Fig. 323.
Fig. 324.
The solution of this difficulty will be better considered by means of a simple and more familiar case. The intersecting vault in its most normal form is plain enough in its application to a square compartment, but becomes difficult when applied to a space longer one way than the other; yet oblong spaces continually present themselves as requiring to be vaulted.
Mathematically this is readily met, and that with perfect accuracy, by making one of the intersecting vaults elliptical instead of circular in its curvature; making, for instance, the narrower arch a semi-ellipse with its longer semi-diameter vertical. This, however, is an unsightly form, and was always rejected, though the natural mode of effecting the object, and though it would give intersecting curves which would be complete and in vertical planes.
The Roman builders solved the problem at the sacrifice of mathematical accuracy, by what is called stilting the narrower arch; that is, raising its springing till its crown becomes level with that of the wider arch. This is a practical solution of the difficulty, but is not a very pleasing one, inasmuch as the line of intersection is most uncouthly twisted, and, in point of fact, begins at considerable height above the springing of the vault (Figs. [324], [325].)
Fig. 325.
To go back, however, to our previous case of the apsidal termination of a vaulted space, it affords a very fair solution of the difficulty by which we were before encountered; for it is clear that the arches on the sides of the octagon may be lifted up till their crowns become level with that of the main vault; and, as the intersecting angles of a polygonal groined vault coincide with its transverse ribs, we have nothing to do but to raise from every angle a transverse rib similar, or very nearly similar, to those of the main vault, and to make the smaller vaults of the octagon to intersect upon them ([Fig. 326]). There will be a little geometrical inaccuracy in the forms of these cells of vaulting; but, as the angle ribs would assume correct lines, these inaccuracies would not seriously offend the eye.