To find the bevels when tangent b" inclines and tangent a" is level, make a c in Fig. 116 equal to a c in Fig. 115. This line will be the base of the two bevels. Upon a, erect the line a w m at right angles to a c; make a w equal to o w in Fig. 115; nect w and c; the bevel at w will be the one to apply to gent b" at n where the wreath is joined to the rail of the flight. Again, make a m in Fig. 116 equal the distance shown in Fig. 115 between w and m, which is the full height over which gent b" is inclined; connect m to c in Fig. 116, and at m is the bevel to be applied to the level tangent a".
Seventh Case. In this case, illus trated in Fig. 117, the upper tangent b" is shown to in cline, and the bot tom tangent a" to be level, over an acute-angle plan. The plan here is the same as that in Fig. 100, where a curve is shown to stretch out from the line of the straight stringer at the bot tom of a flight to a newel, and is large enough to contain five treads, which are gracefully rounded to cut the curve of the central line of rail in 1, 2, 3, 4. This curve also may be used to connect a landing rail to a flight, either at top or bottom, when the plan is acute-angled, as will be shown further on.
To find the bevels— for there will be two bevels necessary for this wreath, owing t o one tangent 1/' being inclined and the other tangent a" being level—make a c, Fig. 118, equal to a c in Fig. 117, which is a line drawn square to the ground line from the newel and shown in all preceding figures to have been used for the base of a triangle containing the bevel. Make a w in Fig. 118 equal to w o in Fig. 117, which is a line drawn square to the inclined tangent b" from w; connect w and c in Fig. 118. The bevel shown at w will be the one to be applied to the joint 5 on tangent b", Fig. 117. Again, make a m in Fig. 118 equal to the distance shown in Fig. 117 between the line representing the level tangent and the line ne 5, which is the height that tangent b" is shown to rise; connect m to c in Fig. 118; the bevel shown at m is to be applied to the end that intersects with the newel as shown at m in Fig. 117.
The wreath is shown developed in Fig. 101 for this case; so that, with Fig. 100 for plan, Fig. 101 for the development of the wreath, and Figs. 117 and 118 for finding the bevels, the method of handling any similar case in practical work Pan be found.
How, to Put the Curves on the Face- Mould. It has been shown how to find the angle between the tangents o f the face-mould, and that the angle is for the purpose of squaring the joints at the ends of the wreath. In Fig. 119 is shown how to lay out t h e curves by means of pins and a string—a very common practice amono. stair-build ers. In this example the face mould has equal tangents as shown at c" and d". The angle between the two tangents is shown at m as it will be required on the face-mould. In this figure a line is drawn from m parallel tothe line drawn from h,which is marked in the diagram as "Directing Ordinate of Section." The line drawn from m will contain the minor axes; and a line drawn through the corner of the section at 3 will contain the major axes of the ellipses that will consti tute the curves of the mould.
The major is to be drawn square to the minor, as shown. Place, from point 3, the circle shown on the minor, at the same distance as the circle in the plan is fixed from the point o. The diameter of this circle indicates the width of the curve at this point. The width at each end is determined by the bevels. The distance a b, as shown upon the long edge of the bevel, is equal to the width of the mould, and is the hypotenuse of a right-angled triangle whose base is 2 the width of the rail. By placing this dimension on each side of n, as shown at b and b, and on each side of h" on the other end of the mould, as shown also at b and b, we obtain the points b 2 b on the inside of the curve, and the points b 1 b on the outside. It will now be necessary to find the elliptical curves that will contain these points; and before this can be done, the exact length of the minor and major axes respectively must be deter mined. The length of the minor axis for the inside curve will be the dis tance shown from 3 to 2; and its length for the outside will be the distance shown from 3 to 1.
To find the length of the major axis for the inside, take the length of half the minor for the inside on the dividers: place one leg on b, extend to cut the major in z, continue to the minor as shown at k. The distance from b to k will be the length of the semi-major axis for the inside curve.