[144] Geometrically, the axiom of the plane is, not that three points determine a figure at all, which follows from the axiom of the straight line, but that the straight line joining two casual points of the plane lies wholly in the plane. This axiom requires a projective method of constructing the plane, i.e. of finding all the triads of points which determine the same projective figure as the given triad. The required construction will be obtained if we can find any projective figure determined by three points, and any projective method of reaching other points which determine the same figure.
Let O, P, Q be the three points whose projective relation is required. Then we have given us the three straight lines PQ, QO, OP. Metrically, the relation between these points is made up of the area, and the magnitude of the sides and angles, of the triangle OPQ, just as the relation between two points is distance. But projectively, the figure is unchanged when P and Q travel along OP and OQ, or when OP and OQ turn about O in such a way as still to meet PQ. This is a result of the general principle of projective equivalence enunciated above ([§§ 108], [109]). Hence the projective relation between O, P, Q is the same as that between O, p, q or O, P′, Q′; that is, p, q and P′, Q′ lie in the plane OPQ. In this way, any number of points on the plane may be obtained, and by repeating the construction with fresh triads, every point of the plane can be reached. We have to prove that, when the plane is so constructed, the straight line joining any two points of the plane lies wholly in the plane.
It is evident, from the manner of construction, that any point of PQ, OP, OQ, OP′ or OQ′ lies in the plane. If we can prove that any point of pq lies in the plane, we shall have proved all that is required, since pq may be transformed, by successive repetitions of the same construction, into any straight line joining two points of the plane. But we have seen that the same plane is determined by O, p, q and by O, P, Q. The straight lines PQ, pq have, therefore, the same relation to the plane. But PQ lies wholly in the plane; therefore pq also lies wholly in the plane. Hence our axiom is proved.
[145] A detailed proof has been given above, Chap. I. 3rd period. It is to be observed that any reference to infinitely distant elements involves metrical ideas.
[146] Cf. [Section A, §§ 115–117.]
[147] Contrast Erdmann, op. cit. p. 138.
[148] Cf. Erdmann, op. cit. p. 164.
[149] Strictly speaking, this method is only applicable where the two magnitudes are commensurable. But if we take infinite divisibility rigidly, the units can theoretically be taken so small as to obtain any required degree of approximation. The difficulty is the universal one of applying to continua the essentially discrete conception of number.
[150] Cf. Erdmann, op. cit. p. 50.