated by equal intervals, as calculated by the help of Pythagoras' theorem, may be made to coincide with one and the same suitably chosen distance (on a solid). The concepts and propositions of Euclidean geometry may be derived from Pythagoras' proposition without the introduction of rigid bodies; but these concepts and propositions would not then have contents that could be tested. They are not "true" propositions but only logically correct propositions of purely formal content.
There are no absolutely definite marks and, moreover, tempera ture, pressure and other circumstances modify the laws relating to position. It is also to be recollected that the structural con stituents of matter (such as atom and electron, q.v.) assumed by physics are not in principle commensurate with rigid bodies, but that nevertheless the concepts of geometry are applied to them and to their parts. For this reason consistent thinkers have been disinclined to allow real contents of facts (reale Tatsachenbe stdnde) to correspond to geometry alone. They considered it pref erable to allow the content of experience (Erfahrungsbestande) to correspond to geometry and physics conjointly.
This view is certainly less open to attack than the one repre sented above; as opposed to the atomic theory it is the only one that can be consistently carried through. Nevertheless it would not be advisable to give up the first view, from which geometry derives its origin. This connection is essentially founded on the belief that the ideal rigid body is an abstraction that is well rooted in the laws of nature.
Foundations of Geometry.—We come now to the question: what is a priori certain or necessary, respectively in geometry (doctrine of space) or its foundations? Formerly we thought everything; nowadays we think—nothing. Already the distance concept is logically arbitrary; there need be no things that corre spond to it, even approximately. Something similar may be said of the concepts straight line, plane, of three-dimensionality and of the validity of Pythagoras' theorem. Even the continuum doctrine is in no wise given with the nature of human thought, so that from the epistemological point of view no greater authority attaches to the purely topological relations than to the others.
Earlier Physical Concepts.—We have yet to deal with those modifications in the space-concept which have accompanied the advent of the theory of relativity. For this purpose we must consider the space-concept of the earlier physics from a point of view different from that above. If we apply the theorem of Pythagoras to infinitely near points, it reads where ds denotes the measurable interval between them. For an empirically-given ds the co-ordinate system is not yet fully determined for every combination of points by this equation. Besides being translated, a co-ordinate system may also be rotated. This signifies analytically : the relations of Euclidean geometry are covariant with respect to linear orthogonal trans formations of the co-ordinates.