The _square_ dome is, in fact, the exact correlation or complement to the square groined vault. Like it, it is generated by the intersection of two barrel vaults of the diameter of the sides of the square; but the parts of such vaults which are retained in the one, are precisely those which are omitted in the other. The angular lines are the same, though in the one case the angles _project_, and in the other they _recede_; and while the groined vault is reduced in its bearing to four points in the corners, the square dome demands for its support the whole line of the walls, which, however, it reduces in height to the level of the springing line; while the other allows them to rise in their centres to the full height of the vault. In some cases, as in the vaulting beneath the tower of Grantham Church, “Welsh” groins are united with the polygonal dome, a form quite applicable to the vaulting of an apse (Fig. 346). [Illustration: Fig. 346.] There is another peculiar feature in the square or other straight-sided dome, viz., that it may be cut by vertical planes, as is the case with the spherical dome. Thus, if we inscribe within the base of a square or triangular dome, another square or triangle whose corners bisect the sides of the original base, and erect upon the sides of this newly-formed figure vertical planes, these will intersect the dome in arched forms, and the parts left will give a new form of vaulting, rising from the angles of the figure, and terminating in an unaltered portion of the original dome. This form was not unfrequently used, especially in vaulting triangular spaces (Figs. 347, 348). [Illustration: Figs. 347, 348.] I have now gone through all the normal varieties (of which I can think) of the round-arched vault, and it is time that I should allude to a great step which, after perhaps the first quarter of the twelfth century was introduced into their construction. I allude to the addition of a moulded rib beneath their intersecting angles. It is clear that these angular lines are both in reality and in effect, the weak points of plain groining. I have mentioned that the Romans fortified them in construction, by using in them stronger material than in the rest of the vault; and the early Norman builders made a feeble attempt to take off from the dulness of the intersection, where it approaches its apex, by artificially sharpening the edge in plastering it; for, without this, it becomes (in a vault where the courses of stone or brick are concealed) almost invisible. The great step in advance which I have now to mention provided both the constructive and the artistic strength which the line required.