This section is from the book "A Treatise On Architecture And Building Construction Vol3: Stair Building, Ornamental Ironwork, Roofing, Sheet-Metal Work, Electric-Light Wiring And Bellwork", by The Colliery Engineer Co.. Also available from Amazon: A Treatise On Architecture And Building Construction.
In Fig. 37 is shown a method of describing the curves of a curtail step. Begin by drawing a circle of a size greater than that required to enclose the required scroll. Let a b c d be such a circle; divide it into 16 equal parts, and let the diameter of the eye of the scroll be equal to e f, in this case 7 inches. The diameter of the eye should always be somewhat greater than the width of the hand rail, which in the present instance is 6 inches. At right angles to b d, and tangent to the eye, draw e' i; from i draw i j, perpendicular to h g; and so proceed, drawing lines at right angles to the successive radial lines; and continue this process to m, or to any point desired. From the points of intersection on the radial lines, with a radius equal to 3 inches, or one-half the width of the rail, describe arcs as shown. Tangent to these arcs, draw in an outline of the curve of the hand rails.
In Fig. 38 is shown a plan (a) of a curtail step with curved steps receding from same. This plan may be drawn from the dimensions shown, taking as much of the scroll delineated in Fig. 37 as is required to fulfil the conditions. At z r, in Fig. 38, is shown the junction of the scroll with the straight rail. Where the scroll has been developed in a diagram, as shown in Fig. 37, the whole or any part of it may be transferred to the plan of the stairway, as was done in Fig. 38. In order to fix the height of the wreathed portion of the scroll, draw an elevation of the risers and treads as at (b). Draw the bottom line of the rail x x through the points on the treads where the center lines of the short balusters intersect; draw the center line c b of the rail; make f s equal to j a in (a). Parallel to the riser lines, draw c a through s; make t i equal to, say, 2 inches, thereby fixing the height of the eye and the level portion of the scroll. Draw t b equal to one-half the depth of the rail; draw b a parallel to the tread line.
Then, as the bottom line of the scrolled rail is 2 inches above the first step i, and the length of a short baluster being 2 feet 2 inches, the height from i to t will be 2 feet 4 inches, when in position. It is desirable to keep the scrolled rail as low as that shown, so that its varying and pleasing curvature will be seen to the best advantage. In the plan (a), a b is the center line of the rail prolonged, and a c b is taken from lines similarly lettered in (b). From b, draw b e tangent to the center line of the rail at e. The joint s d is made at right angles to e b.
From s d to z r is the wreathed portion of the rail, since it both curves and rises. The remaining portion of the scroll, from s d to the eye, is horizontal, as shown by b o in the elevation (b), and in this part of the scroll, three or four of the balusters should be bolted to the step at their bases, and to the scrolled rail at their top. The construction of the veneered riser and scroll block is shown in Fig. 39 (a), and the complete scroll step is shown in (b).