A PENDF.NTIVF. is a portion of a spheric surface, ter minated On two sides by two vertical planes or straight walls, and a horizontal plane at the top.
A pendentive is therefore bounded on all sides by three circular lines, two of which are in vertical planes, and one horizontal.
Ilence, if an apartment is built upon a square or po lygonal plan, and from the interior faces of the walls, concave surfaces, which are portions of the same sphere, and which have one common centre in the axis of the apartment, spring upwards, and towards the axis, until they terminate in a complete horizontal circle, then as many pendentives will be formed as the walls have an gles or sides.
If the springing lines upon the walls are semicircles, the pendentives will be portions of a hemisphere, having the angular points in a great circle passing horizontally through the centre.
If the springing, lines arc segments less than semi circles, the portion of the sphere from which the pen dentives are formed, will be a segment less than a hemi sphere.
The ichnography or 'plan of a pendentive ceiling is the triangular spaces formed lay a square or polygon equal to that of the plan circumscribing a circle, the radius of tt hich is equal to the radius of a circle which terminates the pendentives.
Pelvic mit CS are often employed in ceilings, in place of cylindric or cylindroidic surfaces, as in simple vaults, or in the «imposition of groins. ('lie whole of the inte rior of St Paul's cathedral is vaulted with cores Of this N% Ilk h support the small cupolas. The beautiful inter ion of St Stephen's, Wallbrook, has its dome sup ported upon eight pendentives over right columns, dis posed in the angles of an octagon. The ceilings or prin cipal passages in large edifices arc sometimes arched • ith pendentives, supporting rows of small cupolas, or flat circular ceilings. The apartment which contains the principal stairs is most elegantly celled in this man ner, whether the plan be square or oblong. If the plan he oblong, sphero-cylindric arches may be made in the length of the plan, by which means the pendentives may rise from the sides of a square. When the height will admit, semi-cylindric vaults frill have a more graceful effect than those, the sections of which arc portions less than a semicircle. In this ease, the pendentives will he bounded by four lertical semicircles, the planes of NVIliC In are at right angles to each other. Two of these
• mieireles tt ill be formed upon the planes of the walls, • the other two by the intersection of the two semi lindric vaults, which will form two sphero-cylindric arches.
Thdugh pendentives are usually portions of a spheric surface, they may be portions of an ellipsoid, or of it re gular polygonal dome ; or or any fig-tire, the horizontal sections of It Inch are circles.
Let A RC 1)A be the plan of a pendentive ceiling, and GI 11KLEFG the section and interior elevation, the sec tion being made by a plane parallel to one of the sides. ('late CX X V. Fig. 9.
The pendentives are formed of ribs, which must. be considered as sections of a hemisphere ; the plan A IIC DA must he considered as a portion of the !lend -,pheric base ; the springing ribs Ail', I3MC, CN I), DOA, arc all semicircles of a diameter equal to the side of the plan, and supposed, in the execution, to be raised peril( ndicular to the plan. The ribs which form the ceiling are all formed of arcs \Odell are portions of great circles, their planes bisecting each other in a vertical axis ; and, as tire diameter of the sphere is the diagonal or the plan, half the diameter will be the radius for all the ribs. The interior circle on the plan is the seat of he Liar. The only thing remaining to show, is the length of the ribs, which are all different.
PRou• XI11. To find the length of the ceiling ribs.
Let it br required to find the length of the rib, the scat of which is a beda: draw Yf c parallel to AD : tr.oin Y, with the distance Y 6, describe an arc b e; and ith the distance Y c, describe an arc r f: on the mid (Ile point Z, of the side AD of the plan, describe the arc NVIliCh gives the curve of all the ribs : the concen tric outer circle may be at any distance, according to the breadth or the rib, so as to make it sufficiently strong : draw g h and e i k parallel to AB, cutting the arc at ,f,' and i, and the outer edge at J.: and k : draw I Q r, 711 n p 0, likewise parallel to AB ; and draw the ( dion of the kil) Q r o n, then will Q h k 11 r; Q be the rib required. tri the same mariner ina) all the others bd. found. The ribs ha ing (Mind for one-eightli part ol tb( ceiling, is sufficient to mould the whole, a shown at No. 1, 2, 3, 4.