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 direction. How to find the true direction, or the angle between the tangents a and m shown in this diagram, was demonstrated in [Fig. 89]; and how to find the direction of the tangents c″ and d″ was shown in [Fig. 92].
Fig. 94. Tangents Unfolded to Find Their Inclination.
[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′ m′. By this process the plan tangents are now in one straight line on the floor line, as shown from w to m′. 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.
Fig. 95. Well-Hole Connecting Two Flights, with Two Wreath-Pieces,
Each Containing Portions of Unequal Pitch.
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 d″, c″, and m, down to where it connects with the bottom level tangent, as shown. This gives the pitch or inclination to the tangents over and above the well-hole. The same line is shown in [Fig. 93], folded around the well-hole, from n, where it connects with the flight at the upper 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 c″ and d″, coincides with the pitch-line of the same tangents as presented in [Fig. 92], where they are 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 necessary condition where a joint, as shown at h in [Figs. 93] and [94], is to connect two pieces of wreath as in this example.