At Fig. 159 the plan of the newel and the adjacent hand-rail are repeated, but upon an enlarged scale; and in which bb' is the reduced height of the point b, or is equal to bb' less t r, Fig. 158, and the angle bb' t equals the angle bb'r of Fig. 158. In this plan the actual heights must now be taken. Join t and u; then t u is the level tangent, as also the line of intersection of the cutting plane C and the horizontal plane A. Perpendicular to t u, at a point t or any where above it, draw u' b". Parallel with t u draw b b"'; make b" b"' equal to b b'; join b"' and u'; then the angle b b"' u' is the angle which the plank in position makes with a vertical line, or what is usually termed the plumb-bevil. Perpendicular to b"' u' draw u' u" and b"' b""; make b"' b"" equal to b b"; make u' t' equal to u' t, and u' u" to u' u; join b"" and t'; then b"" t' is the tangent in the cutting plane, the horizontal projection of which is bt. The butt-joint at b"" is drawn square to the tangent b"" t' Parallel to the intersecting line t u' draw ordinates across the plane A from as many points as desirable, and extend them to the rake-line u' b"'; through the points of their intersection with this line, and perpendicular to it, draw corresponding ordinates across the plane C. Make d" d"' equal to d' d, and so in like manner, for all other points, obtain in the plane C for each point in the horizontal plane A its corresponding point in the plane C: in each case taking the distance to the point in the plane A from the line h b" and applying it in the plane C from the rake-line u' b"' For the curves bend a flexible strip to coincide with the several points obtained, and draw the curve by the side of the strip. The point of the mitre is at d"' the mitre-joint is shown at hd"' and d"'c". The line f c" is drawn through c" the most projecting point of the mitre, and parallel to the rake-line u' b"' Additional wood is left attached, extending from h to f; this is an allowance to cover the mitre, which has to be cut vertically; the butt-joint at b"" and the face at fc" are both to be cut square through the plank. The face fc" because it is parallel to the rake-line u' b"', is a vertical face, as well as being perpendicular to the surface of the plank. On it, therefore, lines drawn according to the rake, or like the angle u' b"' b" will be vertical and will give the direction of the mitre-faces. We now have at C the face-mould for the railing over the plan from b to d in A. The mould thus found is that made upon a cutting plane C, passed through the plank, parallel to its face, but at the middle of its thickness. To put it in position, let the plane C be lifted by its upper edge c" and revolved upon the line u' b"' until it stands perpendicular to the plane B. Now revolve both C and B (kept in this relative position during the revolution) upon the line u' b' until the plane B stands perpendicular to the plane A. Then every point upon plane C will be vertically over its corresponding point in the plane A. For example, the point b"" will be vertically over b, t' over t, and so of all other points. To show the application of the face-mould to the plank, make b"' bv equal to half the thickness of the plank; parallel to u' b"' draw bv c, a line which represents the upper surface of the plank, for the line u' b"' is at the middle of the thickness. Through b"" and parallel with b"' u', draw the line c' b"" and extend it across the face-mould; make b"' c' equal to bv c; through c', and parallel with b"" t" draw c' e. Now, m n o' p is an end view of the plank, showing the face view of the butt-joint at b"" Through r, the centre, draw a line parallel with the sides. Then bvi represents the point b""; make bvi ey equal to b"" e; through r, the centre, draw c, r across the face of the joint; then e' r is a vertical line (see Art. 284), parallel and perpendicular to which the four sides of the squared-up wreath are to be drawn as shown. In applying the face-mould to the plank at first, for the purpose of marking by its edges the form of the face-mould, it will be observed that the face-mould is understood to have the position indicated by the line u' b"', or at the middle of the thickness of the plank. By this marking the rail-piece is cut square through the plank, and this cutting gives the correct form of the wreath, but only at the middle of the thickness of the plank. After it is cut square through the plank, then, to obtain the form at the upper and under surfaces, the face-mould is required to be moved endwise, but parallel with the auxiliary plane B, and solar as to bring the face-mould into a position vertically over or under its true position at the middle of the thickness of the plank. For example, the point b"", if the mould were placed at the middle of the thickness of the plank, would be at the height of the point b"'; but when upon the top of the plank, the point b"" would have to be at the height of the point c ', therefore the mould must be so moved that the point b"" shall pass from bv to c; consequently bv c is the distance the mould must be moved, or, as it is technically termed, the sliding distance; hence b"" c', which is equal to bvc, is the distance the mould is to be moved: up when on top, and down when underneath. This is more fully explained in Art. 284.

Fig. 159.