Melchior observed that, for any graph embedded in the real projective plane, the formula formula_34 must equal formula_35, the Euler characteristic of the projective plane. Here formula_36, formula_37, and formula_38 are the number of vertices, edges, and faces of the graph, respectively. Any nontrivial line arrangement on the projective plane defines a graph in which each face is bounded by at least three edges, and each edge bounds two faces; so, double counting gives the additional inequality formula_39. Using this inequality to eliminate formula_38 from the Euler characteristic leads to the inequality formula_41. But if every vertex in the arrangement were the crossing point of three or more lines, then the total number of edges would be at least formula_42, contradicting this inequality. Therefore, some vertices must be the crossing point of only two lines, and as Melchior's more careful analysis shows, at least three ordinary vertices are needed in order to satisfy the inequality formula_41.