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GEOMETRICAL STAIRWAYS AND HANDRAILING The term geometrical is applied to stair ways having any kind of curve for a plan.

The rails over the steps are made con tinuous from one story to another. The resulting winding or twisting pieces are called wreaths.

Wreaths. The construction of wreaths is based on a few geometrical problems—namely, the projection of straight and curved lines into an oblique plane; and the finding of the :Ingle of inclination of the plane into which the lines and curves are projected. This angle is called the bevel, and by its use the wreath is made to twist.

In Fig. 84 is shown an obtuse angle plan; in Fig. 85, an acute-angle plan; and in Fig. 86, a semicircle en closed within straight lines.

Projection. A knowledge of how to project the lines and curves in each of these plans into an oblique plane, and to find the angle of inclination of the plane, will enable the student to construct any and all kinds of wreaths.

The straight lines a, b, c, d in the plan, Fig. 86, are known as tangents; and the curve, the central line of the plan wreath.

The straight line across from n to n is the diameter; and the perpendicular line from it to the lines c and b is the radius.

A tangent line may be defined as a line touching a curve without cutting it, and is made use of in handrailing to square the joints of the wreaths.

In Fig. 86, it is shown that the joints connecting the central line of rail with the plan rails w of the straight flights, are placed right at the springing; that is, they are in line with the diameter of the semi circle, and square to the side tangents a and d.

The center joint of the crown tangents is shown to be square to tangents b and c. When these lines are projected into an oblique plane, the joints of the wreaths can be made to butt square by applying the bevel to them.

All handrail wreaths are assumed to rest on an oblique plane while ascending around a well-hole, either in connecting two flights or in connecting one flight to a landing, as the case may be.

In the simplest cases of construction, the wreath rests on an inclined plane that in clines in one direction only, to either side of the well-hole; while in other cases it rests on a plane that inclines to two sides.

Fig. 87 illustrates what is meant by a plane inclining in one direction. It will be noticed that the lower part of the figure is a reproduction of the quad rant enclosed by the tangents a and b in Fig. 86. The quadrant, Fig. 87, represents a central line of a wreath that is to ascend from the joint on the plan tangent a the height of h above the tangent b.

In Fig. SS a view of Fig. 87 is given in which the tangents e and b are shown in plan, and also the quadrant representing the plan central line of a wreath. The curved line extending from a to h in this figure represents the development of the central line of the plan wreath, and, as shown, it rests on an oblique plane inclining to one side only—namely, to the side of the plan tangent a. The joints are made square to the devel oped tangents a and m of the in clined plane; it is for this purpose only that tangents are made use of in wreath construc tion. They are shown in the figure to consist of two lines, a and which are two adjoining sides of a developed section (in this case, of a square prism), the section being the assumed inclined plane whereon the wreath rests in its ascent from a to h. The joint at h, if made square to the tangent in, will be a true, square butt-joint; so also will be the joint at a, if made square to the tangent a.

In practical work it will be required to find the correct goemetrical angle between the two developed tangents a and la; and here, again, ' it may be observed that the finding of the correct angle between the two developed tangents is the essential purpose of every tangent system of hand railing.