The plans of apartments intended to be covered with cupolas, arc, in general, either square or octangular. The pendentives are likewise commonly equal in num ber to the angles of the walls ; but this is not essential, because, in polygonal plans, arches may be thrown across the angles, to double the number of the sides of the po lygon, still preserving the equal sides. Over the middle of the walls, equal and similar arches may be built, that shall touch those across the angles at the bottom, and have their tops in the same level : Or, instead of walls, piers may be carried to an adequate height upon each angle of the polygon, and return upon either side of it.
Archivolts may then be turned over every two adjacent piers, and the spandrels be filled in to the level of the summits of the arches or archivolts, as before, and the termination will he a circle on the inside, as already stated.
There do not appear any instances among the Roman buildings, of pcndentives or spandrels being supported by four pillars, or by quadrangular or polygonal walls, and which support themselves on a spherical dome or a cylindrical wall. Pendentives rising from pillars, and surmounted with a dome, were originally introduced in the celebrated church of St Sophia at Constantinople. St Paul's, and St Stephen's, Walbrook, London, are beautiful specimens of this sort.
When two or more plain vaults penetrate or intersect each other, with their summits in the same horizontal plane or level, the figure of the intrados, formed by the several branches of the vaults, is called a groin: In other words, a groin is a vault in which two geometrical so lids may be transversely applied, one after another, so that a portion of the groin will have been in contact with the first solid, and the remainder with the second when the first is removed, and that the summit of the one may intersect that of the other. This definition will be found almost universal, as it applies not only to plain vaults intersecting each other, but also to such as are annular, or in the form of semi-cylindric rings, intersected by cy lindric or cylindroidal plain vaults, whose axes tend to that of the annuli ; but it does not include the species used in the chapel of Henry VII. at Westminster, and in King's College chapel at Cambridge, where, instead of the horizontal sections of the curved surfaces presenting exterior right angles, as is generally the case, they pre sent convex arches of circles.
A property common to every kind of groins, is, that the several branches intersect and form arches of equal height upon each inclosing wall, the perpendicular sur face of which is continued on both sides, till intercepted by the intrados of the arches ; consequently the upright of each wall is equal in height to that of the common apex of the arches. This forms a striking and charac teristic difference between domes and groins : the latter is a branched vault, terminating in each branch against the inclosing wall ; whereas the former is a vault without branches, having its curves springing from all points of the wall or walls around the bottom of its circumference, whether upon a polygonal, circular, or elliptical plan. Another property of the dome is, that all its horizontal sections arc similar figures, whether made by the exte rior or interior surface.
Groins are variously denominated, according to the surfaces of the geometrical bodies which form the simple vault, viz.
When the axes of the simple vaults arc in two vertical planes, crossing at right angles to each other, they form a rectangular groin.
When three or more simple vaults of one common height pierce each other, and form a complex vault, in such a manner that if the surfaces of the several solids of which each is formed were respectively applied, one at a time, to succeeding portions of the surface of the complex vault, each portion of the complex vault would come in contact with certain corresponding portions of the surface of each of the solids : the complex surface thus generated is called a multangular groin.
When the several axes of the simple vaults form equal angles around the same point, and when each of the vaults arc of the same width, the surface of the solid is called an equiangular groin.
When the breadths of the cross vaults, or openings of a groined vault arc equal, the groin is said to be equi lateral.
The species of every groin, formed by the intersec tion of two vaults of unequal width, is denoted by two preceding words; the first ending in o, indicates the simple vault of the greater width, and the second ter minating with ic, denotes the simple vault of the less width.