Such a union is conceivable, only if, by some device, the different figures can, at least at some point, be cut to the same shape. The problem to be solved may therefore be stated as the question, whether the summit of a square structure can be converted into a circle corresponding to the rim of the dome it is to support. In SS. Sergius and Bacchus we have the type of a building in which a step was taken in that direction. There, as we have seen, the base upon which the dome rests is formed by a substructure consisting of eight arches arranged in the figure of an octagon; the gaps at the angles, where arch bends away from arch, being filled with masonry to the level of the heads of the arches. Such a base, it is true, does not match the dome as accurately as a round substructure like the Pantheon. Still, an octagon approximates to a circle more closely than a square does. Its contour offers more points of contact to the orbed canopy set upon it, and the gaps at its angles, being comparatively small, can readily be filled up to afford the dome a continuous support. If the fit is not perfect, it is sufficient to secure a decided advance beyond the simpler art, which knew only how to put one round thing upon another round thing. And what beautiful results could be gained by this advance, SS. Sergius and Bacchus in Constantinople and San Vitale at Ravenna are there to prove. But the end was not thus reached. Circular buildings and octagonal buildings are exceedingly beautiful; they should always stay with us. But they are not the most convenient, and cannot become the buildings in general use. For the practical purposes of life, square or oblong halls are in greater demand; such as the basilica, which could serve as a court of justice, a church, a school, a market, or a throne-room. Hence the question still remained, Can the summit of a square structure be turned into a circular base for a dome?