Let it be required to furnish the face-moulds for a circular stairs similar to that shown in Fig. 133.

Preliminary to making the face-moulds it is requisite to make a plan, or horizontal projection of the stairs, and on this to locate the projections of the tangents and develop their vertical projections. For this purpose let bcdefg, Fig. 158, be the horizontal projection of the centre of the rail, and the lines numbered from 1 to 19 be the risers. At any point, a, on an extension of the line of the first riser locate the centre of the newel. On a as a centre describe the two circles; the larger one equal in diameter to the diameter of the newel-cap, the inner one distant from the outer one equal to half the width of the rail. Let the first joint in the hand-rail be located at b, at the fourth riser; through b draw h k tangent to the circle. Select a point, h, on this tangent which shall be equally distant from b and from the inner circle of the newel-cap, measured on a line tending to a; join h and a, and from a point, q, on the line b o describe the curve from b to the point of the mitre of the newel-cap, the curve being tangent, at this point, to the line ah. Select positions for the other joints in the hand-rail as at c, d, e, and f.

Fig. 158.

Through these draw lines tangent to the circle.* Then the horizontal projection of the tangents will be the lines l m, m n, and n p. Now, if a vertical plane stand upon each of these lines, these planes would form a prism not quite complete standing upon the base-plane, A. Upon these vertical planes, C, D, E, F, G, and H, lines may be drawn which at each joint shall be tangent to the central line of the rail. These are the tangents now to be sought. Perpendicular to the tangents at b, c, d, etc., draw the lines b b" c c, dd' e e" ff" gg" and h h"" kk' k k"l l"l l"' etc. As b is at the fourth riser, and the height is counted from the top of the first riser, make b b' equal to three risers. (To avoid extending the drawing to inconvenient dimensions, the heights in it are made only half their actual size. As this is done uniformly throughout the drawing, this reduction will lead to no error in the desired results.) As c is on the eighth riser, therefore make c c' equal to seven risers, and so, in like manner, make the heights dd"ee" and ff' each of a height to correspond with the number of the riser at which it is placed, deducting one riser. These heights fix the location of each tangent at its point of contact with the central line of the rail. But each tangent is yet free to revolve on this point of contact, up or down, as may be required to bring the ends of each pair of tangents in contact; or, to make equal in height the edges of each pair of vertical planes, which coincide after they are revolved on their base-lines into a vertical position; as, for example: the edges k k' and k k" of the planes C and D must be of equal height; so, also, the edges ll' and ll" of the planes D and E must be of equal height. The method of establishing these heights will now be shown.