Another method of ambulatory vaulting in the Gothic period consisted in the subdivision of the apsidal aisles into triangles by adding intermediate supports between each pier forming the outside corners of trapezoidal bays. This method, never had a wide popularity. It was used at a comparatively early date and on a large scale in the cathedral of Notre Dame at Paris (begun 1163) (Plate IV-c), where the triangular bays have no ribs beneath their masonry. It appears with the addition of three half ribs or even a still greater number, in a number of late Gothic churches, especially in Germany,[445] and was also used at Saint Eustache (1532-1637)[446] and Saint Severin[447] in Paris, whose builders may very probably have been influenced by the cathedral church of Notre Dame. In Notre Dame, where there are two ambulatories the doubling of the piers did not do away with the possibility of a central eastern chapel or window in the exterior wall. But in most cases, where there is but one aisle, as, for example, in the Marienkirche at Stargarde (end of fourteenth century) (Plate IV-d) or the old cathedral of Heidelberg,[448] an axial pier prevents this arrangement. Perhaps to avoid this the builders of Saint Steven at Nymwegen and of the cathedral at Brandenburg left the eastern bay trapezoidal so that there might be a central Lady chapel. At Kolin[449] where there is an axial pier in the apse a central chapel off the ambulatory naturally follows. AMBULATORIES WITH MULTIPLE RIBBED VAULTS As has been noted, the late Gothic passion for multiple ribs affected the ambulatory as it did the remainder of the church, and vaults of most complex character are to be found especially in certain German churches. Of these, Güben (Plate IV-e) and the cathedral of Freiburg (second half of the fifteenth century) (Fig. 88) are among the most elaborate.[450] In them, the structural purpose of the rib is totally subordinated to decorative principles and to a desire on the part of the builders to show their knowledge of the intricate problems of stereotomy. With such vaults as these, marking the decline of Gothic architecture, it is not surprising that there was such a complete reaction in vault construction on the part of the succeeding Renaissance builders. [Illustration: FIG. 88.--FREIBURG, CATHEDRAL.] With this discussion of the ambulatory, the study of mediaeval church vaulting is practically complete, but a few paragraphs should be added to give a short account of some unusual eastern terminations and a brief reference to the radiating chapels. Both of these, while presenting no great structural accomplishments, at least show the skill of the builders in meeting any and all requirements imposed by the plan. EXCEPTIONAL EASTERN TERMINATIONS