The domes which we have hitherto considered are exclusively and of necessity carried by circular or other continuous walls. They are consequently supported uniformly throughout their entire circumference, and their use is necessarily limited to the coverings of circular or _quasi_-circular or polygonal buildings. Had no further development been attained, it would ever have been felt to be a sad deficiency in the scope of architectural facilities that the noblest form of covering should be limited to the least usual and, for most purposes, the least convenient form of apartment. We are happily _as far as possible_ from being left in this dilemma. A very simple application of geometrical thought opened a way by which almost _any_ reasonable form of building may be covered by a dome, or by a series or group of domes. I will endeavour, as simply as I am able, to explain this important development. It is a property of the sphere that every possible plane section of it is a _circle_. It follows that every vertical section of a hemispherical or segmental dome assumes the form of a semicircular or segmental arch. If, therefore, a square be inscribed in the base of a dome, and walls be built on that square, and continued up till they meet the dome, they will intersect with it in four semicircles (Fig. 406). If, instead of _walls_, you build _arches_ on the sides of that square, these arches will coincide with the curve of the dome where they meet it, and, if strong enough, will carry the portion of the dome remaining between them. If, again, instead of arches, you suppose the dome intersected on the lines of the inscribed square by _vaults_ at right angles to those sides, the result will be the same. [Illustration: Fig. 406.] In the first case we have a dome, or a portion of one, covering a square apartment; in the second we have the same covering standing on arches open towards the exterior; in the third, we have a dome covering the intersection of two barrel-vaults, just as is more usually done by groining. The process, however, is not limited to a _square_; it is equally applicable to the octagon or any other polygon--indeed, to any figure which can be inscribed in a circle. The following diagrams (Figs. 407, 408, and 409) will tend better to explain this. Nor is it necessary that the inscribed figure should be _complete_, for remnants of the circle may equally well be left between the arches or walls. [Illustration: Fig. 407.] [Illustration: Fig. 408.] Thus, a circular space may be intersected by four vaults of less width than the sides of a square (Fig. 410), leaving portions of the circular walls remaining between them. The dome, again, may as well be _segmental_ in section as _semicircular_, in which case the arches supporting it will also be segmental (Fig. 411). Again, the figure inscribed need not be _equilateral_, so that _oblong_ compartments, such as those customary in the nave of a church, may be domically vaulted.