49.  Fundamental theorem.  Postulate of continuity. We have thus shown that two projective point-rows, superposed one on the other, may have two points, one point, or no point at all corresponding to themselves.  We proceed to show that

If two projective point-rows, superposed upon the same straight line, have more than two self-corresponding points, they must have an infinite number, and every point corresponds to itself; that is, the two point-rows are not essentially distinct.

If three points, A, B, and C, are self-corresponding, then the harmonic conjugate D of B with respect to A and C must also correspond to itself.  For four harmonic points must always correspond to four harmonic points.  In the same way the harmonic conjugate of D with respect to B and C must correspond to itself.  Combining new points with old in this way, we may obtain as many self-corresponding points as we wish.  We show further that every point on the line is the limiting point of a finite or infinite sequence of self-corresponding points.  Thus, let a point P lie between A and B.  Construct now D, the fourth harmonic of C with respect to A and B. D may coincide with P, in which case the sequence is closed; otherwise P lies in the stretch AD or in the stretch DB.  If it lies in the stretch DB, construct the fourth harmonic of C with respect to D and B.  This point D’ may coincide with P, in which case, as before, the sequence is closed.  If P lies in the stretch DD’, we construct the fourth harmonic of C with respect to DD’, etc.  In each step the region in which P lies is diminished, and the process may be continued until two self-corresponding points are obtained on either side of P, and at distances from it arbitrarily small.

We now assume, explicitly, the fundamental postulate that the correspondence is continuous, that is, that the distance between two points in one point-row may be made arbitrarily small by sufficiently diminishing the distance between the corresponding points in the other. Suppose now that P is not a self-corresponding point, but corresponds to a point P’ at a fixed distance d from P.  As noted above, we can find self-corresponding points arbitrarily close to P, and it appears, then, that we can take a point D as close to P as we wish, and yet the distance between the corresponding points D’ and P’ approaches d as a limit, and not zero, which contradicts the postulate of continuity.

50. It follows also that two projective pencils which have the same center may have no more than two self-corresponding rays, unless the pencils are identical.  For if we cut across them by a line, we obtain two projective point-rows superposed on the same straight line, which may have no more than two self-corresponding points.  The same considerations apply to two projective axial pencils which have the same axis.