difficulties which they attempted unsuccessfully to solve with the round arch were wholly removed, as we have seen in St. Denis, by the use of the pointed arch and of groin ribs. 
FIG. 55 In addition to what has already been shown concerning the vaults of St. Denis one further illustration of the flexibility of the Gothic system, in the construction of such vaults, is afforded by the Cathedral of Paris.
The form of each vaulting compartment of an apsidal aisle is, of course, trapezoidal; and where two such compartments adjoin each other concentrically, as they do in the double apsidal aisles of Paris (Fig. 55), there results an awkward difference in the lengths of their sides. The simple arrangement employed at St. Denis and elsewhere might still, of course, be employed; but the disparity in the spans of the arches and in the magnitudes of the cells would, on this arrangement, be very great. To obviate this the architect of the apse of Paris adopted another method whereby all the longitudinal arches are rendered of nearly equal span. This result is obtained by dividing the longest side of the compartment A into two parts by the insertion of a pier, a; and the longest side of the compartment B into three parts by the insertion of two piers, b and c. There are no intersections of the arches, and consequently no common keys in these compartments, each line in the plan representing a complete arch. [1]
In regard to chapels the Gothic apse in France differs considerably in different buildings according as the ecclesiastical, or the communal, influences predominated in the foundation. In the one case chapels are general and
- ↑ For full particulars regarding the mode, at once simple and ingenious, of filling in these triangular spaces, see Viollet-le-Duc, s.v. Voute, p. 512.
