This figure clearly shows that the joint at a of the bottom wreath—owing to the tangent a being level and fore aligning with the level rail of the landing—will be a true butt-joint; and that the joint at h, which connects the two wreaths, will also be a true joint, owing to it being made square to the tangent ?a of the bottom wreath and to the tangent c" of the upper wreath, both tangents having the same inclination; also the joint at it will butt square to the rail of the flight above, owing to it being made square to the tangent d", which is shown to have the same inclination as the rail of the flight adjoining.
As previously stated, the use made of tangents is to square the joints of the wreaths; and in this diagram it is clearly shown that the way they can be made of use is by giving each tangent its true direc tion. How to find the true direction, or the angle between the tangents a and in shown in this diagram, was demonstrated in Fig. S9; and how to find the direction of the tangents c"and d" was shown in Fig. 92.
Fig. 94 is presented to help further toward an understanding of the tangents. In this diagram they are unfolded; that is, they are stretched out for the purpose of finding the inclination of each one over and above the plan tangents. The side plan tangent a is shown stretched out to the floor line, and its elevation a' is a level line. The side plan tangent d is also stretched out to the floor line, as shown by the arc n' ne. By this process the plan tangents are now in one straight line on the floor line, as shown from iv to in'. Upon each one, erect a perpendicular line as shown, and from m' measure to n, the height the wreath is to ascend around the well-hole. In practice, the number of risers in the well-hole will determine this height.
Now, from point n, draw a few treads and risers as shown; and along the nosing of the steps, draw the pitch-line; continue this line over the tangents (1", c", and m, clown to where it connects with the bottom level tangent, as shown. This gives the pitch or inclination to the tangents over and above the w e l l-h o f e. The same line is shown in Fig. 93. folded around the w el l-h o e, from n, where it con nects with the flight at the tip per end of the well-hole, to a, where it connects with the level landing rail at the bottom of the well-hole. It
will be observed that the upper portion, from joint n to joint h, over the tangents and d", coincides with the pitch-line of the same tangents as presented in Fig. 92, where they arc used to find the true angle between the tangents as it is required on the face-mould to square the joints of the wreath at h.
In Fig. 89 the same pitch is shown given to tangent m as in Fig. 94; and in both figures the pitch is shown to be the same as that over and above the upper connecting tangents c" and d", which is a neces sary condition where a joint, as shown at h in Figs. 93 and 94, is to connect two pieces of wreath as in this example.
In Fig. 94 are shown the two face-moulds for the wreaths, placed upon the pitch-line of the tangents over the well-hole. The angles between the tangents of the face-moulds have been found in this figure by the same method as in Figs. S9 and 92, which, if compared with the present figure, will be found to correspond, excepting only the curves of the face-moulds in Fig. 94.
The foregoing explanation of the tangents will give the student a fairly good idea of the use made of tangents in wreath construction. The treatment, however, would not be complete if left off at this point, as it shows how to handle tangents under only two conditions— namely, first, when one tangent inclines and the other is level, as at a and m; second, when both tangents incline, as shown at c" and d".
In Fig. 95 is shown a well-hole connecting two flights, where two portions of unequal pitch occur in both pieces of wreath. The first piece over the tangents a and b is shown to extend from the square end of the straight rail of the bottom flight, to the joint in the center of the well-bole, the bottom tangent a" in this wreath inclining more than the upper tangent b". The other piece of wreath is shown to connect with the bottom one at the joint h," in the center of the well hole, and to extend over tangents c" and d" to connect with the rail of the upper flight. The relative inclination of the two tangents in this wreath, is the reverse of that of the two tangents of the lower wreath. In the lower piece, the bottom tangent a", as previously stated, inclines considerably more than does the upper tangent b" ; while in the upper piece, the bottom tangent c" inclines considerably less than the upper tangent d".