In the above outline of the history of vaulting I have, for the sake of simplicity, omitted two modes actually adopted to avoid the difficulty of oblong groining over naves. The first, which was common in German round-arched churches, was to make the vaulting of the nave simply to comprise _two_ bays of the aisles, thus bringing the main vault equally into squares with those of the aisles. The second was the use of what Dr. Whewell has entitled _sexpartite_ vaulting, and which is common both in France, Germany, and England (Fig. 12). It adopts the system last named, but subdivides the double bay by a triangular slip of vaulting (Figs. 13, 14). The real solution arose, however, from the free and simple use of the pointed arch, which gave the result which is seen at Westminster[7] and in nearly all the vaulted churches of the thirteenth century--the simple groined vault with arches of equal height, though the side arches are sometimes stilted, not from necessity, but merely to afford greater space for clerestory windows. [Illustration: Fig. 13.] [Illustration: Fig. 14.] On this plain and practical result M. Viollet le Duc (of whose admirable essay on vaulting, together with those of Dr. Whewell and Professor Willis, I have made free use) makes the following striking remarks:-- “It had required fifty years for the architects of the end of the twelfth century to arrive, from the still Romanesque vaults of Autun and Vezelay, at this great result; and from this moment the entire construction of religious edifices was derived from the disposition of the vaults; the form and dimension of the pillars--their spacing; the window-openings--their width and height; the position and direction of the buttresses--the importance of their pinnacler; the strength, the number, and curvature of the arched buttresses; the disposing and the carrying off the rain water; the system of covering,--all proceed from the combination of the vaulting. The vaults govern the _ossature_ of the monument to a point to which it would be impossible to raise it otherwise than by commencing rigorously to plan them previously to laying the first courses of the structure. This rule is so well established that if we see a church of the thirteenth century destroyed to the level of the bases, and of which the plan alone remains, we can with certainty trace the plan of the vaults, and indicate the direction of all the arches and their thickness. At the end of the fourteenth century the rigour of the system is still more absolute; we can trace, in examining the base of an edifice, not only the number and direction of the arches of the vaults, and know their strength, but the number of their mouldings and even their profiles. In the fifteenth century it is the arches (mouldings) themselves which descend to the floor, and the pillars are only vertical _fasces_ formed of all the members of these arches. After this, we demand how is it that serious men have been able