36. Definition of projectivity. Two fundamental forms are protectively related to each other when a one-to-one correspondence exists between the elements of the two and when four harmonic elements of one correspond to four harmonic elements of the other.
Fig. 6
37. Correspondence between harmonic conjugates. Given four harmonic points, A, B, C, D; if we fix A and C, then B and D vary together in a way that should be thoroughly understood. To get a clear conception of their relative motion we may fix the points L and M of the quadrangle K, L, M, N (Fig. 6). Then, as B describes the point-row AC, the point N describes the point-row [pg 22] AM perspective to it. Projecting N again from C, we get a point-row K on AL perspective to the point-row N and thus projective to the point-row B. Project the point-row K from M and we get a point-row D on AC again, which is projective to the point-row B. For every point B we have thus one and only one point D, and conversely. In other words, we have set up a one-to-one correspondence between the points of a single point-row, which is also a projective correspondence because four harmonic points B correspond to four harmonic points D. We may note also that the correspondence is here characterized by a feature which does not always appear in projective correspondences: namely, the same process that carries one from B to D will carry one back from D to B again. This special property will receive further study in the chapter on Involution.
38. It is seen that as B approaches A, D also approaches A. As B moves from A toward C, D moves from A in the opposite direction, passing through the point at infinity on the line AC, and returns on the other side to meet B at C again. In other words, as B traverses AC, D traverses the rest of the line from A to C through infinity. In all positions of B, except at A or C, B and D are separated from each other by A and C.
39. Harmonic conjugate of the point at infinity. It is natural to inquire what position of B corresponds to the infinitely distant position of D. We have proved (§ 27) that the particular quadrangle K, L, M, N employed is of no consequence. We shall therefore avail ourselves of one that lends itself most readily to the solution of the problem. We choose the point L so that the triangle ALC is isosceles (Fig. 7). Since D is supposed to be at infinity, the line KM is parallel to AC. Therefore the triangles KAC and MAC are equal, and the triangle ANC is also isosceles. The triangles CNL and ANL are therefore equal, and the line LB bisects the angle ALC. B is therefore the middle point of AC, and we have the theorem