The next form, perhaps, in point of simplicity is an equal-sided polygon,--say, for example, an octagon (Fig. 322). We must here suppose eight cylindrical vaults crossing one another from the opposite sides of the octagon; and it is clear that their intersecting lines will be the diagonals or lines joining the opposite angles of the octagon, which will coincide in position with the transverse ribs. The objection to this form of vaulting is the low proportion of the arches produced by these intersections, which, though more than twice and a half the width of the side arches, only rise to the same height, or about one-fifth of their span,--a defect which will be remedied by a development I shall presently have to describe.[40] Just as the half-dome (as seen in the chapel of the Tower of London)[41] forms a natural covering for an apsidal termination of a barrel vault, so a portion of a polygon, thus vaulted, would _appear_ to be the correlative apsidal termination of a _groined_ vault.[42] A difficulty, however, at once presents itself in the small height of the vault last described, which is not one-half of the height of the semicircular vault which it would have to meet. How, then, is this to be got over? How are the vaults proceeding from the narrow arches of the sides of the octagon to be brought to range in height with the wide vault which spans the whole space (Figs. 323 and 326)? [Illustration: Fig. 322.] [Illustration: Fig. 323.] [Illustration: 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.) [Illustration: Fig. 325.]