Suppose the walls of an art gallery form an n-sided polygon. How many guards are needed and where place them so that the whole gallery is watched by the guards, assuming that these are stationary and can look in all directions? Due to Victor Klee, this question was solved by V. Chvátal [J. Comb. Theory, Ser. B 18, 39-41 (1975; Zbl 0278.05028)] showing that [n/3] guards always suffice, and that this bound is tight. A very short proof of St. Fisk [ibid. 24, 374 (1978; Zbl 0376.05018)] uses a triangularization of the polygon. The graph so obtained may be vertex-3-colored, and the vertices colored in the least frequent color solve the problem. In the present paper this watchman theorem is sharpened for the case of traditional galleries, i.e. where all the walls are parallel to some fixed pair of perpendicular axes (although by linear transformation the orthogonality is irrelevant). For such rectilinear polygonal regions the best bound for the number of guards is reduced to [n/4]. The argument consists in first showing that each such region may be convexly quadrilateralized, i.e. partitioned into convex quadrilaterals by adding nonintersecting lines between vertices. To each such quadrilateral one then adds both diagonals as new edges (without common vertex) yielding a graph which is vertex-4-colorable. Placing a guard on each vertex of this graph which is colored in the least used color solves the problem. The main part of the paper consists of the proof that any rectilinear region may be convexly quadrilateralized. This is done by an induction argument in two steps. First it is shown that any region admitting at least one out of three types of configurations may be reduced to a ”simpler” region. Secondly that any region admitting no such special configurations necessarily has an infinite number of vertices.