[Illustration: Fig. 314.] The first is the prolongation of an arch in a direct line at right angles to its plane (Fig. 313), the second may be conceived as generated by the revolution of an arch upon its vertical axis (Fig. 314). I will keep, for the present, to the development of vaulting from the first of these types. We will first suppose that, while limited by constructive convenience to some moderate span, we have occasion to vault over an area of double that width. The most natural expedient which suggests itself is to divide the space into two widths by an arcade whose top ranges on a level with the springing of the vaulting, and on this and the outer walls to place twin and parallel barrel vaults. This was a system at first largely made use of, as we may see, in some of the covered tanks or piscinæ of the ancients, and in the galleries of the Colosseum. It is clear, however, that this is an imperfect covering for a single room or hall, not only from its severing it too much into two separate areas, but from its placing so much of the covering above the level of side windows, and thus practically reducing the available height of the walls; not to mention its heavy effect. [Illustration: Fig. 315.] Let us see how these imperfections may be obviated. The solution of the question may have arisen from a different and accidental case. Let us suppose two corridors, each covered by a barrel vault, crossing each other at right angles. It is easy to see that these vaults must, by their intersection, generate angles running diagonally from corner to corner of the crossing of the corridors, and that these angles of intersection would assume curves of an elliptical form (Fig. 315). This square of intersection would in fact be found to be vaulted on a system previously unthought of. Let us next suppose _twin_ corridors, severed only by a wall, crossing two other such corridors, all similarly covered by barrel vaults. Instead of the simple intersection of our previous case, we now have a group of four, or _two pairs_ of such intersecting vaults, meeting in the centre on a mere frustum of the partition walls reduced to a square pier, from whose angles spring four of those edges of intersection before described (Fig. 316). [Illustration: Fig. 316.] This, then, contains the solution of the problem under consideration, for, returning to our first case of vaulting a hall of double width, we may, by repeating as many as we may need of these pairs of intersecting or “groined” compartments, such as we have generated by the last process, effect our object in a perfect manner; the vaults being all of equal height, and the two widths being practically united into one, while the walls cease to be stunted of their full height, and room is left in them for windows reaching nearly to their top.