When groining springs from a wall, nearly the same system applies, excepting that one division only of the pier is needed instead of all four. Thus the simplest provision is a mere projecting pilaster, carrying the cross ribs, the wall itself taking the place of the lateral ones. This pilaster may be converted into a shaft or a double shaft, or the rib may be amplified by a central semi-roll moulding, and the whole carried by a triple shaft or other combinations, or a corbel substituted for the pilaster or group of shafts. Thus we have vaulting reduced to a principle which, however plain, is at once artistic and constructionally good, and is susceptible of all degrees of ornamentation. What I have said of doors applies equally to windows, subject to some modifications arising from their practical requirements. The simplest form of an arched opening, going square through the wall, is eminently unsuited to a window; and this is so obvious that it has rarely been used at any period, for the square edges of a thick wall evidently prevent the light from diffusing itself in the interior. The most favourable forms are those in which the jambs are sloped, either directly from the exterior inwards, or from some intermediate point, both inwards and outwards, so as to give the freest scope for the rays of light. In this respect I have nothing to say against the forms customary in the previous style.[19] The Norman windows are of great variety. The simplest, which is prevalent in very homely buildings (as may be seen in many extremely humble churches on the cliff between Dover and Deal), is an opening with no external recess, but splaying at some 45 degrees inward, the glass being flush with the exterior. From this we have every variety of architectural grade; first, a chamfer or moulding added to the exterior; then, two orders, plain or moulded; or a shaft may be substituted for the outer order in the jamb, or the same repeated, as in doorways. Internally, the thickness of the walls continues to be splayed so as to diffuse the light, though in buildings of a high architectural class, mouldings or divided orders (with or without shafts) may enrich the inner angle, or may even take the place of the splay altogether. In domestic windows two or more openings are often used externally, divided by a little column, the whole being internally united into a single opening. These are sometimes comprised on the exterior under a single arch to increase the architectural effect.[20] The same is also used for belfries and other positions where use dictates it. I have now shown you that doors, windows, archways, arcades, and vaulting were generated, as to their architectural treatment, simply by the exercise of logical reasoning.