In Fig. 110 the method of finding the bevels is shown. A line is drawn from qv to c", square to the pitch of the tangents, and turned over to the ground line at h, which point is nected to a as shown. The bevel is at h. To show that equal tangents have equal bevels, the line m is drawn, having the same inclination as the bottom tangent c", but in another direction. Place the dividers on o', and turn to touch the lines d" and m, as shown by the semicircle. The line from o' ton is equal to the side plan tangent w a, and both the bevels here shown are equal to the one already found. They represent the angle of inclination of the plane where on the wreath ascends, a view of which is given in Fig. 111, where the plane is shown to incline equally in two directions. At both ends is shown a section of a rail; and the bevels are applied to show how, by means of them, the wreath is squared or twisted when winding around the well-hole and ascending upon the plane of the section. The view given in this figure will en able the student to understand the nature of the bevels found in Fig. 110 for a wreath having two equally inclined tangents; also for all other wreaths of equally inclined tangents, in that every wreath in such case is assumed to rest upon an in clined plane in its ascent overthe well hole, the bevel in every case being the angle of the inclined plane.
Third Case. In this example, two unequal tangents are given, the upper tangent inclining more than the bottom one. The method shown in Fig. 110 to find the bevels for a wreath with two equal tan gents, is applicable to all conditions of variation in the inclination of the tangents. In Fig. 112 is shown a case where the upper tangent d" inclines more than the bottom one c". The method in all cases is to continue the line of the upper tangent d", Fig. 112, to the ground line as shown at n; from n, draw a line to a, which will be the horizon tal trace of the plane. Now, from o, draw a line parallel to a n, as shown from o to d, upon d, erect a perpendicular line to cut the tangent d", as shown, at in; and draw the line in u o". Make u o" equal to the length of the plan tangent as shown by the arc from o. Put one leg of the dividers on n; extend to touch the upper e, and turn over to 1; connect 1 to o"; the bevel at 1 is to be applied to tangent d". Again place the dividers on 2t; extend to the line it, and turn over to 2 as shown; connect 2 to o", and the bevel shown at 2 will be the one to apply to the bottom tangent c". It will be observed that the line It represents the bottom tangent. It is the same length and has the same inclination. An example of this kind of wreath was shown in Fig.
95, where the upper tangent d" is shown to incline More than the bot tom tangent c" in the top piece ex tending from h" to 5. Bevel 1, found in Fig. 112, is the real bevel for the end 5; and bevel 2, for the end it" of the wreath shown from it" to 5 in Fig. 95.
Fourth Case. In Fig. 113 is shown how to find the bevels for a wreath when the upper tangent inclines less than the bottom tangent. This example is the reverse of the preceding one; it is the condition of tangents found in the bottom piece of wreath shown in Fig. 95. To find the bevel, continue the upper tangent b" to the ground line, as shown at n; connect n to a, which will be the horizontal trace of the plane. From o, draw a line parallel to n a, as shown from o to d; upon d, erect a perpendicular line to cut the continued portion of the upper tangent b" in in; from in, draw the line u o" across as shown. Now place the dividers on u; extend to touch the upper tangent, and turn over to 1, connect 1 to o"; the bevel at 1 will be the one to apply to the tangent b" at h, where the two wreaths are shown connected in Fig. 95. Again place the dividers on 'a; extend to touch the line c; turn over to 2; connect 2 to 0"; the bevel at 2 is to be applied to the bottom tangent a" at the joint where it is shown to connect with the rail of the flight.
Fifth Case. In this case we have two equally inclined tangents over an obtuse-angle plan. In Fig. 102 is shown a plan of this kind ; and in Fig. 103, the development of the face-mould.
In Fig. 114 is shown how to find the bevel. From a, draw a line to a', square to the ground line. Place the dividers on a'; extend to touch the pitch of tangents, and turn over as shown to vi; connect m to a. The bevel at in will be the only one required for this wreath, but it will have to be applied to both ends, owing to the two tangents being inclined.
Sixth Case. In this case we have one tangent inclining and one tangent level, over an acute-angle plan.
In Fig. 115 is shown the same plan as in Fig. 114; but in this case the bottom tangent a" is to be a level tangent. Probably this condition is the most commonly met with in wreath construction at the present time. A small curve is considered to add to the appear ance of the stair and rail; and consequently it has become almost a "fad" to have a little curve or stretch-out at the bottom of the stairway, and in most cases the rail is ramped to intersect the newel at right angles instead of at the pitch of the flight. In such a case, the bottom tangent a" will have to be a level tangent, as shown at a" in Fig. 115, the pitch of the flight being over the plan tangent b only.