125.  Desargues’s theorem concerning conics through four points. Let DD’ be any pair of points in the involution determined as above, and consider the conic passing through the five points K, L, M, N, D.  We shall use Pascal’s theorem to show that this conic also passes through D’.  The point D’ is determined as follows:  Fix L and M as before (Fig. 34) and join D to L, giving on MN the point N’.  Join N’ to B, giving on LK the point K’.  Then MK’ determines the point D’ on the line AA’, given by the complete quadrangle K’, L, M, N’.  Consider the following six points, numbering them in order:  D = 1, D’ = 2, M = 3, N = 4, K = 5, and L = 6.  We have the following intersections:  B = (12-45), K’ = (23-56), N’ = (34-61); and since by construction B, N, and K’ are on a straight line, it follows from the converse of Pascal’s theorem, which is easily established, that the six points are on a conic.  We have, then, the beautiful theorem due to Desargues: 

The system of conics through four points meets any line in the plane in pairs of points in involution.

126. It appears also that the six points in involution determined by the quadrangle through the four fixed points belong also to the same involution with the points cut out by the system of conics, as indeed we might infer from the fact that the three pairs of opposite sides of the quadrangle may be considered as degenerate conics of the system.

127.  Conics through four points touching a given line. It is further evident that the involution determined on a line by the system of conics will have a double-point where a conic of the system is tangent to the line.  We may therefore infer the theorem

Through four fixed points in the plane two conics or none may be drawn tangent to any given line.

[Figure 35]

FIG. 35

128.  Double correspondence. We have seen that corresponding points in an involution form two projective point-rows superposed on the same straight line.  Two projective point-rows superposed on the same straight line are, however, not necessarily in involution, as a simple example will show.  Take two lines, a and a’, which both revolve about a fixed point S and which always make the same angle with each other (Fig. 35).  These lines cut out on any line in the plane which does not pass through S two projective point-rows, which are not, however, in involution unless the angle between the lines is a right angles.  For a point P may correspond to a point P’, which in turn will correspond to some other point than P.  The peculiarity of point-rows in involution is that any point will correspond to the same point, in whichever point-row it is considered as belonging.  In this case, if a point P corresponds to a point P’, then the point P’ corresponds back again to the point P.  The points P and P’ are then said to correspond doubly.  This notion is worthy of further study.