[Illustration: Fig. 332.--View of Crypt, Worcester Cathedral.] The same problem, when applied to a polygon instead of a circle, is open to two different modes of solution. In the one, the main vault is always supposed to run from each _side_ towards the central pillar; in the other, from each _angle_ towards the pillar. I shall, however, have to go more minutely into this when I come to pointed-arch vaulting, to which the last-named system more especially applies.[44] Having now briefly touched upon the most prominent forms of round-arched vaulting in its more normal form, as resulting from the _barrel_ vault and its intersections, I will digress for a short time to consider some of the conditions which relate to what I in my last lecture stated to be the other most simple kind of vault--the _dome_. I do so, however, not with any idea of treating at large on a form which should be made the subject of a separate lecture, but merely to facilitate the explanation of certain indirect influences which it exercised upon ordinary vaulting. A dome in its most typical form stands upon a circular wall; this, however, is by no means a necessary condition. It may in reality cover a square or polygonal space just as well; for, suppose a square or a polygon inscribed within the base of a hemisphere, it is clear, from the properties of a sphere, that vertical planes erected on the sides of such square or polygon will cut the hemisphere in semicircles of the diameter of those sides (Fig. 333). It follows, therefore, that the walls of a square or polygonal building would intersect with a dome in the form of semicircular arches standing on each of its sides; and, consequently, that such a square or polygon will carry a hemispherical dome, or rather the remainder of it left after cutting the base into a square or polygon. [Illustration: Fig. 333.] For our immediate purpose we will limit the case to that in which the inscribed figure is a _square_. [Illustration: Fig. 334.] [Illustration: Fig. 335.]