I have defined the barrel vault as the prolongation of an arch in a direct line at right angles to its plane.[45] But an arch may be prolonged in other than a straight line. Let us, in the previous figure, suppose the arches which rise from the sides of the square to be prolonged, not horizontally, but in a curve rising as it proceeds, and so regulated that the semicircle as it moves forward retains its vertical position, and is guided in its motion by the diagonal lines drawn in the dome. This process at once generates a new form of vault (Fig. 337). For each of the triangular gores of the dome we now substitute a vault, of which every vertical section parallel to the side of the square is a portion of a circle of the same diameter with those raised on the sides, while the angles of the intersection of these newly generated vaults are themselves semicircles. It is a perfectly accurate geometrical figure, none of whose salient lines are other than portions of circles, though the ridge or crown lines now become elliptical. It is a most useful development, as being much stronger than the ordinary groined vault. Oddly enough, it has--so far as I am aware--no suitable name. It is usual to speak of such vaults as being “domed up,” but this is a very rough description. When adapted to the pointed arch, it has been called by Mr. Petit the Angevine vault. I know no better way of describing it than as round-arched vaulting with a raised ridge (Fig. 338). [Illustration: Fig. 337.] [Illustration: Fig. 338.] Now, though less obvious at first sight, the very same processes are applicable either to an oblong, to a tapering four-sided figure, such as the bay of the aisle of an apse, or even to one of the triangular compartments of the apse itself, or of a circle. For, in either case, we have only to cut out the required slice from a hemispherical dome, to draw the diagonal lines from the angles of such form to the apex, and then to substitute for the gores of the dome the vault generated by the motion of the semicircle, produced by the plane of the sides of the figure parallel to itself, and rising under the guidance of the diagonal lines (Fig. 339). This process, it will at once be seen, is capable of solving all the problems of irregular figures which I have enumerated at an earlier stage in my lecture, without the aid of stilting, and without giving intersecting curves, which deviate from the vertical plane, while it avoids the use of the ellipse for any prominent line (Fig. 340). [Illustration: Fig. 339.] [Illustration: Fig. 340.] [Illustration: Fig. 341.]