Although the trapezoidal bay and its variants has been the only one thus far considered in the discussion of ribbed vaulted ambulatories, it was not by any means universally employed. The alternation of square and triangular bays, which had been used as early as the Carolingian period in the royal chapel at Aachen, and in the tenth century at Verona, in groined vaulted ambulatories, also played a considerable rôle after the use of ribs became general. This system afforded a number of structural advantages, the chief one being, of course, that the major bays were square or nearly so, and therefore presented no structural problems not already solved in other portions of the church, while the triangular divisions were of comparatively small size and could be covered in the same manner as in the Romanesque period, with three-part groined vaults, provided the builders wished to avoid attempting ribbed vaults over them. Two general plans are noticeable in the use of this alternate ambulatory system. In the first, which appears at an early date in Saint Martin of Étampes (1165), Saint Remi at Reims (1170-1181), and Notre Dame [Illustration: FIG. 87.--REIMS, SAINT REMI.] at Chalons-sur-Marne (end of twelfth century), the square bays alternate with two triangular bays or, in other words, the ambulatory is first divided into trapezoids by transverse arches and these in turn subdivided into a square and two triangles. This system may be understood from the plan of Saint Remi (Plate III-l) and the interior view of the same church (Fig. 87). Its most noticeable feature is the lack of ribs in the triangular bays, these remaining of simple Romanesque groined type. Exactly the same arrangement appears at Chalons-sur-Marne, except that here the arches into the apse correspond to the flat sides of a polygon, while those opening into the chapels are on a curve in order that the exterior wall of the triforium above them may be a semicircle.[442] In both these churches, the radiating chapels occupy all the space included beneath each group of three outer arches in a manner similar to that described in connection with the cathedral of Auxerre, but in Saint Martin at Étampes, the chapel is limited in width to the span of the central arch, making possible a window in the exterior wall of each of the triangular bays. A very similar arrangement appears in the outer ambulatory of Bourges cathedral (cir. 1195-1215) (Plate IV-a), except that here the chapels are so narrow as to give a reversed trapezoidal character to what would otherwise be a square bay like that at Étampes and Saint Remi. Even though the triangular severies are thus increased in size, the builders have left their vaults unribbed.