The former is, on a _primâ facie_ view, the more natural, but it has the disadvantages of breaking the chief side of the vaulting compartment which rises from the corners into a resalient angle, and also of rendering the main ribs from these angles across to the pillar, in one half of their length _diagonal_ ribs, and in the other _transverse_; and of making one half represent a _projecting_ and the other a _receding_ angle, while the _angle_ ribs of the outer half meet the _transverse_ ribs of the inner half of the vault. These objections are entirely removed by supposing the main vaults to run directly from the angle to the pillar. In either case the ridge which surrounds that half of the vault which springs from the pillar takes the form of an inner octagon. In the first case, the sides of this are parallel to the walls, while in the second they take an intermediate direction; the angles of the inner octagon being opposite the centres of the sides of the outer one, and _vice versâ_. The vaulting compartments which rise from the angles of the great octagon are precisely similar to the opposite ones which rise from the pillar, and the ribs which rise from the angles to the pillar are throughout _transverse_ ribs, while the angle ribs from each side duly meet one another. I exhibit a view of the interior of the Chapter-house, Westminster (Fig. 353), to show the beauty of this form of vaulting. Few forms, in fact, in any style of architecture present such beauties as an octagon vaulted in this manner; and I am happy to think that our London specimen, which has been lost for the last century or more, will now very shortly be restored to its original form and condition. I have already mentioned that in all these forms of vaulting,--that is to say, those with level ridges,--owing to a geometrical error resulting from the use of circular curves for all the ribs, the filling in of the vaulted spaces must be artificially shaped to fit those curves. The use, however, of a form of vaulting analogous to that before described as having _raised_ ridges would obviate this inaccuracy. Suppose, for example, an oblong compartment with pointed arches of similar proportions on all its sides and on its diagonals, and the vaults of each cell generated by the motion of the curve of the surrounding arches towards the point of intersection, guided by the diagonals, we obtain at once a vault with pointed arches and raised ridges, the precise correlative of that before described with round arches and raised ridges, and one in which the filling of the vaulted spaces assumes a systematic and accurately geometrical form (Fig. 354). [Illustration: Fig. 354.] The proportions of the arches are not essentials, but where the narrower arches approach the height of the wider ones, a curious effect is produced on the form of the ridge, which, always elliptical, becomes then so obviously so as to be unpleasing.