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.
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.
To this end let it be observed, that of the horizontal projection of any pair of intersecting tangents, their lengths, from the point of intersection to the points of contact with the circle, are equal; for example: of the two tangents h k and lk, the distances from k, their point of intersection, to b and c, their points of contact with the circle, are equal; and so also cl equals dl, dm equals e m' etc. It will be observed that this equality is not dependent on b,c, d, etc., the points of contact, being disposed at equal distances; for, in this example, they are placed at unequal distances, some being at three treads apart and others at four; and yet while this unequal distribution of the points b, c, d, etc., has the effect of causing the point of contact, as b, c, or e, to divide each whole tangent into two unequal parts, it does not disturb the equality of the two adjoining parts of any two adjacent tangents. Now, because of this equality of the two adjoining parts of a pair of tangents, the height to be overcome in passing from one point of contact to the next must be divided equally between the two; each tangent takes half the distance. Therefore, for stairs of this kind, the arrangement being symmetrical, we have this rule by which to fix the height of the ends of any two adjoining tangents, namely: To the height at the lower point of contact add half the difference between the heights at the two points of contact; the sum will be the required height of the two adjoining ends of tangents. For example: the heights at b and c, two adjacent points of contact, are respectively three and seven risers; the difference is four risers; half this added to three, the height of the lower rise, gives five risers as the height of kk" kk" the height at the adjoining ends of the tangents h k and l k. Again, the heights at c and d are respectively seven and ten risers; their difference is three; half of which, or one and a half risers, added to seven, the height at the lower point of contact, makes nine and a half risers as the heights ll" ll"'at the ends of the adjoining tangents k l and in l. In a similar manner are established the heights of the tangents at; m, n, and p.
* A tangent is a line perpendicular to the radius, drawn from the point of contact.
The rule for finding the heights of tangents as just given is applicable to circular stairs in which the treads are divided equally at the front-string, as in Fig. 158. Stairs of irregular plan require to have drawn an elevation of the rail, stretched out into a plane, upon which the tangents can be located. This will be shown farther on.
The locations of the joints c, d, c, in this example, were disposed at unequal distances merely to show the effect on the tangents as before noticed. In practice it is proper to locate them at equal distances, for then one face-mould in such a stairs will serve for each wreath.
When the tangent at G has been drawn, the level tangent for the landing may be obtained in this manner: As the joint f is located at the eighteenth riser, one riser below the landing, draw a horizontal line at s, one riser above the point f" and at half a riser above this draw the level line at p'; then this line is the level tangent, and p its point of intersection with the raking tangent. Draw the vertical line p,p, and from p draw the tangent p g, which is the horizontal projection of the tangent p' g' on plane H (which, to avoid undue enlargement of the drawing, is reduced in height), where p p" equals p p.
To obtain the horizontal tangent t u at the newel, proceed thus: Fix the point r, in the tangent r k' at a height above b t equal to the elevation of the centre of the newel above the height of a short baluster - for example, from 5 to 8 inches - and draw a line through r parallel to b t; this is a horizontal line through the middle of the height of the newel-cap, and upon which and the rake the easement to the newel is formed. Perpendicular to b t draw r t, and join t and u; then t u is the horizontal tangent.