Fig. 8. Stews the construction of the curtail step, which is that under the scroll ;a b c d represents the veneer which covers the riser ; efg the nosing of the cover, or hori zontal part of the step ; ik/ the face of the string-board ; and m no the projection of the nosing.
Fig. 9. Slims the cover-board for the curtail step ; abed and efgli in dotted lines represent the plan of the scroll ; oftgrs the nosing of the curtail step : tuv s the nosings and the ends of the risers. The circle 1,2, 3, &c. is described from the centre of the scroll, and divided into equal parts, equal to the distances of the balusters from centre to centre ; and lines arc drawn to the centre of the scroll, in order to ascertain the middle of the balusters, by giving a regular gradation to the spaces. The whole of the spiral lines in this and in Fig. 9. are drawn from the same centres as the scroll.
Three heights from the base being given on the surface of a cylinder, or through three given points on its surface, to find a section of the cylinder, the heights of these points from the base being given.
Let ABC, Fig. 1. and 2. Plate CCCXXXIX. be the base of the cylinder, and A, B, C, the scats of the three points, of which the heights are a b c. Draw AD, BG, and CE perpendicular to the chord AC ; make AD equal to a, BG equal to b, and CE equal to c. Produce CA and Ell to meet each other in F. Draw BH parallel to CF, and GH parallel to EF ; and join FH and produce it to I, if necessary. In CF take any point K, and draw KI perpendicular to CF, and KL perpendicular to EF, cut ting EF at M. About K as a centre describe the arc MN, cutting CF at N, and draw NI. From F, with the dis tance FI, describe an arc at L, and join FL. In the arc ABC take any number of points, from which draw lines parallel to FI, meeting the chord CF. From these points in the chord CF draw lines parallel to CE to meet EF. From these points draw lines parallel to FL, equal in length to the lines parallel to Fl ; and through the extre mities draw the curve EPD, which is the section of the cylinder required. The angle which the section now found makes with the plane ACED, is equal to the angle CNI.
Fig. 1. No. 1. is the plan of a part of a geometrical stair with winders and flyers below and above, with the rail marked upon the plan. No. 2. is the falling mould, found in the following manlier. ABC is the semi-circumference of ABC, No. 1. extended in a straight line ; AD and CE are each in length equal to the breadth of one of the flyers extended in the same line ; EF, perpendicular to ED, is the height of twelve steps, viz. equal to the heights of all the windows and two of the fly•is. D..aw Al perpendi cular to AD; make Al and FG cacti equal to the height of a step. Draw Gil perpendicular to FG, and make
GI1 equal to the breadth of one of the flyers Join FH, HI, and Ill. Draw BK perpendicular to AC, cutting III at L. Make LM at the discretion of the workman ; through M draw OMP, meeting II) in P, and produce FH to 0. Make P a, P b, 0 c, 0 d each equal to ID or HF. Divide a P into any number of equal parts, begin ning at a, and P b into the same numbei of equal parts, beginning at P ; and draw lines through the correspond ing points of division, and they will form a parabolic curve.
Divide c 0 and o D in the same manner, and form another parabolic curve ; and thus the curves and the straight line a d will form the under edge of the falling mould. Make MN equal to the thickness of the falling mould, and through the point N draw the upper edge parallel to the lower edge, and thus complete the boundaries of the falling mould.
Make AV any distance, at the discretion of the work man, say 4 inches. Produce BM to K, curing the upper edge of the falling mould at X. Bisect BV in Q. Draw QR and VW parallel to BX, cutting the upper edge of the falling mould at R and W.
Draw KTS parallel to EV. Make KT, TS, respec tively equal to BQ, QV. Draw TU and SY parallel to KM, cutting the under edge of the falling mould at U and Y.
As one of the joints of the rail piece generally stands over the middle of the semicircular plan, and the other joint not over the place where the straight and circular parts meet, but three or four inches advanced into the straight part, in order to make a stronger joint, by using straight bolts or screws instead of circular ones ; in lay ing down the plan of the rail for the face mould, we must first lay down the circular quarter, and then add the straight part, which is shewn by AV in the construction of the falling mould. It is obvious that the falling mould depends in some measure on the fancy of the workman. The reason of raising it above the line I, is, because if the rail were made the same height above the steps over the circular part of the plan as over the straight part, it would approach nearer to the nosings of the steps of the winders than to those of the flyers. It is evident that the ends of the steps might be so narrow in the circular part, as even to cause the rail to coincide with the nosings of the win ders, while the under side of the rail would be two feet and upwards from the nosings, supposing the rail to be one height throughout, both in the straight and circular parts ; the height being supposed to be regulated from the top. The curves a b and cd arc entirely at the fancy of the workman, some making them longer and some shorter.