Such a study will, however, show a decided difference in the elevation of the transverse ribs and consequent shape of the vaults, which is worthy of some notice. If, for example, a triangle be inscribed beneath a number of these transverse arches, it will be found that the angles inside its base vary from about fifty degrees in Saint Germer-de-Fly, Rouen cathedral and Beverley Minster;[220] to fifty-five degrees in Soissons, Amiens, Salisbury, and Milan cathedrals, and Westminster Abbey; and even to sixty degrees in the cathedrals of Cologne and Reims. Moreover there is a great difference in the curve of these same transverse ribs. Those in Saint Germer, Beverley, and Rouen closely approach a semicircle, those in Amiens and Salisbury are much more pointed, but made up of two arcs without, however, a long radius with the resulting flattened appearance to be noted at Cologne and Reims and more decidedly at Milan. All this would seem to indicate that the elevation of these ribbed vaults,--and this is true of six-part and complex vaults as well,--was largely a matter of individual taste with a tendency to favor the form used at Amiens. The reason for the employment of very sharp curves like those of Reims, Cologne and Milan, was doubtless due to the appreciation on the part of the builders of the fact that such curves greatly reduced the outward thrusts, rather than to any idea of beauty of appearance to be gained, for in this they are perhaps inferior to the less pointed examples. RECTANGULAR FOUR-PART RIBBED VAULTS IN CHURCHES WITHOUT SIDE AISLES [Illustration: FIG. 34.--ALBI, CATHEDRAL.] The use of rectangular four-part ribbed vaulting was not confined to churches with side aisles, but appears also in those with a single broad nave. It is the method employed in the Sainte Chapelle at Paris (fin. 1248), where there are simple salient buttresses, and there is a splendid example in the Cathedral of Albi (Tarn) (begun 1282) (Fig. 34), where the nave has a very wide span and is flanked by chapels in two stories between heavy pier buttresses which are thus enclosed in the church in a truly Byzantine manner. In the smaller church of Saint Nicholas at Toulouse these buttress chapels are in but one story and the bays are more nearly square in plan, a compromise between the square and rectangular systems which appears on an even larger scale in the cathedral of Saint Bertrand-des-Comminges (Haute-Garonne) (cir. 1304). As far as construction is concerned these vaults over a single broad nave offer no advance over those in churches with side aisles, not even requiring a scientific system of flying buttresses to offset their outward thrust. Their only importance lies in the very broad space sometimes covered by them.[221] VAULTING WITH ADDED RIBS