+DETAILS.+ German window tracery was best where it most closely followed French patterns, but it tended always towards the faults of mechanical stiffness and of technical display in over-slenderness of shafts and mullions. The windows, especially in the “hall-churches,” were apt to be too narrow for their height. In the fifteenth century ingenuity of geometrical combinations took the place of grace of line, and later the tracery was often tortured into a stone caricature of rustic-work of interlaced and twisted boughs and twigs, represented with all their bark and knots (_branch-tracery_). The execution was far superior to the design. The carving of foliage in capitals, finials, etc., calls for no special mention for its originality or its departure from French types. +PLANS.+ In these there was more variety than in any other part of Europe except Italy. Some churches, like Naumburg, retained the Romanesque system of a second western apse and short choir. The Cistercian churches generally had square east ends, while the polygonal eastern apse without ambulatory is seen in St. Elizabeth at Marburg, the cathedrals of Ratisbon, Ulm and Vienna, and many other churches. The introduction of French ideas in the thirteenth century led to the adoption in a number of cases of the chevet with a single ambulatory and a series of radiating apsidal chapels. +Magdeburg+ cathedral (1208-11) was the first erected on this plan, which was later followed at Altenburg, Cologne, Freiburg, Lübeck, Prague and Zwettl, in St. Francis at Salzburg and some other churches. Side chapels to nave or choir appear in the cathedrals of Lübeck, Munich, Oppenheim, Prague and Zwettl. +Cologne+ +Cathedral+, by far the largest and most magnificent of all, is completely French in plan, uniting in one design the leading characteristics of the most notable French churches (Fig. 141). It has complete double aisles in both nave and choir, three-aisled transepts, radial chevet-chapels and twin western towers. The ambulatory is, however, single, and there are no lateral chapels. A typical German treatment was the eastward termination of the church by polygonal chapels, one in the axis of each aisle, the central one projecting beyond its neighbors. Where there were five aisles, as at Xanten, the effect was particularly fine. The plan of the curious polygonal church of +Our Lady+ (Liebfrauenkirche; 1227-43) built on the site of the ancient circular baptistery at Treves, would seem to have been produced by doubling such an arrangement on either side of the transverse axis (Fig. 142). [Illustration: FIG. 141.--COLOGNE CATHEDRAL. PLAN.] [Illustration: FIG. 142.--CHURCH OF OUR LADY, TREVES.]