In connection with the geometry of cubic curves, the English mathematician James Joseph Sylvester (1814–1897) sought a finite set S of distinct points in the plane, not all in a straight line, possessing the property that the straight line joiniig any two points of S contains at least one more point of S. The property is possessed by some pairs of points in the set S = {A, B, C, …, I) shown in Figure 3.1. (G, H, I are in a straight line by Pappus' theorem). However, there are several other pairs, for example A, D or D, I, such that the lines through them contain no other points of S. Of course, it is conceivable that an unsuccessful attempt to construct such a set of points might be made successful by adding a few strategically placed points. In the attempt of Figure 3.1, the point J, the intersection of AD and GH, might be included in the set in order to correct for the deficiency in the line AD. But, with J added, four new deficient pairs areformed (JB, JC, JE, JF).

Given a finite set of points, not all collinear, let us connect each pair of them by a straight line. If a line contains exactly two points of the set we shall call it an ordinary line.