2 e equal 23, and 1 f equal to 1, 2, &c. and trace the curve through these points, and it will give the angle-rib.

The method of covering spherical domes is, to suppose them polygonal, and the principle the same as the foregoing operation for an octangular dome.

A niche, in carpentry, is the wood-work to be lathed over for plastering. The general construction of niches is with cylindrical backs and spherical heads, called cylindro-spheric niches; the execution of which depends upon the principles of spheric sections.

As every section in a sphere is a circle, and that section passing through its centre is equal, and the greatest that can be formed by cutting the sphere; it is evident, that if the head of a niche is intended to form a spherical surface, the ribs may be all formed by one mould, whose curvature must be equal to that of the greatest circle of the sphere; viz. one passing through its centre; but the same spherical surface may, though not so eligible, be formed by ribs of wood, moulded from the sections of lesser circles, in a variety of ways.

The reason why these latter spherical surfaces are not so eligible as those of greater circles is, because their disposition for sustaining the lath is not so good, and the trouble of moulding them to different circles, and of forming the edges according to different bevels, in order to range them in the spherical surface, is very great, compared with those Lade from great circles.

The disposition of the ribs of niches is generally in a vertical plane, parallel to each other, or intersecting each other in a vertical line. When the line of intersection passes through the centre of a sphere, all the ribs are great circles; but if the line of intersection does not pass through the centre of the sphere, the circles which form the spherical surface are all of different radii. When the ribs are fixed in parallel vertical planes, their disposition is either parallel to the face of the wall, or parallel to a vertical plane, passing through the centre of the sphere, perpendicular to the surface of the wall; but this method is not so eligible for the purposes of lathing.

Another method is, by making the planes of the ribs parallel to the horizon: this is not only attended with great labour in workmanship, but is incommodious for lathing. The various positions in which the ribs of a niche may be placed, are very numerous; but the regular positions, al

ready enumerated, ought to be those to which the carpenter should direct his attention.

To get out the ribs for the head of a niche, all of them being in vertical planes passing through the centre of the sphere.

Fig. 591, No. 1. From the centre C draw the ground-plan of the ribs, and set out as many ribs upon the plan as you intend to have in the head of the niche. With the foot of your compasses in C, and from the ends of each rib, at k and l, draw the small concentric dotted circles round to the centre rib, at o and p, and draw o m, and p n, parallel to a b, the face of the wall; then from r round to s on the plan is the length and sweep of the centre rib to stand over; and from n round to s the length and curve of the rib that stands from b to g; and from m round to s, the curve of the shortest rib, that stands from k to h on the plan.

How to find the bevel of the ends of the back ribs against the front rib.

The back ribs are laid down distinct by themselves, at AB and C from the plan. Take 6 1, in No. 1, and set it to b 1, at B, draw the perpendiculars, and when they intersect the rib, it will show the bevel required. The same operation being done to C, the bevel is found in the same manner.

The places of the back-ribs when fixed upon the frontrib are ascertained by drawing perpen culars, and completing the elevation of the niche No. 2 from the plan.

To find the radius of curvature of the ribs of a spherical niche, when the ribs all meet in a vertical line, which divides the front rib into two equal parts.

Fig. 592, No. 1. Complete the circle, of which the inside of the plan is an arc; produce the middle line of the plan of any rib, as of a b, to meet the opposite side of the circumference in b; on the whole line a b, as a diameter, describe a semicircle, and from the point c, when the ribs intersect, draw a perpendicular to c d, to meet the arc a d at d, which arc is the curve of the rib, whose seat is d. The other rib, as AD, is found in the same manner. No. 2 is the elevation of the niche.

Pendentive cradling, is a cove bracketing, springing from the rectangular walls of an apartment upwards to the ceiling, so as to form the horizontal part of the ceiling into a complete circle or ellipsis.

The proper criterion for such bracketing, if the walls are out by horizontal planes through the coved parts, is, that all the sections through such parts will be portions of circles, or of ellipses, and have their arcs proportioned to the sides of the apartment, so that each section will be a compound figure. Besides having four curvilinear parts, it - will have four other parts, which are portions of the sides of the rectangular apartment: and the axis of the ellipsis will bisect each side of the rectangle.

Fig. 593. Let ABCD be the plan of a room, or stair-case, to be brack

eted, so as to form the surface of a pendentive ceiling; and let A b c D be the section across the diagonal; it is required to find the curvature of the springing ribs ?

Draw C d perpendicular to AC, meeting AC, take the distance from C to the line AC, and set it from Con the line CA, and from this point draw a perpendicular to meet the curve Ale D of the diagonal rib; make the versid sine of the segment A dC equal to this perpendicular, and describe the segment Ad C, which is the springing line required. If from the centre C an arc be described, with a radius equal to the length of the seat of a rib, to meet the seat of the diagonal rib AD; and, if from the point of meeting a perpendicular be drawn to meet the curve A b, the portion of the arc of the diagonal rib, intercepted between A and the perpendicular, will give the length of the rib, corresponding to the seat which was taken.. Fig. 594. The diagonal rib is a semicircle: the operation is exactly the same, and may be described in the same words.

MENSURATION OF CARPENTERS' work.

All large and plain articles in which an uniform quantity of materials and workmanship is expended, are generally measured by the square of 100 superficial feet.

Piles used in the foundations are valued at per piece, and driven by the foot run, according to their diameter, and the quality of the ground.

Keepers and planking are measured by taking the superficial contents in yards or squares.

Plain centreing is measured by the square; but as the ribs and boarding are two different qualities of work, they ought to be measured and valued separately; one dimension of the boarding being taken by girting it round the arch, the other being the length of the vault.

Centreing for groins should be measured and valued as common centreing; but in addition thereto, the angles should be paid for by the foot run, that is, the ribs and boarding ought to be measured and valued separately, according to the exact superficial contents of each; and the angles by the lineal foot for workmanship, in fitting the rib and boards, and for the waste of wood occasioned by the operation.

Wall-plates, lintels, and bond-timbers, are measured by the cubic foot, under the denomination of fir-in-bond.

Naked flooring may either be measured by the square, or by the cubic foot, according to the description of the work, and the quantity of timber employed. In forming an estimate of its value, it should be observed, that in equal cubic quantities of small and large timbers, the small tim