The next process in the construction of a wreath that the handrailer will be called upon to perform, is to find the bevels that will, by being applied to each end of it, give the correct angle to square or twist it when winding around the well-hole from one flight to another flight, or from a flight to a landing, as the case may be.

The wreath is first cut from the plank square to its surface as shown in Fig. 104. After the application of the bevels, it is twisted, as shown in Fig. 105, ready to be moulded; and when i n position, ascending from one end of the curve to the other end, over the inclined plane of the section around the well-hole, its sides will be plumb, as shown in Fig. 106 at b. In this figure, as also in Fig. 105, the wreath a lies in a horizontal position in which its sides appear to be out of plumb as much as the bevels are out of plumb. In the upper part of the figure, the wreath b is shown placed in its position upon the plane of the section, where its sides are seen to be plumb. It is evident, as shown in the relative position of the wreath in this figure, that, if the bevel is the correct angle of the plane of the section whereon the wreath b rests in its ascent over the wellhole, the wreath will in that case have its sides plumb all along when in position. It is for this purpose that the bevels are needed.

Fig. 111. Application of Bevels to Wreath Ascending on Plane Inclined Equally in Two Directions.

Fig. 111. Application of Bevels to Wreath Ascending on Plane Inclined Equally in Two Directions.

Fig. 112. Finding Bevel Where Upper Tangent Inclines More Than Lower One.

Fig. 112. Finding Bevel Where Upper Tangent Inclines More Than Lower One.

A method of finding the bevels for all wreaths (which is considered rather difficult) will now be explained:

First Case

In Fig. 107 is shown a case where the bottom tangent of a wreath is inclining, and the top one level, similar to the top wreath shown in Fig. 9S. It has already been noted that the plane of the section for this kind of wreath inclines to one side only; therefore one bevel only will be required to square it, which is shown at d, Fig. 107. A view of this plane is given in Fig. 108; and the bevel d, as there shown, indicates the angle of the inclination, which also is the bevel required to square the end d of the wreath. The bevel is shown applied to the end of the landing rail in exactly the same manner in which it is to be applied to the end of the wreath. The true bevel for this wreath is found at the upper angle of the pitch-board. At the end a, as already stated, no bevel is required, owing to the plane inclining in one direction only. Fig. 109 shows a face-mould and bevel for a wreath with the bottom tangent level and the top tangent inclining, such as the piece at the bottom connecting with the landing rail in Fig. 94.

Fig. 113. Finding Bevel Where Upper Tangent Inclines Less Than Lower One.

Fig. 113. Finding Bevel Where Upper Tangent Inclines Less Than Lower One.

Second Case

It may be required to find the bevels for a wreath having two equally inclined tangents. An example of this kind also is shown in Fig. 94, where both the tangents c" and d" of the upper wreath incline equally. Two bevels are required in this case, because the plane of the section is inclined in two directions; but, owing to the inclinations being alike, it follows that the two will be the same. They are to be applied to both ends of the wreath, and, as shown in Fig. 105, m the same direction - namely, toward the inside of the wreath for the bottom end, and toward the outside for the upper end.

Fig. 114. Finding Bevel Where Tangents Incline Equally over Obtuse Angle Plan.

Fig. 114. Finding Bevel Where Tangents Incline Equally over Obtuse-Angle Plan.

Fig. 115. Same Plan as in Fig. 114, but with Bottom Tangent Level.

Fig. 115. Same Plan as in Fig. 114, but with Bottom Tangent Level.

In Fig. 110 the method of finding the bevels is shown. A line is drawn from w to c", square to the pitch of the tangents, and turned over to the ground line at h, which point is connected 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' to n 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 whereon 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 enable 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 inclined plane in its ascent over the well-hole, the bevel in every case being the angle of the inclined plane.

Fig. 116. Finding Bevels for Wreath of Fig. 115.

Fig. 116. Finding Bevels for Wreath of Fig. 115.

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 tangents, 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 horizontal 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 m; and draw the line m 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 u; extend to touch the upper tangent d", 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 u; extend to the line h, 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 h 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 bottom tangent c" in the top piece extending from h" to 5. Bevel 1, found in Fig. 112, is the real bevel for the end 5; and bevel 2, for the end h" of the wreath shown from h" to 5 in Fig. 95.