This section is from the book "The American House Carpenter", by R. G. Hatfield. Also available from Amazon: The American House Carpenter.

In order that a more comprehensive idea of the lines given for applying a face-mould may be had, let A, Fig. 165, represent one end of a wreath-piece as it appears when first cut from a plank, and when held up in the position it is to occupy at completion over the stairs. Also, let B represent the corresponding face-mould, laid upon the wreath-piece A in the position which it should have after sliding. And, for the purpose of a clearer illustration, let it be supposed that the two pieces, A and B, are transparent. Then let a' a b d c' e' represent a solid of wedge form, having a triangular level base, a b d, upon the three lines of which stand these three vertical planes, namely: on the line a b the plane a' a b c., upon the line a d the plane a' a d e.', and on the line d b the plane b d e' c'; the top of the solid is an inclined plane, a' c' e', and the vertical line a' a is the edge of the wedge. Now, it will be observed that the point a in the base of the solid is identical with a, the centre of the butt-joint, and the point a' (at the intersection of two vertical planes and the inclined plane of the solid) is vertically over a, and is identical with a', a point in the upper surface of the plank. Also, the inclined plane e' c' a' which forms the top of the solid, coincides with the upper surface of the plank A, from which the wreath-piece has been squared; and the line c' a' (at the angle formed by the inclined plane e' c' a, and the vertical plane a' a b c') coincides with f g' the slide-line drawn upon the top of the plank; also, the line e' a' (at the angle formed by the inclined plane e' c' a' and the vertical plane a, a d e,) coincides with a' k' the tangent line upon the underside of the face-mould after it has been slid to its new position, vertically over its true position at the middle of the thickness of the plank. From a the line a c is drawn parallel with a, c,; so, also, the line a e is drawn parallel with a' e'; consequently the line e c is parallel with e' c'; and the plane e c a is parallel with the plane t' c' a' and coincides with a plane passing-through the middle of the thickness of the plank, and, consequently, is the cutting plane referred to in previous articles, upon which the lines are drawn which give shape to the face-mould. When the face-mould is first laid upon the plank, the line i'j' coincides with i" j", and when in that position, its form marked upon the plank is the form by which the plank is sawed square through; but this gives the form of the wreath, not as it is at the surface of the plank, but as it is at the middle of the thickness of the plank, or upon the plane ace; so that, for example, the line i"j" represents the line ij drawn through a, the centre of the butt-joint; and when the mould B is slid to the position shown in the figure, the line i'j' comes into a position vertically over ij; hence the three lines i' i, a' a, and j'j are each vertical and in a vertical plane, ii' j'j. By these considerations it will be seen that the face-mould B, located as shown in the figure, is in its true position for the second marking, by which the additional cutting is now to be performed vertically. This being established, it will now be shown how to get upon the butt-joint a line in the vertical plane containing the tangent. If the top and bottom lines of the vertical plane a'a bc' be extended, they will meet in the point /, and will extend the plane into a triangle lb c', cutting the upper edge of the butt-joint in f, the end of the tangent, and the point in which the point a' of the underside of the face-mould was located when the mould was first applied to the plank. The line fa on the butt-joint is perpendicular to i j'or i" j". Again, if the top and bottom lines of the plane a'ade' be extended, they will meet in p, and will extend the plane into the triangle p d e', cutting the edge of the butt-joint in h, a point from which, if a line be drawn upon the butt-joint to a, its centre, this line will be in the vertical plane pde',, which plane contains the tangent perpendicular to which the butt-joint is drawn; consequently lines upon the butt-joint parallel to h a will each be in a vertical plane parallel to the vertical tangent plane, and lines drawn upon the butt-joint perpendicular to these lines will be horizontal lines; hence the line h a is the required line by which to square the wreath at the butt-joint. Now, it will be observed that the triangle afa' is like that given in the various figures for obtaining face-moulds, to regulate the sliding of the face-mould and the squaring at the butt-joint. For example, in Fig. 163, the right-angled triangle b"' b"' bvi is the one referred to. This triangle is in a vertical plane parallel to one containing the slide-line; its longer side is a vertical line; one of the sides containing the right angle is equal to half the thickness of the plank, while the other, drawn parallel to the face of the plank, is the distance the face-mould is required to slide. Precisely like this, the triangle a f a' of Fig. 165 is in the vertical plane l b c' containing fg, the slide-line; its longer side, a' a, is a vertical line; fa, one of the sides containing the right angle, is equal to half the thickness of the plank, while the other side, drawn coincident with the surface of the plank, is the distance to slide the face-mould. Therefore the triangle a'f a of Fig. 165 gives the required lines by which to regulate the application of the face-moulds. The relative position of the more important of these lines is geometrically shown in Fig. 166, in which A and B are upon the horizontal plane of the paper, C is in a vertical plane standing on the ground-line b d, and D is a plan of the butt-joint, revolved upon the line i" j" into the horizontal plane, and then perpendicularly removed to the distance ff. The lettering corresponds with that in Fig. 165. The shaded part of D shows the end of the squared wreath. When the blocked piece has been marked by the face-mould in its second application, its edges are to be trimmed vertically as shown in Fig. 167, after which the top and bottom surfaces of the wreath are to be formed from the shape marked on the butt-joints.

Fig. 165.

Fig. 166.

Fig. 167.

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