532 



CARPENTRY. 



Carpentry, 



CXXV. 



Fig. 1,2. 



Fig. S. 



Construe- In order to curve the face of the front rib to the 

 tive cylindric surface of the wall, divide the quadrant 

 ^ ^ number o f equa l p art8 , as six ; and 

 ordinates to MN. In Fig. 1. through X draw 

 a b parallel to PR, cutting CQ at a . make a b 

 equal to CA or CB, Fig. 2, with the semi-greater 

 axis a b t and semi lesser axis a N, describe the qua- 

 drant of an ellipse b KLMN: transfer the divisions 

 on GN, Fig. 2, to a 6, No. 1, and draw the ordi- 

 nates c d e, i k I, &c. to the quarter axis of the el- 

 lipse, cutting the line of the wall at d, g, k, &c. : 

 Extend the arc DN, Fig. 2, along the straight line 

 Fig. 3, from N to e 1 , from N to fl*, from N to / 3 , 

 &c. : draw the ordinates NG, e*d, h*g, fik, Sue. ; 

 and make them respectively equal to NG, e d, hg, 

 Ik, &c. Through the points G, d, g, k, n, &c. 

 draw a curve, and it will complete the envelope 

 mould, which, bent round the under edge of No. 2, 

 from D to n, and from D to m, and a line being 

 drawn by the curved edge, will give the cylindro- 

 spheric line, or line of concourse of the wall and 

 niche ; and, by cutting away the waste stuff* from 

 the front part, the front rib will be formed. 



To form any of the back ribs represented by FN, 

 EM. DL, CK, all tending to the centre O, suppose 

 the rib upon CK to be formed, its thickness is here 

 represented ; but we shall suppose the centre of the 

 rib to be the base : draw v x and TV y perpendicular 

 to CK, from the points where the thickness termi- 

 nates upon the elliptic curve, till they meet the in- 



j eide of the plate at x and y ; continue CK until it 



meet ab; through the points of meeting, and through 

 y, draw y z ; parallel to which, draw x, 8? ; then will 

 CX y be the inner edge of the rib, Z and z the outer 

 edge, the dotted line the base, y z, x, Sf the two 

 sides of the joint at the top. This same rib is shown 



Pig- 4, 5, 6, at Fig. 4. : in the same manner, Fig. 5, 6, and 7, 



7 - will be found. 



Fig 8. is a bevel, which, being applied with the 

 straight edge of it to the inside of the rib Fig. 2, 

 and the wood cut away towards the front out to no- 

 thing on the under edge, at the same angle all round, 

 and brought to its place, will form to the spherical 

 surface of the niche, and to the cylindric surface 

 of the wall. 



Fig. 8. 



Penden- 

 lives. 



A PENDENTIVE is a portion of a spheric surface, 

 terminated 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. 



Hence, if an apartment is built upon a square or 

 polygonal 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 to- 

 wards the axis, until they terminate in a complete 

 horizontal circle, then as many pendentives will be 

 formed as the walls have angles or sides. 



If the springing lines upon the walls are semicir- 

 cles, the pendentives will be portions of a hemi- 

 sphere, having the angular points in a great circle 

 passing horizontally through the centre. 



If the -pringing lines are segments less than semi- 



circles, the portion of the sphere from which the Construe- 

 pendentives are formed, will be a segment less than tive 

 a hemisphere. 



The ichnography or plan of a pendentive ceiling 

 is the triangular spaces formed by a square or poly- 

 gon equal to that of the plan circumscribing a circle, 

 the radius of which is equal to the radius of a circle 

 which terminates the pendeiitives. 



Pendentives are often employed in ceilings, in place 

 of cylindric or cyhndroidic surfaceb, as in simple 

 vaults, or in the composition of groins. The whole 

 of the interior of St Paul's cathedral is vaulted with 

 coves of this kind, which support the small cupolas. 

 The beautiful interior of St Stephens, Wallbrook, 

 has its dome supported upon eight pendentives over 

 eight columns, disposed in the angles of an octagon. 

 The ceilings of principal passages in large edifices 

 are sometimes arched with pendeiitives, suppotting 

 rows of small cupolas, or flat circular ceilings. The 

 apartment which contains the principal stairs is most 

 elegantly ceiled in this manner, whether the plan be 

 square or oblong. If the plan be 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 will have a more graceful ef- 

 fect than thoae, the sections of which are portions 

 less than a semicircle. In this case, the pendentives 

 will be bounded by four vertical semicircles, the 

 planes of which are at right angles to each other. 

 Two of these semicircles will be formed upon the 

 planes of the walls, and the other two by the inter- 

 section of the two semi-cylindric vaults, which will 

 form two sphero-cylmdric arches. 



Though pendentives are usually portions of a 

 spheric surface, they may be portions of an ellipsoid, 

 or of a regular polygonal dome ; or of any figure, 

 the horizontal sections of which are circles. 



Let ABCDA be the plan of a pendentive ceiling, PLATE 

 and GHIKLEFG the section and interior elevation, CXXV. 

 the section being made by a plane parallel to one of S- 

 the sides. 



The pendentives are formed of ribs, which must 

 be considered as sections of a hemisphere ; the plan 

 ABCDA must be considered as a portion of the he- 

 mispheric base ; the springing ribs APB, BMC, 

 CND, DOA, are all semicircles of a diameter equal 

 to the side of the plan, and supposed, in the execu- 

 tion, to be raised perpendicular to the plan. The 

 ribs which form the ceiling are all formed of arcs 

 which are portions of great circles, their planes bi- 

 secting each other in a vertical axis ; and, as the di- 

 ameter of the sphere is the diagonal of the plan, 

 half the diameter will be the radius for all the ribs. 

 The interior circle on the plan is the seat of the kirb. 

 The only thing remaining to shew, is the length of 

 the ribs, which are all different. 



PROB. XIII. To find the length of the ceiling 

 ribs. 



Let it be required to find the length of the rib, 

 the seat of which is a b c d a : draw Yjf e parallel 

 to AD : from Y, with the distance Y 6, describe an 

 arc b e ; and with the distance Y c, describe an arc 

 cf: on the middle point Z, of the side AD of the 

 plan, describe the arc QR, which gives the curve of 



